Construction of wavelets through Walsh functions 9789811363696, 9789811363702

Table of contents :
Preface......Page 6
References......Page 9
Contents......Page 11
About the Authors......Page 14
Introduction......Page 16
References......Page 19
1.1 Walsh Functions......Page 22
1.2 Walsh–Fourier Transform......Page 26
1.3 Haar Functions and Its Relationship with Walsh Functions......Page 29
1.4 Walsh-Type Wavelet Packets......Page 31
1.5.1 Continuous Wavelet Transform......Page 33
1.5.2 Discrete Wavelet System......Page 36
1.5.3 Multiresolution Analysis......Page 37
1.6 Wavelets with Compact Support......Page 38
1.7 Exercises......Page 44
References......Page 45
2.1 Walsh–Fourier Coefficients......Page 47
2.1.1 Estimation of Walsh–Fourier Coefficients......Page 48
2.1.2 Transformation of Walsh–Fourier Coefficients......Page 51
2.2 Convergence of Walsh–Fourier Series......Page 54
2.2.1 Summability in Homogeneous Banach Spaces......Page 59
2.3.1 Approximation by Césaro Means of Walsh–Fourier Series......Page 60
2.3.2 Approximation by Nörlund Means of Walsh–Fourier Series in Lp Spaces......Page 65
2.3.3 Approximation by Nörlund Means in Dyadic Homogeneous Banach Spaces and Hardy Spaces......Page 77
2.4.1 Image Representation and Transmission......Page 90
2.4.2 Data Compression......Page 93
2.4.5 ECG Analysis......Page 94
2.4.6 EEG Analysis......Page 95
2.4.8 Pattern Recognition......Page 96
References......Page 97
3.1 Haar System and Its Generalization......Page 99
3.2 Haar Fourier Series......Page 102
3.3 Haar System as Basis in Function Spaces......Page 106
3.4 Non-uniform Haar Wavelets......Page 108
3.5 Generalized Haar Wavelets and Frames......Page 113
3.6.1 Applications to Solutions of Initial and Boundary Value Problems......Page 114
3.6.2 Applications to Solutions of Integral Equations......Page 116
3.7 Exercises......Page 117
References......Page 118
4.1 Preliminary......Page 119
4.2 Orthogonal Wavelets and MRA in L2(mathbbR+)......Page 123
4.3 Orthogonal Wavelets with Compact Support on mathbbR+......Page 134
4.4 Estimates of the Smoothness of the Scaling Functions......Page 153
4.5 Approximation Properties of Dyadic Wavelets......Page 178
4.6 Exercise......Page 189
References......Page 190
5.1 Multiresolution Analysis on Vilenkin Groups......Page 191
5.2 Compactly Supported Orthogonal p-Wavelets......Page 214
5.3 Periodic Wavelets on Vilenkin Groups......Page 225
5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform......Page 231
5.5 Application to the Coding of Fractal Functions......Page 246
References......Page 254
6.1 Introduction......Page 255
6.2.1 Haar–Vilenkin Mother Wavelet......Page 256
6.3 Approximation by Haar–Vilenkin Wavelets......Page 263
6.4 Covergence Theorems......Page 266
6.5 Haar–Vilenkin Coefficients......Page 268
6.6 Exercises......Page 271
References......Page 272
7.1 Biorthogonal Wavelets on R+......Page 273
7.2 Biorthogonal Wavelets on Vilenkin Groups......Page 283
7.3 Construction of Biorthogonal Wavelets on The Vilenkin Group......Page 289
7.4 Frames on Vilenkin Group......Page 292
7.5 Application to Image Processing......Page 297
References......Page 301
8.1 Introduction......Page 303
8.2 Nonuniform Multiresolution Analysis on Positive Half Line......Page 305
8.2.1 Construction of Nonuniform Multiresolution Analysis on Positive Half Line......Page 311
8.2.2 The Analogue of Cohen's Condition......Page 317
8.3 Nonuniform Wavelet Frames in L2(mathbbR)......Page 323
8.3.1 Necessary and Sufficient Condition......Page 327
8.4 Exercises......Page 334
References......Page 335
9.1 Introduction......Page 336
9.2 Vector-Valued Multiresolution Analysis......Page 337
9.3 Vector-Valued Multiresolution p-Analysis on mathbbR+......Page 340
9.4 The Existence of Orthogonal Vector-Valued Wavelets on mathbbR+......Page 345
9.5 Vector-Valued Nonuniform Multiresolution Analysis......Page 353
9.5.1 Construction of Vector-Valued Nonuniform Multiresolution Analysis......Page 362
9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties......Page 366
9.6.1 Vector-Valued Nonuniform Wavelet Bases......Page 374
9.7 Exercise......Page 376
References......Page 377
A.1 Basic Results of Functional Analysis......Page 378
A.2 Vilenkin Systems......Page 386
A.3.1 The Dyadic Group G......Page 388
A.4 Introduction to Real and Fourier Analysis......Page 391
References......Page 397
Lake Index......Page 398
Index......Page 399

Citation preview

Industrial and Applied Mathematics

Yu. A. Farkov Pammy Manchanda Abul Hasan Siddiqi

Construction of Wavelets Through Walsh Functions

Industrial and Applied Mathematics Editor-in-chief Abul Hasan Siddiqi, Sharda University, Greater Noida, India Editorial Board Zafer Aslan, Istanbul Aydin University, Istanbul, Turkey Martin Brokate, Technical University, Munich, Germany N.K. Gupta, Indian Institute of Technology Delhi, New Delhi, India Akhtar A. Khan, Rochester Institute of Technology, Rochester, USA René Pierre Lozi, University of Nice Sophia-Antipolis, Nice, France Pammy Manchanda, Guru Nanak Dev University, Amritsar, India Zuhair Nashed, University of Central Florida, Orlando, USA Govindan Rangarajan, Indian Institute of Science, Bengaluru, India Katepalli R. Sreenivasan, NYU Tandon School of Engineering, Brooklyn, USA

The Industrial and Applied Mathematics series publishes high-quality research-level monographs, lecture notes and contributed volumes focusing on areas where mathematics is used in a fundamental way, such as industrial mathematics, bio-mathematics, financial mathematics, applied statistics, operations research and computer science.

More information about this series at http://www.springer.com/series/13577

Yu. A. Farkov Pammy Manchanda Abul Hasan Siddiqi •

•

Construction of Wavelets Through Walsh Functions

123

Yu. A. Farkov Department of Applied Information Technology, School of Public Policy Russian Presidential Academy of National Economy and Public Administration (RANEPA) Moscow, Russia

Pammy Manchanda Department of Mathematics Guru Nanak Dev University Amritsar, Punjab, India

Abul Hasan Siddiqi School of Basic Sciences and Research and Centre for Advanced Research in Applied Mathematics and Physics Sharda University Greater Noida, Uttar Pradesh, India

ISSN 2364-6837 ISSN 2364-6845 (electronic) Industrial and Applied Mathematics ISBN 978-981-13-6369-6 ISBN 978-981-13-6370-2 (eBook) https://doi.org/10.1007/978-981-13-6370-2 Library of Congress Control Number: 2019930359 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The book is essentially devoted to the construction of wavelets by Walsh functions and their transforms. This embodies the contemporary research on wavelets defined on positive half line, wavelets on Vilenkin groups, Haar–Vilenkin wavelets, studied by the authors of this book and their collaborators. More precisely, this book deals with wavelet theory on Vilenkin groups, particularly on the Cantor group, which can also be interpreted as wavelet theory on the positive half line. This theory is in the framework of Walsh analysis (dyadic analysis in the case of the Cantor group), which has been actively studied for the last four decades. The foundation of these developments is discussed in books authored by Schipp, Wade, and Simon; Golubov, Efimov, and Skvortsov; Maqusi; Siddiqi; and Beauchamp. Recent research work not included in these books constitute important ingredients of the present volume. These themes are an approximation of special cases of matrix transform of Walsh–Fourier series in dyadic homogeneous Banach spaces and Hardy spaces, Haar–Vilenkin wavelets and their properties, orthogonal and biorthogonal wavelets and frames, nonuniform multiresolution analysis on the positive half line, and vector-valued wavelets on the positive half line. The book contains nine chapters and one appendix. Chapter 1 is devoted to the basic results of Walsh analysis and wavelets on the real line. An exhaustive background for Chaps. 1–3 can be found in Schipp et al. [1]. It has been observed that Walsh functions were used for the transposition of conductors in open wire lines as early as the late 1800s, and a complete system of Walsh functions seems to have been found around 1900. Rademacher functions were introduced to the mathematical world in 1922 by Rademacher. His widow told an audience of mathematicians that Rademacher knew the concept of Walsh functions, but he abandoned the idea as advised by his academic mentors. The concept of wavelets was introduced by the French Geophysicist, Jean Morlet, in 1982. Its literal meaning is “small wave”. Alex Grossmann, French Theoretical Physicist, studied inverse formula for the wavelet transforms. In 1984, the collaboration of Morlet and Grossmann yielded a detailed mathematical study of the continuous wavelet transforms and their various applications, of course, without realizing that similar results had already been obtained 20–50 years earlier by Calderon, Littlewood, v

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Paley, and Franklin. In 1985, Yves Meyer, recipient of the Abel Prize of 2017, accidentally found the then existing literature of wavelets. With the knowledge of the Calderon–Zygmund operators and the Littlewood–Paley theory, he was able to lay the mathematical foundation of wavelet theory. The first major achievement is due to Daubechies, Grossman, and Meyer for constructing a painless non-orthogonal expansion in 1986. Chapter 2 deals with the properties of Walsh–Fourier coefficients, the convergence of Walsh–Fourier series, and an approximation of functions belonging to different spaces by special classes of matrix transform of Walsh–Fourier series. The historical note on most of the themes of this chapter is well documented in the monograph of Schipp, Wade, and Simon. Here, we discuss approximation by matrix transform of Walsh–Fourier series. After the result of Yano for arithmetic and Cesáro means, there were no significant results on this theme till 1992. During the visit of Schipp in January 1992 and Moricz in 1994 to Aligarh Muslim University, India, the problem of generalizing the results of Yano, Jastrebova, and Skvorcov on the rate of approximation by Cesáro means was given top priority. Moricz and Siddiqi published an interesting paper generalizing all earlier results (see Theorems 2.24–2.26). Fridli, Manchanda, and Siddiqi studied approximation by Norlund means in dyadic homogeneous Banach spaces and Hardy spaces and extended the earlier results of Moricz and Siddiqi (see Theorems 2.27–2.29). Chapter 3 presents Haar systems and its generalization, Haar–Fourier series, the relation between Haar and Walsh functions, and applications of Haar wavelets to initial and boundary value problems. Alfred Haar was a Ph.D. student of David Hilbert at Göttingen University. Haar function is the simplest example of mother wavelet. Haar wavelets and its generalizations were studied in detail by Devore et al. [2] and Dubeau et al. [3]. In a recent monograph, Lepik and Hein [4] have studied the applications of Haar wavelets to initial and boundary problems. Chapter 4 is devoted to the construction of dyadic wavelets and frames through Walsh functions. This chapter is based on the research work of Farkov [5–12]. In Sect. 4.1 of this chapter, preliminary results needed for the subsequent chapters are explained. Section 4.2 of this chapter presents the orthogonal wavelets and multiresolution (MRA). Section 4.3 of this chapter is devoted to orthogonal wavelets with compact support on the positive half line. Section 4.4 introduces the estimates of the smoothness of the scaling functions. Section 4.5 presents the approximation properties of dyadic wavelets. Chapter 5 is devoted to orthogonal and periodic wavelets on Vilenkin groups. This chapter is essentially based on the research work of Farkov and his collaborators. As we know, Walsh functions are identified with characters of the Cantor dyadic group. This fact was recognized by Gelfand as back as in 1940. Vilenkin group was introduced in 1947. In a series of papers published after 2005, Farkov and his research collaborators have studied orthogonal and periodic wavelets [6, 13–19]. The results of Chap. 4 and 5 may be extended for results given in [20]. Section 5.1 deals with multiresolution analysis on Vilenkin groups. Section 5.2 presents compactly supported orthogonal p-wavelets. Section 5.3 introduces the periodic wavelets

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on Vilenkin groups. Section 5.4 introduces periodic wavelets related to the Vilenkin– Christenson transform. Section 5.5 presents applications to the coding of fractal functions. Chapter 6 is devoted to Haar–Vilenkin wavelets and their properties. Haar–Vilenkin system was introduced around 1947. The concept of Haar–Vilenkin wavelets was introduced and studied by Manchanda, Meenakshi, and Siddiqi [21, 22]. Chapter 7 is devoted to biorthogonal wavelets and frames on Vilenkin groups. This chapter is based mainly on the research papers of Farkov published between 2008 and 2012; [14, 23, 24]. Section 7.1 introduces biorthogonal wavelets on the positive half line. Section 7.2 presents biorthogonal wavelets on Vilenkin groups. Parseval frames on Vilenkin groups are also presented in this chapter. This chapter is concluded by applications of biorthogonal dyadic wavelets. Chapter 8 deals with wavelets associated with nonuniform multiresolution analysis on the positive half line. This chapter is based on the research of Manchanda, Meenakshi, Siddiqi, and Vikram published during 2012 and 2017 [25–28]. Chapter 9 presents the concept of orthogonal vector-valued wavelets on the positive half line. Section 9.4 presents vector-valued wavelets and wavelet packets associated with the nonuniform multiresolution analysis. The study of the concept of orthogonal vector-valued wavelets on the real line was initiated by Xia and Suter (1996). This concept on the positive half line was initiated by Farkov in 2005 [16]. The chapter contains the results obtained by Manchanda, Meenakshi, Siddiqi, and Vikram in the recent past [29–32]. The appendix contains topics on the basic results of functional analysis, Vilenkin systems, Vilenkin–Pontryagin class of functions, and real and Fourier analysis including pointwise, uniform, and absolute convergence. The book will serve as a reference material for research in Fourier analysis. The book, however, could also be used for a special course for graduate and advanced undergraduate students of mathematics and engineering. We take this opportunity to express our gratitude to our research groups. We also like to express our sincere gratitude to the chancellor of Sharda University, P. K. Gupta, for extending full support during the preparation of this book. We also take this opportunity to thank Pooja and Mamta for meticulously preparing the final manuscript. We take this opportunity to thank the World Scientific Publishing Company for using results published in International Journal of Wavelets, Multiresolution and Information Processing. Moscow, Russia Amritsar, India Greater Noida, India

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References 1. Schipp, F., Wade, W. R., & Simon, P. (1990). Walsh series. Adam Hilger. 2. Pfander, G. E., & Benedetto J. J. (2000). Periodic wavelet transforms and periodicity detection. SIAM Journal of Applied Mathematics, 62(4), 1329–1368. 3. Dubeau, F., Elmejdani, S., & Krantini, R. (2004). Nonuniform haar wavelets. Applied Mathematics and Computation, 159, 675–693. 4. Lepik, U., & Hein, H. (2014). Haar wavelets with applications. Berlin: Springer. 5. Farkov, Yu. A., & Protasov V. Yu. (2006). Dyadic wavelets and refinable functions on a half line. Matematicheskii Sbornik, 197(10), 129–160. English translation, Sbornik: Mathematics, 197, 1529–1558. 6. Farkov, Yu. A. (2005). Orthogonal wavelets with compact support on locally compact abelian groups. Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 193–220. English translation, Izvestia: Mathematics, 69(3), 623–650. 7. Farkov, Yu. A. (2009). On wavelets related to Walsh series. Journal of Approximation Theory, 161, 259–279. 8. Farkov, Yu. A. (2010). Wavelets and frames based on Walsh-Dirichlet type kernels. Communications in Mathematics and Applications, 1, 27–46. 9. Farkov, Yu. A., & Stroganov, S. A. (2011). The use of discrete dyadic wavelets in image processing. Russian Mathematics (Iz. Yuz), 55(7), 47–55. Original Russian text published in Izvestiya. Uchebnykh Zavednic. Mathematica, 7, 57–66. 10. Farkov, Yu. A. (2012). Periodic wavelets in Walsh analysis. Communication in Mathematics and Applications, 3(3), 223–242. 11. Farkov, Yu. A., & Borisov, M. E. (2012). Periodic dyadic wavelets and coding of fractal functions. Russian Mathematics (Izvestiya Vuz. Matematika), 56(9), 46–56. 12. Farkov, Yu. A. (2014). Wavelet expansions on the Cantor group. Mathematical Notes, 96(6), 996–1007. 13. Farkov, Yu. A. (2007). Orthogonal wavelets on direct products of cyclic groups. Matematicheskie Zametki, 82(6), 934–952. English translation, Mathematical Notes, 82(6), 843–859. 14. Farkov, Yu. A. (2008). Multiresolution analysis and wavelets on Vilenkin groups. Facta Univers. (Nis) ser.: Elec. Engineering, 21(3), 309–325. 15. Farkov, Yu. A. (2009). Biorthogonal wavelets on Vilenkin groups. Tr. Mat. Inst. Steklova, 265(1), 110–124. English translation, Proceedings of the Steklov Institute of Mathematics, 265(1), 101–114. 16. Farkov, Yu. A. (2005). Orthogonal p-wavelets on R+. In Proceedings of International Conference on Wavelets and Spilnes, St. Petersburg, Russia, July 3–8 (p. 426). St. Petersberg: St. Petersberg University Press. 17. Farkov, Yu. A. (2011). Discrete wavelets and the Vilenkin Chrestensen transform. Mathematical Notes, 89(6), 871–884. 18. Farkov, Yu. A. (2011). Periodic wavelets on the p-adic Vilenkin group. P-Adic Numbers, Ultrametric Analysis, and Applications, 3(4), 281–287. 19. Farkov, Yu. A., & Rodionov, E. A. (2011). Algorithms for wavelet construction on Vilenkin groups. P-Adic Numbers, Ultrametric Analysis, and Applications, 3(3), 181–195. 20. Krivoshein, A. V., Protasov, V. Yu., & Skopina, M. A. (2016). Multivariate wavelet frames. Berlin: Springer. 21. Manchanda, P., Meenakshi, & Siddiqi, A. H. (2008). Haar-Vilenkin wavelet. The Aligarh Bulletin of Mathematics, 27(1), 59–73. 22. Manchanda, P., & Meenakshi. (2009). New classes of wavelets. In A. H. Siddiqi, A. K. Gupta, & M. Brokate (Eds.), Proceedings of Conference on Modeling of Engineering and Technological Problems, New York (vol. 1146, pp. 253–271).

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23. Farkov, Yu. A., Maksimov A. Yu., & Stroganov, S. A. (2011). On biorthogonal wavelets related to the Walsh functions. International Journal of Wavelets, Multiresolution and Information Processing, 9(3), 485–499. 24. Farkov, Yu. A. (2012). Examples of frames on the Cantor dyadic group. Journal of Mathematical Sciences, 187(1), 22–34. 25. Manchanda, P., & Sharma, V. (2013). Wavelet packets with nonuniform multiresolution analysis on positive half-line. Asian-European Journal of Mathematics, 6(1), 1350007. 26. Meenakshi, Manchanda, P., & Siddiqi, A. H. (2012). Wavelets associated with nonuniform multiresolution analysis on positive half line. International Journal of Wavelets, Multiresolution and Information Processing, 10(2), 1250018, 1–27. 27. Sharma, V., & Manchanda, P. (2015a). Nonuniform wave packet frames in L2(R). Indian Journal of Industrial and Applied Mathematics, 6(2), 139–152. 28. Sharma, V., & Manchanda, P. (2015b). Nonuniform wavelet frames in L2(R). Asian European Journal of Mathematics, World Scientific, 8(2), 1550034(1–15). 29. Manchanda, P., & Sharma, V. (2012). Orthogonal vector valued wavelets on R+. International Journal of Pure and Applied Mathematics, 75(4), 493–510. 30. Manchanda, P., Meenakshi, & Siddiqi, A. H. (2014). Construction of vector-valued nonuniform wavelets and wavelet packets. Special Issue of Indian Journal of Industrial and Applied Mathematics. 31. Meenakshi, Manchanda, P., & Siddiqi, A. H. (2014). Wavelets associated with vector valued non-uniform multiresolution analysis. Applicable Analysis: An International Journal, 93(1), 84–104. 32. Meenakshi & Manchanda, P. (2017). Vector-valued nonuniform wavelet packets. Numerical Functional Analysis and Optimization. Published online: https://doi.org/ 10.1080/01630563.2017.1355814 (Taylor and Francis).

Contents

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2 Walsh–Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Walsh–Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Estimation of Walsh–Fourier Coefficients . . . . . . . . . 2.1.2 Transformation of Walsh–Fourier Coefficients . . . . . . 2.2 Convergence of Walsh–Fourier Series . . . . . . . . . . . . . . . . . 2.2.1 Summability in Homogeneous Banach Spaces . . . . . . 2.3 Approximation by Transforms of Walsh–Fourier Series . . . . 2.3.1 Approximation by Césaro Means of Walsh–Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Approximation by Nörlund Means of Walsh–Fourier Series in Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Approximation by Nörlund Means in Dyadic Homogeneous Banach Spaces and Hardy Spaces . . . . 2.4 Applications to Signal and Image Processing . . . . . . . . . . . . 2.4.1 Image Representation and Transmission . . . . . . . . . . 2.4.2 Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Quantization of Walsh Coefficients . . . . . . . . . . . . . .

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1 Introduction to Walsh Analysis and Wavelets . . . . . . . . . . . 1.1 Walsh Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Walsh–Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Haar Functions and Its Relationship with Walsh Functions 1.4 Walsh-Type Wavelet Packets . . . . . . . . . . . . . . . . . . . . . 1.5 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Continuous Wavelet Transform . . . . . . . . . . . . . . 1.5.2 Discrete Wavelet System . . . . . . . . . . . . . . . . . . . 1.5.3 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . 1.6 Wavelets with Compact Support . . . . . . . . . . . . . . . . . . . 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.4 Signal Processing . . 2.4.5 ECG Analysis . . . . 2.4.6 EEG Analysis . . . . 2.4.7 Speech Processing . 2.4.8 Pattern Recognition 2.5 Exercises . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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3 Haar–Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Haar System and Its Generalization . . . . . . . . . . . . . . . . 3.2 Haar Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Haar System as Basis in Function Spaces . . . . . . . . . . . 3.4 Non-uniform Haar Wavelets . . . . . . . . . . . . . . . . . . . . . 3.5 Generalized Haar Wavelets and Frames . . . . . . . . . . . . . 3.6 Applications of Haar Wavelets . . . . . . . . . . . . . . . . . . . 3.6.1 Applications to Solutions of Initial and Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Applications to Solutions of Integral Equations . . 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Construction of Dyadic Wavelets and Frames Through Walsh Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Orthogonal Wavelets and MRA in L2 ðR þ Þ . . . . . . . . 4.3 Orthogonal Wavelets with Compact Support on R þ . . 4.4 Estimates of the Smoothness of the Scaling Functions 4.5 Approximation Properties of Dyadic Wavelets . . . . . . 4.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Orthogonal and Periodic Wavelets on Vilenkin Groups . . 5.1 Multiresolution Analysis on Vilenkin Groups . . . . . . . . 5.2 Compactly Supported Orthogonal p-Wavelets . . . . . . . . 5.3 Periodic Wavelets on Vilenkin Groups . . . . . . . . . . . . . 5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application to the Coding of Fractal Functions . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Haar–Vilenkin Wavelet . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Haar–Vilenkin Wavelets . . . . . . . . . . . . . . 6.2.1 Haar–Vilenkin Mother Wavelet . . . 6.3 Approximation by Haar–Vilenkin Wavelets

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235 235 236 236 243

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6.4 Covergence Theorems . . . . 6.5 Haar–Vilenkin Coefficients 6.6 Exercises . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

xiii

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7 Construction Biorthogonal Wavelets and Frames . . . . . . . . 7.1 Biorthogonal Wavelets on R þ . . . . . . . . . . . . . . . . . . . . 7.2 Biorthogonal Wavelets on Vilenkin Groups . . . . . . . . . . 7.3 Construction of Biorthogonal Wavelets on The Vilenkin Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Frames on Vilenkin Group . . . . . . . . . . . . . . . . . . . . . . 7.5 Application to Image Processing . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Wavelets Associated with Nonuniform Multiresolution Analysis on Positive Half Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Nonuniform Multiresolution Analysis on Positive Half Line . . 8.2.1 Construction of Nonuniform Multiresolution Analysis on Positive Half Line . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Analogue of Cohen’s Condition . . . . . . . . . . . . . . 8.3 Nonuniform Wavelet Frames in L2 ðRÞ . . . . . . . . . . . . . . . . . . 8.3.1 Necessary and Sufficient Condition . . . . . . . . . . . . . . . 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Orthogonal Vector-Valued Wavelets on R þ . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Vector-Valued Multiresolution Analysis . . . . . . . . . . . . . . . . . 9.3 Vector-Valued Multiresolution p-Analysis on R þ . . . . . . . . . 9.4 The Existence of Orthogonal Vector-Valued Wavelets on R þ 9.5 Vector-Valued Nonuniform Multiresolution Analysis . . . . . . . 9.5.1 Construction of Vector-Valued Nonuniform Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . . . 9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Vector-Valued Nonuniform Wavelet Bases . . . . . . . . . 9.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Yu. A. Farkov is a distinguished scientist and professor at the Russian Presidential Academy of National Economy and Public Administration (RANEPA), Moscow, Russia. Earlier, he worked as a professor and head of the Department of Mathematics at the Russian State Geological Prospecting University, Moscow, Russia, from 1997 to 2014; a professor and head of the Department of Higher Mathematics at Dubna International University, Dubna, Russia, from 1996 to 2000; an associate professor at the Department of Higher Mathematics and Mathematical Modelling, Moscow State Geological Prospecting Academy, from 1988 to 1997; and a lecturer at the Department of Higher Mathematics, Moscow State University of Mechanical Engineering, Environmental and Chemical Engineering Institute, Moscow, from 1981 to 1988. In 2013, Prof. Farkov received his DSc-Doctor Fiz.- Mat. Nauk from the People’s Friendship University of Russia on “Optimal Methods of Approximation of Function by Generalized Polynomials and Wavelets”. In 1981, he received his Ph.D. degree in Mathematics from Moscow Region Pedagogical Institute, on topic “Investigations of Asymptotic and Approximation Properties of Faber–Erokhin Basis Functions”. From 1977 to 1980, he did his postgraduate studies at the Moscow Electro-Technical Engineering Institute; and in 1975, he received his BSc degree in Mathematics from Uralsk Pedagogical Institute, Kazakhstan. Professor Farkov is on the editorial boards of the journals American Journal of Computational Mathematics, Communications in Mathematics and Applications and International Journal of Education. He is a member of the Moscow Mathematical Society and American Mathematical Society. His research interests include mathematical analysis, wavelet theory, dyadic analysis and approximation theory. Pammy Manchanda is a senior professor of mathematics at Guru Nanak Dev University, Amritsar, India. She has attended and delivered talks and chaired sessions at reputed academic conferences and workshops across the world, including ICIAM (1999–2015) and ICM since 2002. She was invited twice to the Industrial Mathematics Group of Professor Helmut Neunzert, Kaiserslautern University,

xv

xvi

About the Authors

Germany, and visited the International Centre for Theoretical Physics (a UNESCO institution) at Trieste, Italy, many times to carry out her research activities. She was the joint secretary of the Indian Society of Industrial and Applied Mathematics (ISIAM) from 1999 until 2016 and after that, she is the secretary of the society. She has been actively engaged in organizing international conferences by the society. She is the managing editor of the Indian Journal of Industrial and Applied Mathematics (by ISIAM) and a member of the editorial board of the Springer’s book series, Industrial and Applied Mathematics. Professor Manchanda has published 44 research papers in several international journals of repute, edited 2 proceedings of international conferences of ISIAM and co-authored 3 books. Abul Hasan Siddiqi is a distinguished scientist and Adjunct Professor at the School of Basic Sciences and Research and Coordinator at the Centre for Applied Mathematics and Physics at Sharda University, Greater Noida, India. He was also a visiting consultant at the International Centre for Theoretical Physics (ICTP), Trieste, Italy; Sultan Qaboos University, Muscat, Oman; MIMOS, Kuala Lumpur, Malaysia; and professor at several reputed universities including Aligarh Muslim University (Aligarh, India) and King Fahd University of Petroleum & Minerals (Dhahran, Saudi Arabia). He has a long association with ICTP (a UNESCO institution) in several capacities: short-time visitor, long-duration visitor, regular associate, guests of the director and senior associate. He was awarded the German Academic Exchange Fellowship thrice to carry out mathematical research in Germany. He has published more than 100 research papers jointly with his research collaborators, 13 books and edited proceedings of 17 international conferences, as well as supervised 29 Ph.D. scholars. He is the founder secretary and the elected president of the Indian Society of Industrial and Applied Mathematics (ISIAM), which celebrated its silver jubilee year in January 2016. He is the editor-in-chief of the Indian Journal of Industrial and Applied Mathematics (published by ISIAM) and the Springer’s book series, Industrial and Applied Mathematics.

Introduction

The trigonometric Fourier series has played a very significant role in solving the problems of science and technology. The concept of non-trigonometric Fourier series such as Haar–Fourier series and Walsh–Fourier series was introduced by Haar [1] and Walsh [2], respectively; Kaczmarz, Steinhaus, and Paley studied some aspects of Walsh system between 1929 and 1931. Nowadays, Paley’s modification, which is defined as the product of Rademacher functions, is known as Walsh functions [3]. A major breakthrough came when N. J. Fine submitted his Ph.D. dissertation to Pennsylvania University in 1946, which was published subsequently in the Transactions of the American Mathematical Society in 1949. Fine [4] introduced the concept of dyadic group G and proved that Walsh functions are its characters. Haar introduced, in 1909, a class of functions closely related to Walsh functions in response to a question posed by David Hilbert, namely “Does there exist an orthonormal system such that the Fourier series of every continuous function converges uniformly?” He showed that the Fourier series with respect to the orthonormal system introduced by him, now known as the Haar system, of continuous functions converges uniformly on [0,1]. Walsh [2] observed that Haar and Walsh systems are Hadamard transforms of each other. The Walsh and Haar systems perform all usual applications of orthonormal systems, for example, data transmission, multiplexing, filtering, image enhancement, and pattern recognition. An elegant presentation of these developments is given in Schipp et al. [5]; also, see Beauchamp [6], Siddiqi [7], Maqusi [8], and Golubov et al. [9]. In the early eighties, wavelet analysis was introduced and developed by Morlet, Daubechies, Mallat, Meyer, and Coifman et al. [10–15]. Also, see Walnut [16], Christensen [17], Siddiqi [18], and Krivoshein et al. [19]. Wavelet analysis is a refinement of Fourier analysis. A natural question arises: What is the relation between Haar Fourier analysis, Walsh–Fourier analysis, and wavelet analysis? It is known that Haar function is the simplest example of wavelet and Walsh function is the simplest example of wavelet packet. This book focuses on the construction of wavelets by Walsh functions and their transforms. An interpretation of Walsh functions as characters of the Cantor dyadic group was proposed by I. M. Gelfand, and Vilenkin [20] defined a broad class of locally xvii

xviii

Introduction

compact abelian groups (now called Vilenkin groups), containing the Cantor group as a special case. Fine [4] independently examined Walsh functions as characters of the Cantor dyadic group. Recall that for a given p [ 2, the Vilenkin group Gp is defined as the weak direct product of a countable number of the cyclic groups of order p equipped with the discrete topology (for p ¼ 2, we have the locally compact Cantor group). A detailed bibliography of wavelets on locally compact abelian groups is given in [21, 22], where the group Gp is interpreted as an additive group of the field Fp ðtÞ of formal Laurent series. According to [21], the specifics of constructions of wavelets on Vilenkin groups and on the additive group of the field of p-adic numbers Qp are determined by the fact that these groups contain compact subgroups. On the real line R, a classical example of wavelet bases is the Haar system (see, for example, [23]). Haar wavelets for the group Qp were constructed in [24], and on the groups Gp , in [25, 26] (for p ¼ 2, see [27, 28]). An analysis of a general construction of Haar wavelets is given in a recent paper [29]. In the space L2 ðQp Þ, any orthogonal wavelet basis consisting of bandlimited functions is a modification of the Haar system [30]. As distinct from the group Qp , on Vilenkin groups, one may construct orthogonal compactly supported wavelets that are substantially different from Haar wavelets (see examples in [25–27]). We also note that the Fourier transform of the Haar ^ ¼ /, and hence, scaling function / on the group Gp coincides with this function, / Haar wavelets on Gp coincide with Kotelnikov–Shannon wavelets (for more details on this phenomenon, see [21]). As usual, by Z, Z þ , N, and C, we shall denote, respectively, the sets of integers, positive integers, natural and complex numbers. The group Gp consists of the sequences x ¼ ðxj Þ for which xj 2 f0; 1; . . .; p  1g for any j 2 Z, where only a finite number of xj with negative indexes may be nonzero. For any nonzero sequence x ¼ ðxj Þ from Gp , there exists a unique integer number k ¼ kðxÞ such that xk 6¼ 0 and xj ¼ 0 for all j\k. The group operation on Gp is defined as the coordinatewise addition modulo p, for all p, ðzj Þ ¼ ðxj Þ þ ðyj Þ , zj ¼ xj þ yj ðmod pÞ for all j 2 Z and the topology is defined by the complete system of neighborhoods of the origin, Ul ¼ fðxj Þ 2 Gp : xj ¼ 0 for j  lg; ; 2 Z Let R þ :¼ ½0; 1Þ. With the help of the mapping k : Gp ! R þ , kðxÞ ¼

X

xj pj ; x ¼ ðxj Þ 2 Gp ;

j2Z

we interpret in the standard way the group Gp on the half line R þ so that to the addition operation on Gp there corresponds the operation  to be defined below, and to the Haar measure on Gp there corresponds the Lebesgue measure on R þ

Introduction

xix

[5, 9, 31, 32]. The integral and fractional parts of a number x are denoted by [x] and x, respectively. For any m 2 Z, we denote by hmi ¼ p the remainder of the division of m by p. Further, for any x 2 R þ , we set xj ¼ h½p j xip ; xj ¼ h½p1j xip ; j 2 N:

ð1Þ

These numbers are the digits of the p-adic expansion X X x¼ xj pj1 þ xj pj j[0

j\0

(for a p-adic rational x, we get an expansion with a finite number of nonzero terms). It is easily seen that ½x ¼

1 X

xj pj1 ; fxg

j¼1

1 X

xj pj ;

j¼1

and moreover, for any x 6¼ 0, there exists a number k ¼ kðxÞ such that xk 6¼ 0 and xj ¼ 0 for all j [ k. The p-adic addition operation p on R þ is defined by x p y ¼

X X hxj þ yj ip pj1 þ hxj þ yj ip pj ;

ð2Þ

j[0

j\0

where xj , yj are calculated by (1). As usual, we denote by ⊖p the inverse operation of p (for p ¼ 2, these operations coincide). Let  ¼ 2 . A function w is called an orthogonal wavelet in L2 ðR þ Þ if the functions wj;k :¼ 2j=2 wð2 j x  kÞ; j 2 Z; k 2 Z þ form an orthonormal basis for L2 ðR þ Þ. In general, a function w is called a wavelet in L2 ðR þ Þ if the following condition is satisfied Z 0\ Rþ

2 ^ jwðxÞj

dx \1 x

^ is the Walsh–Fourier transform of a function w (for the definition and some where w properties of this transform, see Chap. 2). The continuous wavelet transform of a function f 2 L2 ðR þ Þ with an analyzing wavelet w is defined by Z 1=2 f ðxÞwððx  bÞ=aÞdx; a [ 0; b 2 R þ Ww f ða; bÞ ¼ a Rþ

An analogue of Grossmann–Morlet’s formula for this transform is proved in [33]; for the transform Ww ðf Þ with f 2 L2 ðGp Þ, see [5, 34]. First examples of

xx

Introduction

orthogonal compactly supported wavelets on the group G2 were constructed by Lang [27, 35, 36] by using the group analogues of well-known theorems of Mallat [14] and Cohen [37] on multiresolution analysis (MRA). Lang wavelets on the half line R þ are constructed by using a function w 2 L2 ðR þ Þ satisfying the scaling equations of the form /ðxÞ ¼

n 2X 1

ck /ð2x  kÞ; x 2 R þ

ð3Þ

k¼0

A complete description of the orthogonal wavelets and scaling functions on R þ , whose masks are Walsh polynomials, was obtained in [38]; in [25, 38–42], these results were extended and algorithms for the construction of biorthogonal Parseval wavelets and frames for the group Gp were presented. Examples of step scaling functions on the group Gp are given in [43–46]. For any p  2, necessary and sufficient conditions for which solutions / of the scaling equation /ðxÞ ¼

n pX 1

ck /ðpx  kÞ; x 2 R þ

k¼0

generate an MRA in L2 ðR þ Þ are provided in [47, 48]; these results are replicated in Chap. 8 of [49]. A complete description of MRA wavelets in L2 -spaces on local fields of positive characteristic (including the field Fp ððtÞÞ and, hence, the group Gp ) is obtained in [50]; in connection with these results, see [43–46, 51, 52].

References 1. Haar, A. (2010). Zur Theorie Der Orthogonal Funktionen Systeme. Mathematische Annalen, 69, 331–371. 2. Walsh, J. L. (1923). A closed set of normal orthogonal functions. American Journal of Mathematics, 45, 5–24. 3. Paley, R. E. A. C. (1932). A remarkable system of orthogonal functions. Proceedings of the London Mathematical Society, 34, 241–279. 4. Fine, N. J. (1949). On the Walsh functions. Transactions of the American Mathematical Society, 65(3), 372–414. 5. Schipp, F., Wade, W. R., & Simon, P. (1990). Walsh series. Adam Hilger. 6. Beauchamp, K. G. (1975). Walsh functions and their applications. New York: Academic Press. 7. Siddiqi, A. H. (1978). Walsh function. Aligarh: AMU. 8. Maqusi, M. (1981). Applied Walsh analysis. London, Philadelphia, Rheine: Heyden. 9. Golubov, B. I., Efimov, A. V., & Skvortsov, V. A. (2008). Walsh series and transforms. Moscow: Urss. English translation of 1st ed., 1991, Dordrecht: Kluwer.

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10. Coifman, R. R., Meyer, Y., Quake, S., & Wickerhauser, M. V. (1993). Signal processing and compression with wavelet packets progress. In Wavelets analysis and applications (pp. 77–93). Gif-Sur-Yvette: Frontiers. 11. Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, 41, 909–996. 12. Daubechies, I. (1992). Ten lectures on wavelets. Philadelphia: SIAM. 13. Mallat, S. (1989). Multiresolution approximations and wavelets. Transactions of the American Mathematical Society, 315, 69–88. 14. Mallat, S. (1999). A wavelet tour of signal processing. New York, London: Academic Press. 15. Meyer, Y. (1992). Wavelets and operators. Cambridge: Cambridge University Press. 16. Walnut, D. F. (2002). Wavelet analysis. Basel: Birkhauser. 17. Christensen, O. (2003). An introduction to frames and Riesz bases. Basel: Birkhaeuser. 18. Siddiqi, A. H. (2018). Functional analysis with applications. Berlin: Springer Nature. 19. Krivoshein, A. V., Protasov, V. Yu., & Skopina, M. A. (2016). Multivariate wavelet frames. Berlin: Springer. 20. Vilenkin, N. Ya. (1947). On a class of complete orthonormal systems. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 11, 363–400. English translation, 1963, American Mathematical Society Translations, 28 (Ser. 2), 1–35. 21. Benedetto, J. J., & Benedetto, R. L. (2004). A wavelet theory for local fields and related groups. The Journal of Geometric Analysis, 14, 423–456. 22. Benedetto, J. J., & Benedetto R. L. (2009). The construction of wavelet sets, wavelets and multiscale analysis. In J. Cohen et al. (Ed.), Theory and Applications. Selected Papers Based on the Presentations at the International Conference on Wavelets: Twenty years of Wavelets, De Paul University, Chicago, IL, USA, May 15–17 (pp. 17–56). New York: Springer (Alied and Numerical Harmonic Analysis). 23. Novikov, I. Ya., Protasov, V. Yu., & Skopina, M. A. (2011). Wavelet theory (Moscow, 2006). Providence: AMS. 24. Kozyrev, S. V. (2002). Wavelet analysis as a p-adic spectral analysis. Izvestiya: Mathematics, 66, 367–376. 25. Farkov, Yu. A. (2007). Orthogonal wavelets on direct products of cyclic groups. Matematicheskie Zametki, 82(6), 934–952. English translation, Mathematical Notes, 82(6), 843–859. 26. Farkov, Yu. A. (2008). Multiresolution analysis and wavelets on Vilenkin groups. Facta Univers. (Nis) ser.: Elec. Engineering, 21(3), 309–325. 27. Lang, W. C. (1996). Orthogonal wavelets on the cantor dyadic group. SIAM Journal on Mathematical Analysis, 27(1), 305–312. 28. Sendov, B. I. (1997). Multiresolution analysis of functions defined on the dyadic topological group. East Journal on Approximations, 3(2), 225–239. 29. Novikov, I. Ya., & Skopina, M. A. (2012). Why are haar bases in various structures the same. Mathematical Notes, 91(6), 895–898. 30. Evdokimov, S., & Skopina, M. (2015). On orthogonal p-adic wavelet bases. Journal of Mathematical Analysis and Applications, 424(2), 952–965. 31. Agaev, G. H., Vilemkin, N. Ya., Dzhafarli, G. M., & Rubinstein, A. I. (1981). Multiplicative systems of functions and analysis on 0 dimensional groups. ELM, Baku (in Russian). 32. Edwards, R. E. (1982). Fourier series: A modern introduction (2). Berlin: Springer. 33. Farkov, Yu. A. (2008). Walsh function and the continuous wavelet transform. In Proceedings on V International Symposium on Fourier Series and Their Applications, Rostov na, Donu, Tsvvr (pp. 27–32). 34. Lukashenko, T. P. (1994). Wavelets on topological groups. Izv. Ross. Akad. Nauk Ser. Mat., 58(3), 88–102. English translation, 1995, Izvestiya: Mathematics, 44(3), 515–529. 35. Lang, W. C. (1998a). Fractal Multiwavelets related to the Cantor dyadic group. International Journal of Mathematics and Mathematical Sciences, 21, 307–317.

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36. Lang, W. C. (1998b). Wavelet analysis on the Cantor dyadic group. Houston Journal of Mathematics, 24, 533–544. 37. Cohen, A. (1990). Ondelettes analysis multiresolutions et filtres miroir en quadrature. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 7, 439–459. 38. Farkov, Yu. A., & Protasov V. Yu. (2006). Dyadic wavelets and refinable functions on a half line. Matematicheskii Sbornik, 197(10), 129–160. English translation, Sbornik: Mathematics, 197, 1529–1558. 39. Farkov, Yu. A. (2005). Orthogonal wavelets with compact support on locally compact abelian groups. Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 193–220. English translation, Izvestia: Mathematics, 69(3), 623–650. 40. Farkov, Yu. A. (2007). Biorthogonal dyadic wavelets on ℝ+. Russian Mathematical Surveys, 62, 1198. Translation from Usp. Mat. Nauk, 62(6), 189–190. 41. Farkov, Yu. A. (2009). Biorthogonal wavelets on Vilenkin groups. Tr. Mat. Inst. Steklova, 265(1), 110–124. English translation, Proceedings of the Steklov Institute of Mathematics, 265(1), 101–114. 42. Farkov, Yu. A., Lebedeva, E. A., & Skopina, M. A. (2015). Wavelet frames on Vilenkin groups and their approximation properties. International Journal of Wavelets, Multiresolution and Information Processing, 13(5). https://doi.org/10.1142/5021. 43. Berdnikov, G. S., & Lukomskii, S. F. (2014). N-valid trees in wavelet theory on Vilenkin groups. http://arxiv.Org/abs/1412.309v1. 44. Lukomskii, S., & Vodolazov, A. (2014). Non-Haar MRA on local fields of positive characteristics. http://arxiv.org/abs/1303.5635v/. 45. Lukomskii, S. F. (2014). Step refinable functions and orthogonal MRA on P-adic Vilenkin groups. Journal of Fourier Analysis and Applications, 20(1), 42–65. 46. Lukomskii, S. F. (2015). Riesz multiresolution analysis on zero-dimensional groups. Izvestiya: Mathematics, 79(1), 145–176. 47. Farkov, Yu. A. (2005). Orthogonal p-wavelets on R+. In Proceedings of International Conference on Wavelets and Spilnes, St. Petersburg, Russia, July 3–8 (p. 426). St. Petersberg: St. Petersberg University Press. 48. Farkov, Yu. A. (2009). On wavelets related to Walsh series. Journal of Approximation Theory, 161, 259–279. 49. Debnath, L., & Shah, F. A. (2015). Wavelet transforms and their applications (2nd ed.). New York: Birkhauser/Springer. 50. Behera, B., & Jahan, Q. (2012). Wavelet packets and wavelet frame packets on local fields of positive characteristic. Journal of Mathematical Analysis and Applications, 395, 1–14. 51. Behera, B., & Jahan, Q. (2015). Characterization of wavelets and MRA wavelets on local fields of positive characteristic. Collectanea Mathematica, 66(1), 33–53. 52. Berdnikov G., Kruss, Iu., & Lukomskii, S. (2017). On orthogonal systems of shifts of scaling function on local fields of positive characteristic. http://arxiv.org/1503.08600.

Chapter 1

Introduction to Walsh Analysis and Wavelets

1.1 Walsh Functions The trigonometric Fourier series has played a very significant role in solving problems of science and technology. The concept of non-trigonometric Fourier series such as Haar–Fourier series and Walsh–Fourier series were introduced by Haar [1] and Walsh [2], respectively; Kaczmarz, Steinhaus, and Paley studied some aspects of Walsh system between 1929 and 1931. Nowadays, Paley’s modification, which is defined as the product of Rademacher functions is known as the Walsh function [3]. A major breakthrough came in N. J. Fine’s dissertation of Ph.D. submitted to Pennsylvania University in 1946, which was published subsequently in Transactions of American Mathematical Society in 1949. Fine [4] introduced the concept of dyadic group G and proved that Walsh functions are its characters. Haar introduced in 1909 a class of functions closely related to Walsh functions in response to a question posed by David Hilbert, namely “Does there exist an orthonormal system such that the Fourier series of every continuous function converges uniformly?” He showed that the Fourier series with respect to the orthonormal system, introduced by him, now known as the Haar system, of continuous functions converge uniformly on [0, 1]. Walsh [2] observed that Haar and Walsh systems are Hadamard transforms of each other. The Walsh and Haar systems perform all usual applications of orthonormal systems, for example, data transmission, multiplexing, filtering, image enhancement, and pattern recognition. An elegant presentation of these developments are given in Schipp, Wade, Simon [5]; also see Beauchamp [6], Siddiqi [7], Maqusi [8], and Golubov, Efimov, and Skvortsov [9]. In the early 80s, wavelet analysis was introduced and developed by Morlet et al. [10–17]. Also see Walnut [14], Christensen [16], Siddiqi [7, 18]. In this chapter, we briefly discuss Walsh function, its integration besides dyadic group, properties of Walsh–Fourier transform, Haar function and its relationship with Walsh function, Walsh-type wavelet packet, and wavelet analysis. As usual, Z, Z+ , N, and C denote, respectively, the sets of integers, positive integers, natural numbers, and complex numbers. © Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2_1

1

2

1 Introduction to Walsh Analysis and Wavelets

The Walsh system {wn : n ∈ Z+ } on R+ is defined as follows: w0 (x) = 1, wn (x) =

k 

(w1 (2 j x))ν j , n ∈ N,

j=0

x ∈ R+ , where R+ = [0, ∞), where the ν j are the coefficients of the decomposition n=

k 

νj2j,

ν j ∈ {0, 1}, νk = 1, k = k(n),

j=0

and the function w1 (x) is defined on [0, 1] by the formula  w1 (x) =

1, x ∈ [0, 1/2) −1, x ∈ [1/2, 1),

and is extended to R+ by periodicity: w1 (x + 1) = w1 (x) for all x ∈ R+ . Walsh polynomials are finite linear combinations of the Walsh functions. An arbitrary Walsh polynomial of order n can be written in the following form: w(x) =

n 

c j w j (x)

j=0

where c j are complex coefficients. For more information about the properties of Walsh polynomials and their role in the dyadic harmonic analysis see for example, [5, 7]. We shall denote the integer and the fractional parts of a number x ∈ R+ , by [x] and {x}, respectively. For x ∈ R+ and j ∈ N, we define the number x j , x− j ∈ {0, 1} as follows: x j = [2 j x](mod 2),

x− j = [21− j x](mod 2).

These are the digits of the binary expansion x=

 j 0 such that | f (t ⊕ u) − f (t)| ≤ ε for all u ∈ R + which satisfy |u| < 2−n . Clearly,    

0

2n

  fˆ(y)ψt (y)dy − f (t) = |(S2n f )(t) − f (t)|  2−n | f (t + u) − f (t)|du ≤ 2n =ε

Therefore,



2n

0

fˆ(y)ψt (y)dy = f (t).

lim

n→∞ 0

 ∞   ∞   But  | fˆ(y)|dy → 0 as n → ∞, since f is integrable. fˆ(y)ψt (y)dy  ≤ 2n 2n This proves (b) part of Theorem 1.3. Relationship between Fourier and Walsh–Fourier Transform: fˆ = Walsh– Fourier transform of f is defined above and Fourier transform of f is  F(ω) =

∞ −∞

f (x)e−iωx d x

(1.6)

By the inverse of Walsh–Fourier transform 

∞

f (x) =

fˆχ (x, ω)dω

(1.7)

F(ω)eiωx dω

(1.8)

0

and by the inverse of Fourier transform 

∞

f (x) = 0

Substituting f (x) from (1.7) in (1.6), we get  F(ω) = 0

∞



∞

 ˆ f (τ )χ (x, τ )dτ e−iωx d x

0

Interchanging order of integration of (1.9) yields

(1.9)

8

1 Introduction to Walsh Analysis and Wavelets



∞

F(ω) =

fˆ(τ )E(τ, −ω)dω,

0

where E(τ, −ω) is defined as either the Walsh–Fourier transform of e−iωx or equivalently as the Fourier transform of Walsh function χ (x, ω) (see Sect. 1.2). On the other hand, Walsh–Fourier transform of f can be expressed in terms of its Fourier transform.

1.3 Haar Functions and Its Relationship with Walsh Functions Let In (x) = I ( p, n), where I ( p, n) = [ p2−n , ( p + 1)2−n ], 0 ≤ p < 2n , n, p ∈ N

(1.10)

I ( p, n) is called a dyadic interval. For each x ∈ [0, 1), In (x) is called the dyadic interval of length 2−n . Let l0 denote the collection of sequences a = {an }n∈N such that an ∈ R. For each 0 < p < ∞, let l p denote the set of all sequences a ∈ l0 (space of real sequences) such that a p =

∞ 

1/ p |an | p

is finite.

(1.11)

n=0

Let l∞ denote the collection of a ∈ l0 such that a∞ = supn∈N |an | is finite.

(1.12)

It may be recalled that l p , 1 ≤ p ≤ ∞ is a Banach space and for {ak }, {bk } ∈ l0 (Abel’s transformation) ∞  k=0

⎛ ⎞ n−1  k n   ⎝ ⎠ ak bk = a j (bk − bk+1 ) + bn aj. k=0

j=0

For each n ∈ N set = 2−n/2 wk ak(n) j

(1.13)

j=0



j 2n

 ,

for 0 ≤ j, k < 2n , j, k ∈ N . A matrix of form )2n−1 A(n) = (ak(n) j,k=0 , n ∈ N j

(1.14)

1.3 Haar Functions and Its Relationship with Walsh Functions

9

is called a Hadamard–Paley matrix. Since w j (k/2n ) = wk ( j/2n ), by (properties of Walsh function)), each Hadamard– Paley matrix A(x) is real and symmetric 2n × 2n matrix and furthermore, it is orthogonal matrix: n 2 −1

ak(n) a (n) ji i

=

i=0

n 2 −1

2−n wk

i=0

=

1 2n



wj 

n 2 −1 

wk w j =

I (i,n)

i=0



1

i 2n



wk w j .

0

The Hadamard transform H is defined on l0 as follows: Given b = {bk }k∈N ∈ l0 , define H b = bk−1 H bk , for all k ∈ N , H b ∈ l0 , H b0 = b0 ; For 0 ≤ k < 2n , n ∈ N set H b2n +k =

n 2 −1

ak(n) b2n + j j

(1.15)

j=0

Since the Hadamard–Paley matrices are symmetric and orthogonal, it is clear that H takes l0 onto l0 and satisfies H H (b) = b. that is, if a is a Hadamard transform of b then b is Hadamard transform of a. The concept of Hadamard transform can be extended to function sequences like n 2 −1

b2n +k H ( f 2n +k ) =

k=0

n 2 −1

k=0

=

n 2 −1

j=0

=

n 2 −1

⎛

⎞

b2n +k ⎝

ak(n) . f 2n + j ⎠ j

n 2 −1

⎛ ⎝

j=0 n 2 −1

⎞

ak(n) b2n +k ⎠ j

f 2n + j

j=0

H (b2n + j ) f 2n + j

j=0

for any function sequence { f k } and {bk } ∈ l0 . In particular, n 2 −1

k=2n−1

bk H ( f k ) =

n 2 −1

H (bk ) f k

(1.16)

k=2n−1

Property (1.16) is called Hadamard transform self adjoint. We define below the Haar system and indicate that the Walsh and Haar systems are Hadamard transforms of each other, see [5, pp. 22–23].

10

1 Introduction to Walsh Analysis and Wavelets

The Haar system {h n } is defined as follows: Set h 0 = 1. For n, k ∈ N with 0 ≤ k < 2n define h n on [0, 1] by ⎧ n/2 ⎨ 2 , if x ∈ I (2k, n + 1) h 2n +k (x) = −2n/2 , if x ∈ I (2k + 1, n + 1) ⎩ 0, other wise h 2n +k (x + 1) = h 2n +k (x). Each Haar function is continuous from the right and the Haar system {h n } is orthonormal on [0, 1). The following relation expresses Haar functions in terms of Walsh functions. 2  1 = √ [wm (k/2n ) = wm+1 (k/2n )]wk (x). 2 2n−1 k=0 n−1

h 2n +k =

h kn (x)

1.4 Walsh-Type Wavelet Packets A wavelet packet is a generalization of wavelets. Walsh function is an example of wavelet packet. Walsh-type wavelet packet is a generalization of Walsh function and is a special case of wavelet packet. We mention here some beautiful results of the Ph.D. thesis of Morten and Nielsen submitted to Washington University, Saint Louis, Missouri, May 1999 and approved by the committee chaired by Prof. M. Victor Wickerhauser. Definition 1.1 (Nonstationary Wavelet Packets) Let (ϕ, ψ) be the scaling function ( p) ( p) and wavelet associated with a multiresolution analysis, and let (F0 , F1 ), p ∈ N be a family of bounded operators on l2 (N ) of the form (Fε( p) a)k =



an−2k h ( p) , ε = 0, 1

n∈N

(Fε( p) a)k =



an h ε( p) (n − 2k), ε = 0, 1

n∈Z ( p)

( p)

( p)

with h 1 = (−1)n h 0 (1 − n) a real value sequence in l1 (N ) such that each (F0 , ( p) F1 ) is a pair of conjugate quadrature filters (CQFs), see below. Let {h n } ∈ l1 (N )) be a real valued sequence, and let gk = (−1)k h 1−k for k ∈ Z . Define the operators H, G : l2 (N ) → l2 (N ) by

1.4 Walsh-Type Wavelet Packets

11

(H a)k =



an h n−2k

n∈N

(Ga)k =



an gn−2k

n∈N

The filters H and G are called a pair of CQF if 2H H ∗ = 2GG ∗ = 1, H I = 1, where I = (1, 1, 1, ...1) H ∗G + G∗ H = 1 H G∗ = G H ∗ = 0 of the form ( p)

( p)

p

with h 1 (n) = (−1)n h 0 (1 − n) a real-valued sequence in l 1 (N ) such that (F0 , ( p) F1 ) is a pair of CQFs. H ∗ and G ∗ are defined as 

(Ha∗ )k =

a j h k−2 j

j∈Z

(G a∗ )k =



a j gk−2 j

j∈Z

We define the family of nonstationary wavelet packets {wn }∞ n=0 recursively by letting w0 = ϕ, w1 = ψ, and then for n ∈ N w2n (x) = 2



( p)

h 0 (q)wn (2x − q)

(1.17)

q∈Z

w2n+1 (x) = 2



( p)

h 1 (q)wn (2x − q), wher e 2 p ≤ n < 2 p+1

(1.18)

q∈Z

Definition 1.2 (Basic Stationary wavelet packets). Let (ϕ, ψ) be the scaling function and wavelet associated with a multiresolution analysis, with associated CQFs {h n } p and {gn }. The functions {wn }n generated by Definition (1.1) by letting {h 0 } = {h n } ( p) and {h 1 } = {gn } for all p ∈ N are called basic stationary wavelet packets. Definition 1.3 The CQFs given by h 0 = h 1 = 21 , (−1)k h 1−k are called the Haar filters.

h k = 0 otherwise, and gk =

Definition 1.4 The Walsh system {wn }∞ n=0 is defined recursively on [0, 1) by w0 (x) = χ[0,1] (x) and w2n+ε (x) = wn (2x) + (−1)ε wn (2x − 1), ε = 0, 1, n = 0, 1.......

12

1 Introduction to Walsh Analysis and Wavelets

Definition 1.5 (Walsh type wavelets packets). Let {wn }n≥0,k∈Z be a family of non( p) stationary wavelet packets constructed by using a family {h n }∞ p=1 of finite filters ( p)

in Definition 1.1. If there exists a constant J ∈ N such that h n is the Haar filter for every p ≥ J and w1 has compact support then {wn }n≥0 is called a family of Walsh-type wavelet packets. It may be observed that Walsh-type wavelet packets resemble Walsh functions. Interesting properties of Walsh-type wavelet packet expansions are proved by M. Nielson [21]. Theorem 1.5 The Walsh-type wavelet packet expansion of any f ∈ L p (R), 1 < p < ∞, converges a.e..

1.5 Wavelet Analysis Wavelets meaning small waves were introduced in the early 80s. Wavelet theory dealing with various aspects of wavelets is the outcome of a multidisciplinary endeavor that brought together mathematicians, physicists, and engineers. Wavelet theory is a refinement of Fourier analysis which enables to simplify the description of a cumbersome function in terms of a small number of coefficients. For a detailed account of wavelet analysis and applications, we refer to Daubechies [10], Christensen [16], Frazier [15], Mallat [22] and Siddiqi [18]. We present here a few basic results.

1.5.1 Continuous Wavelet Transform Definition 1.6 A function ψ ∈ L 2 (−∞, ∞) is called a wavelet if it has zero average, that is,  ∞ ψ(t)dt = 0 (1.19) −∞

and

 Cψ = 2π

∞

−∞

2 ˆ |ψ(ω)| dω < ∞ |ω|

(1.20)

ˆ where ψ(ω) is the Fourier transform of ψ(t). Remark 1.1 Very often, ψ is called wavelet without (1.20). Condition (1.20) is essential for proving inversion result. Many researchers take the following condition in place of (1.20): ψ(x) is continuous and has exponential decay [ψ(x) ≤ Me−c|x| for some constants c and M.]

(1.21)

1.5 Wavelet Analysis

13

Remark 1.2 If ψ ∈ L 2 (−∞, ∞) ∩ L 1 (−∞, ∞) then (1.20) implies (1.19). Equation (1.20) is called the wavelet admissibility condition. Lemma 1.1 Let φ be a nonzero n-times, n ≥ 1, differentiable function such that φ (n) ∈ L 2 (−∞, ∞). Then ψ(x) = φ n (x) is a wavelet. Proof From the property of the Fourier transform (see for example, [18]) ˜ ˜ |ψ(ω)| = |ω|k |φ(ω)|. Then  Cψ =

∞

−∞ ∞

 = =

−∞  −1

2 ˆ |ψ(ω)| dω |ω| 2 ˆ |ω|2k |φ(ω)|

|ω|

dω

2 ˆ |ω|2k−1 |φ(ω)| dω +

1

≤

2π(φ2L 2 

It is also clear that

∞

−∞

φ (k) 2L 2 )

+

 ψ(x)d x =

 |ω|>1

2 ˆ |ω|2k |φ(ω)| dω |ω|

0, b ∈ R (1.23) ψ ψa,b (t) = |a| a where ψ is a fixed function, often called mother wavelet, then Tψ f (a, b) =< f, ψa,b >= inner product of f and ψa,b . Remark 1.3 The following properties can be verified keeping in mind the properties of inner product. (a) Tψ (α f + βg)(a, b) = αTψ f (a, b) + βTψ g(a, b) for any α, β ∈ R. (b) Tψ (Sc f )(a, b) = Tψ f (a, b − c), where Sc is a translation   operator defined by Sc f (t) = S(t − c). (c) Tψ (Dc f )(a, b) = √1c Tψ f (a/c, b/c), where c is a positive number and Dc is the dilation operator defined by Dc f (t) = (1/c) f (t/c). (d) Tψ φ(a, b) = Tφ ψ(1/a, −b/a), a = 0, where ψ and φ are two wavelets.

1.5 Wavelet Analysis

15

(e) T Aψ A f (a, b) = Tψ f (a − b), where A is defined by Aψ(t) = ψ(−t) (f) TSc ψ( f (a, b)) = Tψ f (a, b + ca) (g) (TDc ψ f )(a, b) = √1(c) (Tψ f )(ac, b), c > 0 Remark 1.4 Tψ f (a, b) is a function with scale (frequency) a and the spatial (time) b. The plane defined by the variables (a, b) is called the scale-space or time–frequency plane. It measures the variation of f in a neighborhood of b . For a compactly supported wavelet, the value of Tψ f (a, b) depends upon the value of f in a neighborhood of b of size proportional to the scale a. At small scales, Tψ f (a, b) provides localized information such as localized regularity of f (x). Few important results like Parseval’s formula, isometry, and inverse formula for wavelet transformation are mentioned below whose proofs can be found in [10, 18].

1.5.2 Discrete Wavelet System We discretize continuous wavelet transforms by choosing a = 2− j , b = k2− j , j, k ∈ N. Definition 1.8 A function ψ ∈ L 2 (−∞, ∞) is a wavelet if the family of functions ψ j,k (t) defined by j (1.24) ψ j,k (t) = 2 2 ψ(2 j t − k) is an orthonormal basis in L 2 (R), that is, 

∞

−∞



ψ j,k (t)ψ j  ,k  (t)dt = 1 i f j = j , k = k



= 0 other wise

be defined by the condition We say that the functions ψ j,k j, k ∈ N , form the wavelet system associated to the function ψ. It may be observed that the wavelet admissibility condition (1.5.2) is a necessary condition for ψ j,k (t) to be a wavelet system. Definition 1.9 Wavelet coefficients of a function f ∈ L 2 (−∞, ∞), denoted by d j,k , are defined as the inner product of f with ψ j,k (t), that is  d j,k =< f, ψ j,k (t) >=

∞

−∞

f (t)ψ j,k (t)dt

(1.25)

16

1 Introduction to Walsh Analysis and Wavelets

The series



< f, ψ j,k (t) > ψ j,k (t)

(1.26)

j∈N k∈N

is called wavelet series of f . The expression 

< f, ψ j,k (t) > ψ j,k (t)

j∈N k∈N

is called the wavelet representation of f . A characterization of Lipschitz α class, 0 < α < 1 in terms of the wavelet coefficients is given below: Theorem 1.6 f ∈ Li p α, that is, there is a constant K > 0 such that | f (x) − f (y)| ≤ k|x − y|α , 0 < α < 1, if and if only | d j,k |≤ K 2−( 2 +α) 1

(1.27)

For proof we refer [12, 14, 18].

1.5.3 Multiresolution Analysis Multiresolution Analysis introduced by Mallat and Meyer in 1989 provides a general method to construct wavelet orthonormal basis. It is considered as the most important ingredient of wavelet theory. Definition 1.10 (Multiresolution Analysis-[13]) A Multiresolution Analysis (MRA) is a sequence {V j } of closed subspaces of L 2 (−∞, ∞) such that (i) (ii) (iii) (iv) (v)

..... ⊂ V−1 ⊂ V0 ⊂ V1 ......  V = L 2 (−∞, ∞)  j j∈Z V j j∈Z = 0 f (x) ∈ V j if and only if f (2− j x) ∈ V0 , f (x) ∈ V0 if and only if f (x − m) ∈ V0 for all m ∈ N and • there exists a function φ ∈ V0 called scaling function, such that the system {ψ(t − m)m∈N } is an orthonormal basis in V0 .

Construction of a wavelet from MRA: For a given MRA {V j } in L 2 (R) with scaling function φ, a wavelet is obtained in the following manner: Let the subspace W j of L 2 (R) be defined by the condition V j ⊕ W j = V j+1 , V j ⊥W j ∀ j.

1.5 Wavelet Analysis

17

j

Since 2 2 J j , where J j is the dyadic dilation operator defined as J j ( f )(x) = f (2 j x), we have J j (V1 ) = V j+1 . Thus, V j+1 = J j (V0 ⊕ W0 ) = J j (V0 ) ⊕ J j (W0 ) = V j ⊕ J j (W0 ) This gives W j = J j (W0 ) f or all j ∈ Z . By conditions (i) to (iii) we obtain an orthogonal decomposition L 2 (R) =



W j = W1 ⊕ W2 ⊕ W3 ⊕ ... ⊕ Wn ⊕ ...

j∈N

We can find a function ψ ∈ W0 such that {ψ(t − m)m∈Z } is an orthonormal basis in W0 . Such ψ is a wavelet and it is called wavelet associated with the MRA {V j } then ψ(x) =



an (−1)n φ(2x + n + 1),

(1.28)

n∈Z



where an =

∞

−∞

φ

x  2

φ(x − n),

(1.29)

is a wavelet.

1.6 Wavelets with Compact Support Daubechies Wavelets Daubechies (see for details [10, 12] and Pollen in [17]) has constructed, for an arbitrary integer N, an orthonormal basis for L 2 (R) of the form 2 j/2 ψ(2 j x − k), j, k ∈ Z having the following properties: The support of ψ N is contained in [−N + 1, N ]. To emphasize this point, ψ is denoted by ψ N . 

∞

−∞

 ψ N (x) d x =

∞

−∞

 xψ N (x) d x = . . . =

∞

−∞

x N ψ N (x) d x = 0

(1.30)

18

1 Introduction to Walsh Analysis and Wavelets

ψ N (x) has γ N continuous derivatives, where the positive constant γ is about 1/5

(1.31)

Theorem 1.7 (Daubechies) There exists a constant K such that for each N = 2, 3, . . . , there exists an MRA with the scaling function φ and an associated wavelet ψ such that 1. φ(x) and ψ(x) belong to C N . 2. φ(x) and ψ(x) are compactly supported and both suppφ and suppψ(x) are contained in [−K N , K N ]. ∞ ∞ ∞ 3. −∞ ψ N (x) d x = −∞ xψ N (x) d x = . . . = −∞ x N ψ N (x) d x = 0. We refer to [18] for a proof of the theorem. Here, we present a construction of the Daubechies scaling function and wavelet on [0, 3] due to Pollen(for details, see [17]). Theorem 1.8 The function ψ defined as ψ(x) = −bφ(2x) + (1 − α)φ(2x − 1) − (1 − b)φ(2x − 2) + aφ(2x − 3) (1.32) satisfies the following conditions: supp ψ(x) ⊂ [0, 3] 

∞

−∞



∞

−∞

 ψ(x)ψ(x − k) d x =

1k=0 0 k = 0.

ψ(x − k)φ(x) d x = 0 for all k ∈ Z .

(1.33)

(1.34)

Thus, {2 j/2 ψ(2 j t − k)} j∈Z ,k∈Z is an orthonormal basis in L 2 (R). Lemma 1.2 ([17], p. 90) For every x ∈ D, we have 

ψ(x − k) = 1

(1.35)

k∈Z

 √  3− 3 + k ψ(x − k) = x 2 k∈Z

(1.36)

Lemma 1.3 If x ∈ D and 0 ≤ x ≤ 1, then √ 1+ 3 2φ(x) + φ(x + 1) = x + 2

(1.37)

1.6 Wavelets with Compact Support

19

√ 3− 3 2φ(x + 2) + φ(x + 1) = x + 2 √ −1 + 3 φ(x) − φ(x − 2) = x + 2

(1.38)

(1.39)

Lemma 1.4 For 0 ≤ x ≤ 1 and x ∈ D, the following relations hold:

φ

φ

φ

1+x 2 2+x 2

φ

φ



= aφ(x)

√ 2+ 3 = bφ(x) + ax + 4



√ 3 = aφ(1 + x) + bx + 4

 = aφ(1 + x) − ax +

1 4

√ 3−2 3 = aφ(2 + x) − bx + 4

φ





3+x 2

4+x 2

0+x 2

5+x 2

 = bφ(2 + x)

N Lemma 1.5 Suppose that m(ξ ) = k=M ak e−ikξ is a trigonometric polynomial such that (1.40) |m(ξ )|2 + |m(ξ + π )|2 = 1 for all ξ ∈ R m(0) = 1

(1.41)

 π π . m(ξ ) = 0 for ξ ∈ − , 2 2

(1.42)

Then the infinite product θ (ξ ) =

∞ 

m(2− j ξ )

(1.43)

j=1

converges almost uniformly. The function θ (ξ ) is thus continuous. Moreover, it belongs to L 2 (R). The function φ given by φˆ = √12π θ (ξ ) has the support contained in [M, N ] and is a scaling function of an MRA. In particular, {φ(x − k)} is an ONB in L 2 (R). The function ψ(x) defined by

20

1 Introduction to Walsh Analysis and Wavelets

ψ(x) = 2

N 

(−1)k a¯k φ(2x + k + 1)

(1.44)

k=M

is a compactly supported wavelet with suppψ ⊂

 M−N −1 2

,

N −M−1 2

.

Lemma 1.6 Let D = ∪ j∈Z D j where D j = {k2− j /k ∈ Z } (D is a ring, that is, sums, difference and product of elements of D are also in D. It is a dense subset of R). Then, there exists a unique function φ : D → R having the following properties: φ(x) = aφ(2x) + (1 − b)φ(2x − 1) + (1 − a)φ(2x − 2) + bφ(2x − 3) (1.45) 

φ(k) = 1

(1.46)

k∈Z

φ(d) = 0 if d < 0 or d > 3 where a =

√ 1+ 3 , 4

b=

√ 1− 3 . 4

(It is clear that

1 2

(1.47)

< a < 1 and − 41 < b < 0).

Lemma 1.7 The function φ defined in Lemma 1.6 extends to a continuous function on R which we also denote by φ: This continuous function φ has the following properties:  ∞

−∞

and



φ(x) d x = 1 

∞

φ(x)φ(x − k) d x =

−∞

(1.48)

1k=0 0 k = 0.

(1.49)

In other words, φ is a scaling function. Proof of Lemma 1.7 Let K is a nonlinear operator acting on the space of functions on R. Let us define K ( f ) for x ∈ [0, 1] by the following set of conditions:

0+x K( f ) 2

K( f )

1+x 2





2+x K( f ) 2

K( f )

3+x 2

 = a f (x)

 = b f (x) + ax +

√ 2+ 3 4

√ 3 = a f (1 + x) + bx + 4

 = b f (1 + x) − ax +

 1 4

1.6 Wavelets with Compact Support



4+x K( f ) 2

21



 = a f (2 + x) − bx +

K( f )

5+x 2

√ 3−2 3 4

 = b f (2 + x)

K ( f )(y) = 0 for y ∈ / [0, 3]. This definition gives two values of K ( f ) at the points 0, 1/2, 1, 3/2, 2, 5/2, 3. us denote by φ j , j = 0, 1, 2, . . ., the continuous, piecewise linear functions are which on D j equals φ. The function K (φ j ) is well defined at each point and infact, K (φ j ) = φ j+1 . Let x ∈ [0, 3] and j > 0, then we immediately get from the definition of K ( f ) φ j+1 (x) − φ j (x) = K (φ j )(x) − K (φ j−1 )(x) = η(φ j (y) − φ j−1 (y))

(1.50)

where η = a or η = b and y ∈ R is a point depending on x. Since K ( f )(x) = 0 for x∈ / [0, 3] and max(|a|, |b|) = a, from (1.50), we get φ j+1 − φ j ∞ ≤ aφ j − φ j−1 ∞ so by induction, we get φ j+1 − φ j ∞ ≤ aφ1 − φ0 ∞ . Since φ1 − φ0  is finite, the sequence {φ j } converges uniformly to a continuous function which is denoted by φ. This proves the first part of the theorem. We know that [17, p. 92, 4.6], holds for x ∈ R. Since  supp φ ⊂ [0, 3] for each x ∈ R, there are atmost three nonzero terms in the k∈Z φ(x − k). For a positive integer M, let M  φ(x − k) FM (x) = k=−M

From (1.48), we conclude that |FM (x)| ≤ C for some constant C and FM (x) = 1 i f |x| ≤ M − 3 = 0 if |x| ≥ M + 3 Thus, for every integer M

22

1 Introduction to Walsh Analysis and Wavelets

 2(M − 3) − 12C ≤

∞

−∞

FM (x) d x ≤ 2(M − 3) + 12C

(1.51)

From the definition of FM , we also conclude that 

∞ −∞

 FM (x) d x = (2M + 1)

∞ −∞

φ(x) d x

(1.52)

Since (1.51) and (1.52) hold for every positive integer M, making M → ∞, we obtain (1.48). To prove (1.49), let  Lk =

∞

−∞

φ(x)φ(x − k) d x

(1.53)

Since suppφ ⊂ [0, 3], we find that L k = 0 for |k| ≥ 3

(1.54)

L k = L −k .

(1.55)

It is clear that By a change of variable, we see that, for any l, m, n ∈ Z 

∞

−∞

φ(2x − m)φ(2x − 2l − n) d x =

1 L 2l+n−m 2

(1.56)

Substituting value of φ(x) given by (1.52) into (1.53) for l = 0, 1, 2 and applying (1.54) and (1.55), we obtain the following equations: (a(1 − a) + b(1 − b))L 0 = (1 − ab)L 1 + (a(1 − a) + b(1 − b))L 2 2L 1 = (a(1 − a) + b(1 − b))L 0 + (1 − b)L 1 +((1 − b)2 + (1 − a)2 + b2 )L 2 2L 2 = abL 1 + (a(1 − a) + b(1 − b))L 2 From given values of a and b (see Lemma 1.6), we have a(1 − a) + b(1 − b) = 0. Thus, the above system of equations becomes 0 = (1 − ab)L 1 2L 1 = (1 − ab)L 1 + ((1 − b)2 + (1 − a)2 + b2 )L 2 2L 2 = abL 1

(1.57)

1.6 Wavelets with Compact Support

23

This implies that L 1 = L 2 = 0. Thus L k = 0 for k = 0. By (1.48) and (1.49), we can compute  1= =

∞

−∞



 φ(x) d x =

∞ −∞

φ(x).



φ(x − k) d x

k∈Z

L k = L 0.

k∈Z

This proves (1.49). Proof of Theorem 1.7 Suppψ(x) ⊂ [0, 3] follows immediately from (1.32) and the fact that suppφ ⊂ [0, 3]. To obtain (1.33), substitute ψ(x) given by (1.32) into left-hand side of (1.33). We obtain (1.33) using (1.56), (1.48), (1.57) and values of a and b. To obtain (1.34), we proceed similarly but substitute both ψ(x) given by (1.32) and φ(x) given by (1.45) into left-hand side of (1.34). It follows directly from (1.33) and (1.34) that 2 2j ψ(2 j t − k) j∈Z ,k∈Z is orthogonal.

1.7 Exercises 1.1 Show that the characteristic function of this interval [0, x] is W -continuous for x ∈ (0, 1) if and if x ∈ Q (set of rational numbers). 1.2 Let f = χ[0,1/2] . Show that f is W -continuous but f (|x|) is not continuous on G (Dyadic group). 1.3 Show that the map x → |x| is continuous from G to [0, 1] if the metric d(x, y) = |x !− y| is used on [0, 1]. Examine whether it is continuous if the dyadic metric is used instead. 1.4 Show that Walsh system is orthonormal. 1.5 Prove that Walsh–Fourier transform is W -continuous and f (ω) → 0 as ω → ∞. 1.6 Find the integral of Walsh polynomial of degree 2. 1.7 Let h n (x) denote Haar system and let wn (x) denote Walsh system. What is the relation between h n (x) and wn (x)? 1.8 Discuss relationship between Fourier and Walsh–Fourier transform. 1.9 Explain the concept of Walsh-type wavelet packet. 1.10 Draw the graph of the oldest known mother wavelet, the Haar wavelet dates back to 1910 ⎧ ⎨ 1 if 0 ≤ t < 1/2 ψ(t) = −1 if 1/2 ≤ t < 1 ⎩ 0, otherwise 1.11 Write down the scaling function φ associated with Haar wavelet.

24

1 Introduction to Walsh Analysis and Wavelets

1.12 Draw the graph of ψ(22 t) and ψ(2t), where ψ(t) and φ(t) denotes respectively Haar wavelet and Haar scaling function. 1.13 Draw the graph of ψ(t − 3), ψ(t + 3), φ(t − 4) and φ(24 t). 1.14 Let ψ(t) be the Haar mother wavelet. Then show that ψ j,k (t) = 2 j/2 ψ(2 j t − k) (R).  is an orthonormal system in L 2 1.15 Let f ∈ L 2 (R) and Pr ( f ) = j 0 there is a g ∈ P (it denotes the set of Walsh polynomials) such that  f − g L 1 < ε, by the fact that Walsh polynomials are dense in L 1 . 

1

 f wn =

0

as g(n) ˆ = 0 for large n. Therefore,

1

( f − g)wn for large n,

0

lim sup| fˆ(n)| ≤  f − g L 1 < ε

n→∞

implying the desired result. It may be observed that for f ∈ L 2 and Walsh–Fourier coefficients ck ’s of f we have  1 ∞  2 ck = | f (x)|2 d x. k=0

0

This is known as Parseval’s formula.

2.1.1 Estimation of Walsh–Fourier Coefficients We present below the results concerning estimates of the growth of Walsh–Fourier coefficients of functions from various classes such as L 1 , L p , continuous, functions of bounded variation, and absolutely continuous. For functions defined on I = [0, 1] or dyadic intervals [0, 2n ] which are W -continuous (continuous W ),

2.1 Walsh–Fourier Coefficients

29

˙ − f (x)|/x ∈ I, 0 ≤ y ≤ 1} w( f, I ) = sup{| f (x +y) is called the local moduli of continuity. For 1 ≤ p < ∞ and each f ∈ L p w( p) ( f, I ) = sup 0≤yT

= S1 + S2 + o(1). Given ε > 0, choose T so that ck  < ε for k  > T . Then S2 < ε. For S1 write k = 2v + k  , where 0 ≤ v ≤ r , 0 ≤ k  ≤ T and r is defined by 2r ≤ n ≤ 2r +1 . Then 1  1 + log2n = o(1). |ck  | ≤ M n + 1 v=0 k  =0 n+1 r

S1 ≤ Therefore,

1 n+1

n k=0

r

k|bk | = o(1) which completes the proof of Theorem 2.6.

Theorem 2.7 Let A = (am k ) be a regular matrix f ∈ BV [0, 1] and Bk (x) denote the sequence {kck ψk (x)} where ck are Walsh–Fourier coefficients of f . Then for every x ∈ [0, 1]

32

2 Walsh–Fourier Series ∞ 

lim

m→∞

if and only if



lim

m→∞

where Jk (x) =

x 0

am,k Bk (x) = 0

(2.4)

amk k Jk (x) = 0,

(2.5)

k=0

k=0

wk (t) dt, k = 0, 1, 2, . . ., in every 0 < δ < t ≤ 1.

Proof It can be seen that ∞ 

amk Bk (x) =

k=0

= = = =

∞  k=0 ∞  k=0 ∞  k=0 ∞  k=0 ∞  k=0 ∞ 

amk kck wk (x) 

1

amk k

f (t)wk (x)ψk (t) dt

0



1

amk k

˙ dt, by Theorem1.4.1 [82], also f (t)ψk (x +t)

0



1

amk k

˙ f (x +t)ψ k (t) dt

0

˙ k (t)]10 − amk k[ f (x +t)J  amk k

1

˙ integration by parts, Jk (t) d f (x +t),

0

k=0



∞ 1

=− 0

˙ amk k Jk (t) d f (x +t),

k=0

 ˙ as ∞ k=0 amk k f (x +t)Jk (t) = 0 by virtue that Jk (1) and Jk (0) are zero. Thus  1 ∞ ∞  ˙ | amk Bk (x) = − amk k Jk (t)d f (x +t). 0

k=0

k=0

 ˙ Let K m (t) = ∞ k=0 amk k Jk (t) and φx (t) = f (x +t). In order to prove the theorem we a k J (x) = 0 holds then for every f ∈ BV [0, 1] have to show that if limm→∞ ∞ mk k k=0 and for every x ∈ [0, 1],  lim

n→∞ 0

1

K m (t) dφx (t) = 0

and conversely. Condition (2.6) is equivalent to the following condition:

(2.6)

2.1 Walsh–Fourier Coefficients



1

lim

m→∞ 0

33

K m (t) dφx (t) = 0 for every f ∈ BV [0, 1]

(2.7)

for every x ∈ [0, 1] and for 0 < δ < 1. For f ∈ BV [0, 1] and x ∈ [0, 1], given any ε > 0 there exists a δ > 0 such that 

δ

|dφx (t)|
0. k=0 k k Theorem 2.10 Let ( f (x) − c)/(u − x0 ) be absolutely integrable in interval |u − x0 | < δ for some δ > 0. Then the Walsh–Fourier series of f (x) converges to c at the point x0 . Theorem 2.11 If f (x) is continuous and if its modulus of continuity satisfies w(δ, f ) = o(log δ −1 )−1 as δ → 0, then its Walsh–Fourier series converges to f (x) uniformly. Theorem 2.12 Let f (x) ∈ Li pα, α > 21 . Then the Walsh series of f (x) converges absolutely. Theorem 2.13 Let f (x) be a continuous function of period 1. Let 2  n+1

u n (x) =

p=1

  p

 ˙ ˙ p + 1))/2n+1  . − f (x +(2 p −1  f (x)+ 2n+1

Then lim u n (x) = 0 uniformly in x

n→∞

implies that lim Sk ( f, x) = f (x) uniformly in x.

k→∞

Theorem 2.14 Let f (x) satisfy  1 1 | f (x + t) − f (x − t)| p d x dt < ∞, 2 ≥ p ≥ 1. t 0 o Then, the Walsh–Fourier series of f (x) converges almost everywhere. Theorem is a p-periodic function defined on [0, 1] and 2.15 If f (t)   ∞ f (x) = ∞ then f 1(t) = ∞ where n=0 cn wn (t), n=0 |cn | < ∞, n=0 dn wn (t), ∞ |d | < ∞ provided inf | f (t)| > 0. n t n=0 Theorem 2.16 If 1 < p < ∞ and f ∈ L p then ||Sn f − f || L p → 0, as n → ∞. Theorem 2.17 For f ∈ L p , p > 1, Sm f converges to f almost everywhere.

36

2 Walsh–Fourier Series

Theorem 2.18 For f ∈ L 1 and satisfying 1 w(1) ( f, δ) = o(log )−1 as δ → 0, δ ||Sn f − f || L 1 → 0 as n → ∞. We prove here Theorems 2.9–2.12 and 2.16, for the proof of other theorems, we refer to [1, 2]. It may be remarked that short proof of Theorem 2.15 is available using results from Banach Algebra. In the proof of Theorem 2.9, we require the following lemma. Lemma 2.1 For all u such that 0 < u < 1, |Dk (u)| < 2/u. Proof Suppose that 2−n ≤ u < 2−n+1 , and write k in the form k = p2n + q, 0 ≤ q ≤ 2n . We have wr (u)ws (u) = ws+r (u) provided that r and s have dyadic expansions without any exponents common to both, and Dk (u) =

k−1 

p−1 2 −1   n

wr (u) =

r =0

wr 2n +1 (u) +

r =0 i=0

q−1 

p−1 2 −1   n

=

w2n (u)wi (u) +

r =0 i=0

= D2n (u)

w p2n +1 (u)

i=1 q−1 

w p.2n (u)wi (u)

i=0 p−1 

wr (2n u) + w p (2n u)Dq (u)

r =0

= D2n (u)D p (2n u) + w p (2n u)Dq (u). Since D2n (u) contains 1 + w2n−1 (u) = 1 + w1 (2n−1 u) as a factor, and u < 1, D2n (u) = 0, hence

1 2

≤ 2n−1

|Dk (u)| = |Dq (u)| ≤ q ≤ 2n < 2/u. This proves the lemma. Proof of Theorem 2.9 We have ∞  k=0

ck wk (u) =

n−1  (ck − ck+1 )Dk (u) + ak Dn (u) k=0

by Abel’s transformation. By Lemma 2.1 and the fact that ck → 0 as k → ∞,  ∞ k=0 ck wk (x) converges uniformly in δ ≤ u < 1. Proof of Theorem 2.10 We have

2.2 Convergence of Walsh–Fourier Series

sk (x0 , f ) − c = =

 1 0

37

( f (u) − c)Dk (x0 + u) du 

|u−x0 | 0, we can choose δ small enough so that the first integral does not exceed δ/2; with δ thus chosen and fixed, the second integral can be made less than ε/2 for k > k0 , (δ, ε) = k0 (ε). This completes the proof of the theorem. Proof of Theorem 2.11 By Theorem 2.8, s2n (x) → f (x) uniformly. It is sufficient to consider for any given k = 2n + k  , 0 ≤ k  < 2n , the difference 

1

s2n +k  (x) − s2n (x) =

˙ f (u) du. w2n (x)w2n (u)Dk  (x +u)

(2.15)

0

We know that for p < 2n , w p (2−(n+1) ) = 1 so that ˙ −(n+1) ) = Dk  (z); also w2n (u +2 ˙ −(n+1) ) = −w2n (u). Dk (z +2 By invariance of integration, we get 

1

s2n +k  (x) − s2n (x) = −

˙ f (u +2 ˙ −(n+1) ) du w2n (x)w2n (u)Dk  (x +u)

(2.16)

0

By adding (2.15) and (2.16), we get  2(s2n +k  (x) − s2n (x)) = 0

or

1

˙ ˙ ˙ −(n+1) )} du w2n (x +u)D f (u) − f (u +2 k  (x +u){

38

2 Walsh–Fourier Series

2|s

2n +k 

˙ −(n+1) )| − s (x)| ≤ max0≤u≤1 | f (u) − f (u +2



1

2n

˙ |Dk  (x +u)| du

0

≤ w(2−(n+1) , f )L k ≤ w(2−n , f )log(2n ) = 0(1). This proves the theorem. ∞ Proof of Theorem 2.12 ∞ By Theorem 2.1, k=0 ck wk (h)wk (x) is Walsh–Fourier ˙ if u=0 ck wk (x) is Walsh–Fourier series of f (x). By Parseval’s series of f (x +h) formula  1 ∞  ˙ − f (x)]2 d x = [ f (x +h) ck2 (1 − wk (h))2 . 0

u=0

Set h = 2−(n+1) . Then wk (2−(n+1) ) = −1 for 2n ≤ k < 2n+1 . Hence 2n+1 −1

 ck2 ≤

k=2n

1

˙ −(n+1) ) − f (x)]2 d x. [ f (x +2

0

˙ − f (x)| ≤ w(δ, f ), we have Since | f (x +h) 2n+1 −1

ck2 ≤ [w(2−(n+1) , f )]2 ≤ A2−nα ,

k=2n

where A is constant. By Schwarz’s inequality 2n+1 −1 k=2n

⎛ |ck | ≤ ⎝

2n+1 −1

⎞1/2 ⎛ ck2 ⎠

⎝

k=2n

2n+1 −1

⎞1/2 12 ⎠

k=2n

≤ A1/2 2−n(α−1)/2 . Since α > 1/2, the right-hand side of the above inequality is the nth term of a convergent series, which completes the proof. Proof of Theorem 2.16 For ε > 0, choose a Walsh polynomial P such that  f − P L p < ε. Since Sn P = P for n large, it is clear that  f − Sn f  L p ≤  f − P L p + §n P − Sn f  L p . By Corollary 6 of Sect. 3.3, [2] gives the desired result.

2.2 Convergence of Walsh–Fourier Series

39

2.2.1 Summability in Homogeneous Banach Spaces Definition 2.1 Let X be a Banach space with norm .. It is called a dyadic homogeneous Banach space if P ⊆ X ⊆ L 1 (G) hold, where P denotes the set of Walsh polynomials. (2.17)  f  L 1 ≤  f  X , ∀ f ∈ X. ˙ t ∈ [0, 1). For Tx f (t) = f (x +t), Tx f  =  f  X , x ∈ [0, 1). P is dense in X , that is,

(2.18)

P¯ = X.

L p (1 ≤ p < ∞), H (dyadic Hardy space), and C(G) are homogeneous spaces, for details, see [2, 4]. Definition 2.2 A sequence {Pn , n ∈ N } of functions in C(G) is called an approximate identity (for dyadic convolutions) if Pn  = O(1) , as n → ∞.  Pn dμ = 1, n ∈ N

(2.19) (2.20)

G

 lim

n→∞

G\Ik (o)

|Pn | dμ = 0 for eachk ∈ N .

{D2n , n ∈ N } and {K n , n ∈ N }, where K n = approximate identity.

1 n

n k=1

(2.21)

Dk , n ∈ N are examples of

n The following theorem  shows that in any homogeneous Banach space S2 f and σn f , where σn f = n1 nk=1 sk f , converge to f in norm as n → ∞.

Theorem 2.19 Let {Pn , n ∈ N } be an approximate identity and X be a homogeneous Banach space. Then lim Pn ∗ f − f  = 0 for all f ∈ X.

n→∞

For proof of this theorem, we refer to [2, p. 158].

40

2 Walsh–Fourier Series

2.3 Approximation by Transforms of Walsh–Fourier Series Let A = (amn ), m = 1, 2, 3, . . . , n = 1, 2, 3, . . . be an infinite matrix of real numbers. Then a transform of a sequence {sn }, y = {yn } is the sequence defined by the equation ∞  ym = amn xn , m = 1, 2, 3, . . . . n=1

Let sn f = then

n−1

k=0 ck wk (x),

where ck is the kth Walsh–Fourier coefficient of f ∈ L 2 , tm ( f, x) =

∞ 

amn sn ( f, x)

n=1

is called the matrix mean of Walsh–Fourier series. The general problem is to evaluate tm f − f  X for different X and special cases of tm . For X = Li p α and {tm }(C, 1) mean, namely, 1 for n < m m = 0 for n ≥ m.

amn =

 β +n β for n ≥ 1 and A0 = 1, β = −1, −2, . . .. n For X = Li p(α, p), Li p α, Li p j (t) and (C, 1) and (C, β) means the degree of approximation, that is, evaluation of tm f − f  X was carried out between 1947 and 1963. A comprehensive account of this development is presented in [1]. In Sect. 2.3.1, approximation by Cesàro mean of f ∈ Li p α is presented. Case of Nörlund means of Walsh–Fourier series in L p space is studied in Sect. 2.3.2. The same problem in homogeneous Banach spaces is presented [4] in Sect. 2.3.3. β



For amn = An =

2.3.1 Approximation by Césaro Means of Walsh–Fourier Series Fejer kernel or kernel for (C, 1) summability for Walsh–Fourier series is defined as 1 K k (x, u) = Dr (x, u). k r =1 k

This can be written as

2.3 Approximation by Transforms of Walsh–Fourier Series

41

1 ˙ Dr (x +u) k r =1 k

˙ = K k (x, u) = K k (x +u)

˙ = dyadic rational) by setting K k (u) = K x (0, u). (x +u (C, 1) mean of order k is defined as 

1

σk ( f, x) =

˙ f (u) du. K k (x +u)

0

Walsh–Fourier series of f (x) is said to uniformly summable by (C, 1) mean to f (x) if σn (x, f ), converges uniformly to f (x). For any nonnegative integer n and any real α(> −1), we define Aαn =

(α + 1)(α + 2) . . . (α + n) , Aα0 = 1. n!

The kernel for (C, α) summability is defined by K n(α) (x) =

1

n−1 

Aαn−1 k=0

Aαn−k−1 wk (x), n = 1, 2, 3, . . . .

Lemma 2.2 (i) For n ≥ 0, 0 ≤ k  ≤ 2n , (2n + k  )K 2n +k  (u) = 2n K 2n (u) + k  D2n (u) + w2n (u)k  K k  (u). (ii) K 2n (u) ≥ 0 for n ≥ 0. Proof (i) By definition and by the relation Dn (t) = D2n (t) + w2n (t)Dk  (t), where k = 2n + k  , with 0 ≤ k  < 2n , we have 2 



n

(2n + k  )K 2n +k  (u) =

r =1

Dr (u) +

k 

D2n +q (u)

q=1 

k  {D2n (u) + w2n (u)Dq (u)} = 2 K 2n (u) + n

q=1 



= 2 K 2n (u) + k D2n (u) + w2n (u) n

k  q=1 

Dq (u)

= 2n K 2n (u) + K  D2n (u) + w2n (u)k K k  (u).

42

2 Walsh–Fourier Series

(ii) Take k  = 2n in part (i), then 2n+1 K 2n+1 (u) = (1 + w2n (u))2n K 2n (u) + 2n D2n (u). Since D2n (u) ≥ 0, 1 + w2n (u) ≥ 0, K 1 (u) = 1. By the principle of induction, the result holds for all n. The failure of the estimate of Fejér kernel has created great difficulties in proving the analogue of Fejér approximation theorem. For several years, it was an open problem. We present the proof of this result by Yano [1, 2], and a generalization of Yano’s result by Siddiqi and Gupta [1]. Theorem 2.20 If f (x) ∈ Li pα, 0 < α < 1, then σn (x; f ) − f (x) = O(n −α ), where σn (x, f ) denote the arithmetic mean of the Walsh–Fourier series of f (x). We require the following lemmas in the proof of this theorem. Lemma 2.3 Let K n (t) be the Fejér kernel for the Walsh functions and let I p(n) denote the interval p.2−n ≤ t < ( p + 1)/2−n , 0 ≤ p < 2n . Then for n ≥ 2; 2n + 1 (t ∈ I0(n) ), 2 =0,1,...,n−1 ), (ii) K 2n (t) = 2n−r −2 (t ∈ I2(n),r n n (iii) K 2 (t) = 0 elsewhere in (0, 1). (i) K 2n (t) =

(2.22)

Lemma 2.4 Under the assumption of Theorem 2.1 σ2n (x, f ) − f (x) = O(2−αn ).

(2.23)

Proof of Lemma 2.3(i) This lemma holds clearly for n = 2. Suppose (2.22) holds for n ≥ 2, then 1 + ψ2n (t) 1 K 2n (t) + D2n (t). (2.24) K 2n+1 (t) = 2 2 By using this relation, we show that (2.22) holds for n + 1. We know that K 2n+1 (t) (n+1) ψ2n is constant on I p(n+1) , 0 ≤ p < 2n+1 , on I0(n+1) and I1(n+1) , D2n (t) = 2n , on I2m (n+1) (n+1) n−1 (t) = 1, and on I2m+1 , w2n (t) = −1. Hence, on I0 , K 2n+1 (t) = K 2n (t) + 2 = (2(n−1) /2 + 2n+1 ) = (2n+1 + 1)/2. So that (i) is true for (n + 1). On I1(n+1) , K 2n+1 (t) = 2n−1 , so that (ii) is true for (n + 1), with r = 0. For 2−n ≤ t < 1, D2n (t) = ⊂ I2nr −1 for 1 ≤ r < n + 1, we have K 2n+1 (t) = K 2n (t) = 0, hence if t ∈ I2(n+1) r 2n−(r −1)−2 = 2(n+1)−r −2 , and (ii) is true for 0 ≤ r < n + 1. Now (iii) follows from the fact that the integral of K 2(n+1) (t) over (0, 1) is 1, that K 2(n+1) (t) is nonnegative and that the integral over the interval specified in (i) and (ii) (with n replaced by n + 1) is 2n+1 + 1 1 2−(n+1) { + 2n−1 + · · · + 1 + } = 1. 2 2

2.3 Approximation by Transforms of Walsh–Fourier Series

43

This completes the proof of the lemma. Proof of Lemma 2.4 We have  σ2n (x; f ) − f (x) =

1

˙ − f (x)]K 2n (t) dt. [ f (x +t)

0

Then by Lemma 2.1 σ2n (x; f ) − f (x) =

 −n 2n + 1 2 ˙ − f (x)] dt [ f (x +t) 2 0  2−n+i+2−n n−1  n−i−2 + 2 [ f (x + t) − f (x)] dt. 2−n+ j

i=0



2−n

σ2n (x, f ) − f (x) = 2n

o(t α ) dt +

0

= o(2−αn ) +

n−1 

 2n−i−2

i=0 n−1 

−n+2−n

2−n+i

o(t α ) dt

2n−i−2 o(2α(−n+i+1) )2−n

i=0

= o(2−αn ) + o(2αn )

n−1 

2−(1−α)i

i=0

= o(2−αn ). This completes the proof of the lemma. Proof of Theorem: Let n = 2n 1 + 2n 2 + . . . + 2nr , (n 1 > n 2 > n 3 . . . > n r ≥ 0), n (0) = n, n 1 = n − 2n 1 , n (i) = n (i−1) − 2ni , where i = 2, 3, . . . , r − 1, and n (r ) = 0. Then by Lemma 2.2, we get  σn (x, f ) − f (x) =

1

˙ − f (x)]K n (t) dt [ f (x +t)

0

=

 r 1  ni 1 (i−1) ˙ − f (x)]ψn−n 2 [ f (x +t) (t)K 2ni (t) dt n i=1 0  r 1  (i) 1 (i−1) ˙ − f (x)]ψk−k + n [ f (x +t) (x)D2in (t) dt n i=1 0

= Pn + Q n , say. By Lemma 2.3 and the fact |ψn (x)| = 1, we get

(2.25)

44

2 Walsh–Fourier Series

 r 1  ni 1 ˙ − f (x)|K 2ni (t) dt 2 | f (x +t) n i=1 0  r n   1   1 1 2ni 2−αni = O 2(1−α)i = O n n i=1 i=1

|Pn | ≤

=

(2.26)

1 O(2(1−α)n 1 ) = O(n −α ). n

On the other hand, by the property of D2ni (t) stated in the proof of Lemma 2.3  1   n i  [ f (x +t) ˙ − f (x)]D2ni (t)| dt| ≤ 2  0

2−n

O(t α ) dt = O(2−αni ).

(2.27)

0

By definition n (i) < 2ni . Therefore  r  r     1 1 |Q n | ≤ O n (i) 2−αni = O 2(1−α)ni = O(n α−x ). n n i=1 i=1

(2.28)

By (2.25), (2.26), and (2.27), the theorem is proved. Theorem 2.20 can be generalized in the following manner. (β)

Theorem 2.21 If f (x) ∈ Li p α, then for any β > α, σn (x, f ) − f (x) = O(n −α ), (β) where σn (x, f ) denotes the (C, β) mean of the Walsh–Fourier series of f (x). A function f (x) is said to belong Li p j (t) class if the following condition is satisfied:   f  = sup t>0,x

| f (x + t) − f (x)| f (t)

 < ∞,

where j (t) is a positive and nondecreasing function defined on the interval (0, 1). By taking j (t) = t α , we get Li pα class. This class was first introduced by A. H. Siddiqi and further studied by M. Izumi and S. Izumi, see [1] for references. Theorem 2.22 If f (x) ∈ Li p j (t), then σn (x, f ) − f (x) = O( j (1/n)) where 1 −2 j (u)u du ≤ A j (t)t −1 , as t → 0 and j (t) is subadditive. t Theorem 2.23 If f (x) ∈ Li p j (t) then for 0 < β < 1 σn(β) (x,

  −β f ) − f (x) = O n

2N

j (1/t)t

β−1

 dt ,

0

1 where t j (u)u −2 du ≤ At −1 j (t) as t → 0 and N > n, 2 N −1 < n < 2 N , j (t) is subadditive.

2.3 Approximation by Transforms of Walsh–Fourier Series

45

2.3.2 Approximation by Nörlund Means of Walsh–Fourier Series in L p Spaces This section is devoted to the study of approximation by Nörlund means for Walsh– Fourier series of a function in L p and in particular, in Li p(α, p) over the unit interval (0, 1) where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞, by L p , we mean C W , the collection of uniformly W -continuous functions over (0, 1). The discussion in this section is based on paper of Moricz and Siddiqi [5]. As special cases, we obtain the earlier results by Yano [12, 13], Jastrebova [14] and Skvorcov [10] on the rate of approximation by Cesáro means. We denote by Pn the collection of Walsh polynomials of order less than n, that is, function of the form P(x) =

n−1 

ck wk (x),

k=0

where n ≥ 1 and {ck } is any sequence of real(or complex) numbers. Denote by the finite σ -algebra generated by the collection of dyadic intervals of the form

 m

Im (k) = [k2−m , (k + 1)2−m ), k = 0, 1, . . . , 2m − 1,  where m ≥ 0. It is not difficult to see that the collection of m -measurable functions on I coincides with P2m , m ≥ 0. We will study approximation by means of Walsh polynomials in the norm L p = L p (I ), 1 ≤ p < ∞, and C W = C W (I ). We remind the reader that C W is the collection of functions f : I → R that are uniformly continuous from the dyadic topology of I to the usual topology of R, or in short, uniformly W -continuous. The dyadic topology is generated by the union m for m = 0, 1, . . .. As is known (see, e.g., [2, p. 9]), a function belongs to C W if and only if it is continuous at every dyadic irrational of I , is continuous from right on I , and has a finite limit from the left on (0, 1], all these in the usual topology. Hence, it follows immediately that if the periodic extension of a function f from I to R with period 1 is classically continuous, then f is also uniformly W -continuous on I. The converse statement is not true. For example, the Walsh functions wk belong to C W , but they are not classically continuous for k ≥ 1. For the sake of brevity in notation, we agree to write L ∞ instead of C W and set   f p =

1

1/ p | f (x)| p d x

, 1 ≤ p < ∞,

0

 f ∞ = sup{| f (x)| : x ∈ I }. After these preliminaries, the best approximation of a function f ∈ L p , 1 ≤ p ≤ ∞, by polynomials in Pn is defined by

46

2 Walsh–Fourier Series

E n ( f, L p ) = in f P∈P n  f − P p . Since Pn is a finite-dimensional subspace of L p for any 1 ≤ p ≤ ∞, this infimum is attained. From the results of [2, pp. 142 and 156–158], it follows that L p is the closure of the Walsh polynomials when using the norm ., 1 ≤ p ≤ ∞. In particular, C W is the uniform closure of the Walsh polynomials. Next, define the modulus of continuity in L p , 1 ≤ p ≤ ∞ of a function f ∈ L p by w p ( f, δ) = sup |t| 0, where τt means the dyadic translation of t: ˙ x, t ∈ I. τt f (x) = f (x +t), Finally, for each α > 0, Lipschitz classes in L p are defined by Li p(α, p) = { f ∈ L p : w p ( f, δ) = O(δ α )asδ → 0}. Unlike the classical case, Li p(α, p) is not trivial when α > 1. For example, the function f = w0 + w1 belongs to Li p(α, p) for all α > 0 since w p ( f, δ) = 0 when 0 < δ < 2−1 . Main Results Given a function f ∈ L 1 , its Walsh–Fourier series is defined by ∞ 

 ck wk (x), where ak =

1

f (t)wk (t).

0

k=0

The nth partial sums of series in (2.29) are sn ( f, x) =

n−1 

ck wk (x), n ≥ 1.

k=0

As is well known,

 sn ( f, x) =

1

˙ f (x +t)D n (t) dt,

0

where Dn (t) =

n−1  k=0

wk (t), n ≥ 1,

(2.29)

2.3 Approximation by Transforms of Walsh–Fourier Series

47

is the Walsh–Dirichlet kernel of order n. Let {qk : k ≥ 0} be a sequence of nonnegative numbers. The Nörlund means for series (2.29) are defined by n 1  qn−k sk ( f, x), Q n k=1

tn ( f, x) = where Qn =

n−1 

qk ,

n ≥ 1.

k=0

We always assume that q0 > 0 and lim Q n = ∞.

(2.30)

n→∞

In this case, the summability method generated by {qk } is regular if and only if qn−1 = 0. Qn

lim

n→∞

(2.31)

As to this notion and result, we refer the reader [[2, 4, 5] and references therein]. We note that in the particular case when qk = 1 for all k, these tn ( f, x) are the first arithmetic or (C, 1)-means. More generally, when β



qk = A k =

β +k k



β

for k ≥ 1 and q0 = A0 = 1,

where β = −1, −2, . . . , the tn ( f, x) are the (C, β)-means for series (2.29). The representation  1 ˙ f (x +t)L (2.32) tn ( f, x) = n (t) dt 0

plays a central role in the sequel, where L n (t) =

n 1  qn−k Dk (t), n ≥ 1, Q n k=1

(2.33)

is so-called Nörlund kernel. Theorem 2.24 Let f ∈ L p , 1 ≤ p ≤ ∞, let n = 2m + k, 1 ≤ k ≤ 2m , m ≥ 1, and let {qk : k ≥ 0} be a sequence of nonnegative numbers such that n−1 n γ −1  γ q = O(1) for some 1 < γ ≤ 2. γ Q n k=0 k

(2.34)

48

2 Walsh–Fourier Series

If {qk } is nondecreasing, then m−1 5  j 2 qn−2 j w p ( f, 2− j ) + O{w p ( f, 2−m )}, tn ( f ) − f  p ≤ 2Q n j=0

(2.35)

while if {qk } is nonincreasing, then tn ( f ) − f  p ≤

m−1 5  (Q n−2 j +1 − Q n−2 j+1 +1 )w p ( f, 2− j ) + O{w p ( f, 2−m ))}. 2Q n j=0

(2.36) Clearly, condition (2.34) implies (2.30) and (2.31). We note that if {qk } is nondecreasing, in sign qk ↑, then nqn−1 = O(1) Qn

(2.37)

is a sufficient condition for (2.34). In particular, (2.37) is satisfied if qk  k β or (logk)β for some β > 0. Here and in sequel, qk  rk means that the two sequences {qk } and {rk } have the same order of magnitude; that is, there exist two positive constants C1 and C2 such that C1rk ≤ qk ≤ C2 rk for all k large enough. If {qk } is nonincreasing, in sign qk ↓, then condition (2.34) is satisfied if, for example, (i) qk  k −β for some 0 < β < 1, or (ii) qk  (logk)−β for some 0 < β.

(2.38)

Namely, it is enough to choose 1 < γ < min(2, β −1 ) in case (i), and γ = 2 in case (ii). Theorem 2.25 Let {qk : k ≥ 0} be a sequence of nonnegative numbers such that in case qk ↑ condition (2.37) is satisfied, while in case qk ↓ condition (2.38) is satisfied. If f ∈ Li p(α, p) for some α > 0 and 1 ≤ p ≤ ∞, then ⎧ if 0 < α < 1, ⎨ O(n −α ) tn ( f ) − f  p = O(n −1log n) if α = 1, ⎩ if α > 1. O(n −1 )

(2.39)

Now we make a historical comments. The rate of convergence of (C, β)-means for functions in Li p(α, p) was first studied by Yano [13] in the cases when 0 < α < 1, β > α and 1 ≤ p ≤ ∞, and then by Jastrebova [14] in the case when α = β = 1 and

2.3 Approximation by Transforms of Walsh–Fourier Series

49

p = ∞. Later on, Skvorkov [10] showed that these estimates hold for 0 < β ≤ α as well, and also studied the cases when α = 1, β > 0 and 1 ≤ p ≤ ∞. In their proofs, the above authors rely heavily on the specific properties of the binomial coefficients β Ak . In 1963, Watari [11] proved that a function f ∈ L p belongs to Li p(α, p) for some α > 0 and 1 ≤ p ≤ ∞ if and only if E n ( f, L p ) = O(n −α ). Thus for 0 < α < 1 the rate of approximation to functions f in Li p(α, p) by tn ( f ) is as good as the best approximation. For complete references, we refer to [1, 2, 4–6]. Auxiliary Results Yano proved that the Walsh–Fejér kernel K n (t) =

n n−1  1 k Dk (t) = (1 − )wk (t), n ≥ 1, n k=1 n k=0

is a quasi-positive, and K 2m (t) is even positive. These facts are formulated in the following. Lemma 2.5 Let m ≥ 0 and n ≥ 1; then K 2m (t) ≥ 0 for all t ∈ I , 

1



1

|K n (t)| dt ≤ 2 and

0

K 2m (t) dt = 1.

0

A Sidon-type inequality proved by Schipp and Moricz implies that the Nörlund kernel L n (t) is also quasi-positive. More exactly, C = [O(1)]1/γ , 2γ /(γ − 1) in the next lemma, where O(1) is from (2.34). Lemma 2.6 If condition (2.34) is satisfied, then there exists a constant C such that 

1

|L n (t)| dt ≤ C, n ≥ 1.

0

Now, we give a specific representation of L n (t), interesting in itself. Lemma 2.7 Let n = 2m + k, 1 ≤ k ≤ 2m , and m ≥ 1; then Q n L n (t) = −

m−1  j=0

−

m−1  j=0

r j (t)w2 j −1 (t)

j 2 −1

i(qn−2 j+1 +i − qn−2 j+1 +i+1 )K i (t)

i=1

r j (t)w2 j −1 (t)2 j qn−2 j K 2 j (t)

50

2 Walsh–Fourier Series

+

m−1 

(Q n−2 j +1 − Q n−2 j+1 +1 )D2 j+1 (t)

j=0

+Q k+1 D2m (t) + Q k rm (t)L k (t).

(2.40)

Proof We have Q n L n (t) =

m 2 −1

qn−i Di (t) + qn−2m D2m (t) +

j m−1 −1  2

qn−2 j −i (D2 j +i (t) − D2 j+1 (t))

j=0 i=0

+

qn−i Di (t)

i=2m +1

i=1

=

m 2 +k

m−1 

⎛

⎞

⎝

qn−2 j −i ⎠ D2 j+1 (t)

j=0

j 2 −1

i=0

+qn−2m D2m (t) +

k 

qn−2m −i D2m +i (t).

(2.41)

i=1

As is well known(see,e.g., [2, p. 46]), D2m +i (t) = D2m (t) + rm (t)Di (t), 1 ≤ i ≤ 2m .

(2.42)

Furthermore, by (1.1) of [5], it is not difficult to see that w2 j−1−l (t) = w22 j −1 (t)wl (t), 0 ≤ l < 2 j . Hence, we deduce that D2 j+1 (t) − D2 j +i (t) = r j (t)

j 2 −1

wl (t) = r j (t)

l=i

2 j −i−1

w2 j −1−l (t)

l=0

= r j (t)w2 j −1 (t)D2 j −i (t), 0 ≤ i < 2 j .

(2.43)

Substituting (2.42) and (2.43) into (2.41) yields Q n L n (t) = −

m−1  j=0

+

m−1 

r j (t)w2 j −1 (t)

j 2 −1

qn−2 j +1 D2 j −i (t)

i=0

(Q n−2 j +1 − Q n−2 j+1 +1 )D2 j+1 (t)

j=0

+Q k+1 D2m (t) + Q k rm (t)L k (t).

(2.44)

2.3 Approximation by Transforms of Walsh–Fourier Series

51

Performing a summation by part gives j 2 −1

j 2 −1

qn−2 j −1 (t)D2 j −i (t) =

i=0

i K i (t)(qn−2 j+1 +i − qn−2 j+1 +i+1 ) + 2 j K 2 j (t)qn−2 j .

i=1

Substituting this into (2.44) results in (2.40). Lemma 2.8 If g ∈ P2m , f ∈ L p , where m ≥ 0 and 1 ≤ p ≤ ∞, then for 1 ≤ p < ∞ 1/ p p   ˙ − f (x)] dt  d x rm (t)g(t)[ f (x +t) 0 0  1 −1 −m ≤ 2 w p ( f, 2 ) |g(t)| dt, 

1

   

1

(2.45)

0

while for p = ∞   sup |

1

˙ − f (x)] dt : x ∈ I rm (t)g(t)[ f (x +t)

0 −1

≤ 2 w∞ ( f, 2

−m



1

)



|g(t)| dt.

(2.46)

0

Proof Since g ∈ P2m , it takes a constant value, say gm (k) on each dyadic interval Im (k), where 0 ≤ k < 2m . We observe that if t ∈ Im (k) then t + 2−m−1 ∈ Im (k). We will prove (2.45). By Minkowski’s inequality in the usual and in the generalized form, we obtain that 

1

0

   

1 0

1/ p p  ˙ − f (x)] dt  d x rm (t)g(t)[ f (x +t)

 m −1 p 1/ p   1 2   −m−1 ˙ ˙ ˙ gm (k) [ f (x +t) − f (x +t +2 )] dt  d x   0  Im+1 (2k)

 =

k=0

≤

m 2 −1

|gm (k)|

≤

k=0

≤

m 2 −1

1

 |gm (k)|

 Im+1 (2k)

0

k=0 m 2 −1



1

˙ − f (x +t ˙ +2 ˙ −m−1 )| dt | f (x +t)

˙ − f (x +t ˙ +2 ˙ −m−1 )| p d x | f (x +t)

p

1/ p dx

1/ p dt

0

|gm (k)|2−m−1 w p ( f, 2−m ).

k=0

This is equivalent to (2.45). Inequality to (2.46) can be proved analogously.

52

2 Walsh–Fourier Series

Proofs of Theorem 2.24 We carry out the proof of Theorem 2.21 for 1 ≤ p < ∞. The proof for p = ∞ is similar and even simpler. By (2.32) and (2.40) and the usual Minkowski inequality, we may write that 

1

Q n tn ( f ) − f  p = 0

≤

   

 m−1  

+

0 1

0

j=0

   

m−1 

1/ p p  ˙ − f (x)] dt  d x r j (t)g j (t)[ f (x +t)

1

0

 m−1  

1 0

j=0

+

1/ p p   ˙ Q n L n (t)[ f (x +t) − f (x)] dt  d x

1

   

1

0

1/ p p  ˙ − f (x)] dt  d x r j (t)g j (t)[ f (x +t)

(Q n−2 j +1 − Q n−2 j+1 +1 )

j=0



1

× 0

   

1

0



1/ p p   ˙ − f (x)] dt  d x D2 j+1 (t)[ f (x +t)    

1

+Q k+1 0

 +Q k 0

1

0

 1   

1

0

1/ p p   ˙ D2m (t)[ f (x +t) − f (x)] dt  d x

1/ p p  ˙ − f (x)] dt  d x rm (t)L k (t)[ f (x +t)

A1n + A2n + A3n + A4n + A5n , say, where g j (t) = w2 j −1 (t)

j 2 −1

i(qn−2 j+1 +i − qn−2 j+1 +i+1 )K i (t),

i=1

h j (t) = w2 j −1 (t)2 j qn−2 j qn−2 j K 2 j (t), 0 ≤ j < m. Applying Lemma 2.5, in the case when qk ↑ we get that 

1

|g j (t)| dt ≤

0

j 2 −1

i|qn−2 j+1 +i − qn−2 j+1 +i+1 |

i=1 2  j

= 2(2 qn−2 j − j

i=1

while in the case when qk ↓

qn−2 j+1 +i ) ≤ 2 j+1 qn−2 j ,

(2.47)

2.3 Approximation by Transforms of Walsh–Fourier Series

⎛



1

|g j (t)| dt ≤ 2 ⎝

0

2 

53

⎞

j

qn−2 j+1 +i − 2 j qn−2 j ⎠

i=1

≤ 2(Q n−2 j+1 +i − Q n−2 j+1 +i+1 ). Thus by Lemma 2.8 in the case qk ↑ m−1 

2 j qn−2 j w p ( f, 2− j ),

(2.48)

(Q n−2 j+1 +i − Q n−2 j+1 +i+1 )w p ( f, 2− j ).

(2.49)

A1n ≤

j=0

while in the case qk ↓ A1n ≤

m−1  j=0

By virtue of Lemmas 2.5 and 2.8 again, we again that A2n ≤ 2−1

m−1 

2 j qn−2 j w p ( f − 2− j ).

(2.50)

2 j qn−2 j ≤ Q n−2 j+1 +i − Q n−2 j+1 +i+1 .

(2.51)

j=0

Obviously, in the case qk ↓



Since D2m (t) =

2m if t ∈ [0, 2−m ), 0 if t ∈ [2−m , 1),

(see,e.g., [2, p. 7]), by the generalized Minkowski inequality, we find that A3n ≤

m−1 

(Q n−2 j+1 +i − Q n−2 j+1 +i+1 )

j=0



m−1 

1

D2 j+1 0

≤



1

×

1/ p | f (x + t) − f (x)| p d x

dt

0

(Q n−2 j+1 +i − Q n−2 j+1 +i+1 )w p ( f, 2− j ),

(2.52)

j=0

A4n ≤ Q k+1 w( f, 2−m ). Clearly, in the case qk ↑

(2.53)

54

2 Walsh–Fourier Series

Q n−2 j+1 +i − Q n−2 j+1 +i+1 ≤ 2 j qn−2 j .

(2.54)

Finally, by Lemmas 2.6 and 2.8, in a similar way to the above, we deduce that A5n ≤ 2

−1

Q k w p ( f, 2

−m



1

)

|L k (t)| ≤ C Q n w p ( f, 2−m ).

(2.55)

0

Combining (2.57)–(2.55) yields (2.53) in the case qk ↑ and (2.54) in the case qk ↓. Proof of Theorem 2.25 Case(a).qk ↑. We have n − 2 j ≥ 2m−1 for 0 ≤ j ≤ m − 1. Consequently, for such j  s 2j 2 j qn−2 j (n − 2 j + 1)qn−2 j Q n−2 j +1 ≤ C2 j−m+1 , = Qn Q n−2 j +1 Qn n − 2 j + 1 where C equals O(1) from (2.37). Since f ∈ Li p(α, p), from (2.35), it follows that tn ( f ) − f  p =

m−1 O(1)  j 2 qn−2 j 2− jα + O(2−mα ) Q n j=0

= O(1)2−m

m 

2 j−α

j=0

⎧ ⎨ O(2−mα ) if 0 < α < 1, = O(m2−m ) if α = 1, ⎩ O(2−m ) if α > 1. This is equivalent to (2.39). Case (b).qk ↓. For example, we consider case (i) in (2.38). Then Q n  n 1−β . This time we have n − 2 j+1 ≥ 2m−1 for 0 ≤ j ≤ m − 2. Since f ∈ Li p(α, p), from (2.36) it follows that tn ( f ) − f  p ≤

m−2 5  j 2 qn−2 j +1 w p ( f, 2− j ) 2Q n j=0

5 + w p ( f, 2−m ) + O{w p ( f, 2−m )} 2 m−2 O(1)  j = 2 qn−2 j +1 2− jα + O(2−mα ) Q n j=0

2.3 Approximation by Transforms of Walsh–Fourier Series

55

m−2 O(1)2−mβ  j (1−α) 2 + O(2−mα ) n 1−β j=0 ⎧ −1 m(1−α) ) if 0 < α < 1, ⎨ O(n 2 if α = 1, = O(n −1 m) ⎩ if α > 1. O(n −1 )

=

Clearly, this is equivalent to (2.39). Case (ii) in (2.38) can be proved analogously. Concluding Remarks and Problems (A) We have seen that (2.34) is satisfied when qk = (k + 1)β for some β > −1, and Theorems 2.24 and 2.25 apply. If qk increases faster than a positive power of k, the relation (2.34) is no longer true in general. But the case, for example, when qk grows exponentially is not interesting, since the condition (2.31) of regularity is not satisfied. On the other hand, the case when β = −1 is of special interest. Problem 1. Find substitutes of (2.36) and (2.39) when qk = (k + 1)−1 . In this case, the tn ( f ) are called the logarithmic means for series (2.29). (B) It is also of interest that Theorems 2.21 and 2.22 remain valid when qk  k β φ(k),

(2.56)

where β > −1 and φ(k) is a positive and monotone(nondecreasing or nonincreasing) function in k, slowly varying in the sense that lim

k→∞

φ(2k) = 1. φ(k)

It is not difficult to check that in this case Q n  n 1+β φ(n). (C) Now, we turn to the so-called saturation problem concerning the Nörlund means tn ( f ). We begin with the observation that the rate of approximation by tn ( f ) to functions in Li p(α, p) cannot be improved too much as α increases beyond 1. Indeed, the following is true. Theorem 2.26 If {qk } is a sequence of nonnegative numbers such that lim in f q2m −1 > 0,

m→∞

(2.57)

and if for some f ∈ L p , 1 ≤ p ≤ ∞, t2m ( f ) − f  p = O(Q −1 2m ) as m → ∞,

(2.58)

56

2 Walsh–Fourier Series

then f is constant. We note that condition (2.34) is correctly satisfied if qk ↑ or qk ↓ and lim qk > 0. Proof Since by the definition E 2m ( f, L p ) ≤ t2m ( f ) − f  p , and by a theorem of Watari [1963], Moricz and Siddiqi [5] obtained s2m ( f ) − f  p ≤ 2E 2m ( f, L p ), and it follows from (2.58) that s2m ( f ) − f  p = O(Q −1 2m ) as m → ∞.

(2.59)

A simple computation gives that Q 2m {s2m ( f, x) − t2m ( f, x)} =

m 2 −1

(Q 2m − Q 2m −k )ak wk (x).

k=1

Now (2.58) and (2.59) imply that lim 

m→∞

m 2 −1

(Q 2m − Q 2m −k )ak wk (x) p = 0.

k=1

Since .1 ≤ . p , for any p ≥ 1 it follows that lim

m→∞

|(Q 2m − Q 2m − j )a j |  2m −1    1     = lim  w j (x) (Q 2m − Q 2m −k )ak wk (x) d x  m→∞  0  k=1

≤ lim  m→∞

m 2 −1

(Q 2m − Q 2m −k )ak wk (x)1 = 0.

k=1

Hence, by (2.57), we conclude that a j = 0 for all j ≥ 1. Therefore, f = a0 is constant. In particular case when qk = 1 for all k, the tn ( f ) are the (C, 1)-means for series (2.29) defined by n 1 sk ( f, x), n ≥ 1, σn ( f, x) = n k=1

2.3 Approximation by Transforms of Walsh–Fourier Series

57

and Theorem 2.23 is known (see, e.g., [2, p. 191]). It says that if for some f ∈ L p , 1 ≤ p ≤ ∞, σ2m ( f ) − f  p = O(2−m ) as m → ∞, then f is necessarily constant. Problem 2. How can one characterize those functions f ∈ L p such that σn ( f ) − f  p = O(n −1 )for some 1 ≤ p ≤ ∞?

(2.60)

We conjecture that (2.60) holds if and only if ∞ 

∞ 

2m w p ( f, 2−m ) < ∞, or equivalently

m=0

w p (k −1 ) < ∞.

k=1

The “if” part can be proved in the same manner as in the case when w p ( f, δ) = O(δ α ) for some α > 1 (cf. [2, p. 190]). The proof(or disproof) of the “only if” part is a problem.

2.3.3 Approximation by Nörlund Means in Dyadic Homogeneous Banach Spaces and Hardy Spaces We have introduced Dyadic Homogeneous spaces in Sect. 2.2.3. In the previous section (Sect. 2.3.2), we have discussed the approximation properties of Walsh– Norlund ¨ means in L p spaces. As indicated, this study embraced earlier results in the area by Yano, Jastrebova, and Skvorcov on the rate of approximation by Césaro means. The main objective of this section is to present extension of results of Sect. 2.3.2 which have been studied by Fridli, Manchanda and Siddiqi [4]. We present here approximation by Walsh–Norlund ¨ means in dyadic homogeneous Banach spaces, and to dyadic Hardy spaces H p ; p < 1. Let us list some basic but useful properties of homogeneous Banach spaces. The first one is about the norm of convolution operators. If h ∈ L 1 and f ∈ X , then h ∗ f  X ≤ h1  f  X , where ∗ stands for dyadic convolution (h ∗ f )(x) = The modulus of continuity in X is defined by

(2.61)

1 0

˙ f (t) dt. h(x +t)

w( f, δ, X ) = sup μ 1 the Hardy space H p coincides with the Lebesgue space L p , H 1 is a homogeneous Banach spaces which is a proper subspace of L 1 , and for p < 1 the H p spaces are homogeneous Banach spaces. In several cases, it is useful to have other characterizations of the H p norm. One of them relies on the quadratic variation of martingales. For a martingale f , let the  2 1/2 , where dn f = f n+1 − quadratic variation of f be defined as Q f = ( ∞ n=0 |dn f | ) f n . Then f ∈ H p if and only if Q f ∈ L p (0 < p < ∞), and  f  H p ≈ Q f  L p (see [7]). The concept of modulus of continuity and best approximation in H p can be defined in the same way as in homogeneous Banach spaces. Moreover, by using quadratic variation, it is easy to see that for any f ∈ H p , and n ∈ N we have

2.3 Approximation by Transforms of Walsh–Fourier Series

59

 f − S2n f  H p ≈ Q( f − S2n f ) L p = min Q( f − p) L p ≈ E 2n ( f, H p ). p∈P 2n

(2.64) The Lebesgue measure of a measurable set A will be denoted by |A|, and throughout the paper C will stand for an absolute positive constant not necessarily the same in different occurrences. Results We will Norölund means when the generating sequence (qk ) is nonnegative, and monotone. The Norölund means of the Walsh–Fourier series of a martingale f is defined by n 1  qn−k f (n ∈ N), tn f = Q n k=1  where Q n = n−1 k=0 qk . We may suppose that q0 > 0. This implies Q n > 0, n ∈ N, as well. The following theorem is a generalization of the result of Moricz and Siddiqi in [5] on the approximation properties of Nörlund means in L p spaces Theorem 2.27 Let X be a homogeneous Banach space, f ∈ X , and n, m ∈ N, with 2m ≤ n < 2m+1 . (i) If (qk ) is a monotonically increasing sequence of positive real numbers, then ⎞ ⎛ ⎛ ⎞ m−1 2l+1 −1  5 1 ⎝ ⎝  qn− j ⎠ ω( f, 2−l , X ) + qn− j ω( f, 2−m , X )⎠ . tn f − f  X ≤ 2 Q n l=0 m l j=2 j=2

(ii) If (qk ) is a monotonically decreasing sequence of nonnegative real numbers, then ⎛ tn f − f  X ≤

m−1 

⎛

5 1 ⎝ ⎝ 2 Q n l=0

2l+1 −1 

⎞ qn− j ⎠ ω( f, 2−l , X )+

j=2l

⎞ m (n − 2 )q n− j +C qn− j (1 + log+ n )ω( f, 2−m , X )⎠ . m qn−s m s=2 j=2 n 

We note that the statements of Theorem 2.27 can be stated in several equivalent forms. For instance, as a consequence of (2.63), we receive an equivalent form of the3 above theorem if ω( f, 2−l , X ) is replaced by E 2 l( f, X )(l ∈ N). Another version relies on a property of the modulus itself. However, the dyadic modulus of continuity has properties different from the classical one. It is easy to see that similarly to the classical case the inequality ω( f, 2−l , X ) ≤ 2ω( f, 1/j, X ) (2l ≤ j ≤ 2l+1 ). Then, for example, the right side of statement (i) is equivalent to n 5 1  qn− j ω( f, 1/j, X ). 2 Q n j=1

60

2 Walsh–Fourier Series

Remark 2.6 As it is easy to see that in the proof, in case the Walsh system will be a basis in X the statement in (i) of Theorem 2.27 holds without the monotonicity of (qk ). So the statements of Theorem 2.27 are of interest only if the Walsh system is not a Schauder basis X . Examples of homogenous Banach spaces in which the Walsh system is not a basis include L 1 , H 1 , the dyadic VMO space, the space of dyadically continuous functions, and nonreflexive Orlicz spaces. In our second theorem, we are concerned about the approximation properties of Nörlund means in H p (0 < p < 1) spaces. It shows that, even though these spaces are not homogeneous, the results received are similar to those in Theorem 2.27. Theorem 2.28 Let (qk ) be a sequence of nonnegative real numbers. (i) If (qk ) is monotonically increasing, and 0 < p < 1 then ⎛ tn f − f  H p ≤ C p ⎛ ⎝

m−1 

⎛

1 ⎝ ⎝ Q n l=0

n 

2l+1 −1 

⎞p

j=2l

⎞p qn− j ⎠ E 2l ( f, H p ) + p

⎞1/ p

qn− j ⎠ E 2m ( f, H p )⎠ p

j=2m

( f ∈ H p , n, m ∈ N, 2m ≤ n < 2m+1 ). (ii) Suppose that (qk ) is monotonically decreasing and there exists r > 1 such that n−1 n r −1  r q = O(1) Q rn j=0 n

(n ∈ N).

(2.65)

If p > r/(2r − 1) then tn f − f  H p

⎞1/ p ⎛ ⎛ ⎞p m−1 2l+1 −1 1 ⎝ ⎝  p p ≤ Cp qn− j ⎠ E 2l ( f, H p )⎠ + O(E 2m , H p )) Q n l=0 l j=2

( f ∈ H p , n, m ∈ N, 2m ≤ n < 2m+1 ). Auxiliary results In the proof of (ii) of Theorem 2.27, we will need a Sidon-type inequality. Theorem A [4] Let n ∈ N, and e j ( j = 1, . . . , n) be real numbers. Then    n    n    |c j | +    . cj Dj ≤ C |c j | 1 + log −1 n  n  j=1  s=1 |cs | j=1 1

(2.66)

2.3 Approximation by Transforms of Walsh–Fourier Series

61

A multiplier theorem will be used in the proof of (ii) of Theorem 2.28. It shows that certain Hörmander–Mihlin-type multipliers are bounded on H p . For any real sequence, ϕ = (ϕk ) and martingale Tψ f is given by φk f (k) = ϕk fˆ(k) T

(k ∈ N).

Let the difference sequence Δϕ be defined as Δϕk = ϕk − ϕk+1 (k ≥ 0). Then the following theorem holds. Theorem B [4] Let 1 < r ≤ 2, p > r/(2r − 1), and (ϕ j ) be a real sequence. If (ϕ j ) is bounded and satisfies ⎛ 2l ⎝

2l+1 −1  k=2l

⎞1/r

|ϕ j | ⎠ 2l r

≤C

(l ∈ N),

(2.67)

then the Walsh multiplier operator Tϕ is bounded on H p . Lemma 2.9 Let g ∈ P2m (m ∈ N) and f ∈ X . Then τm g ∗ f  X ≤

1 g1 ω( f, 2−m , X ). 2

Proof Multiplying g by τm results in shifting the spectrum by 2m . Hence, ( τm g)(k) = 0 whenever k ∈ / {2m , . . . , 2m+1 − 1}. Since (τm g ∗ f )(k) = ( τm g)(k) fˆ(k), we have that τm g ∗ f = τm g ∗ (S2m+1 f − S2m f ). Thus it follows from (2.61) that τm g ∗ f  X ≤ g1 S2m+1 f = S2m f  X . By the approximation properties of the S2n f (n ∈ N) partial sum noted in introduction, we conclude 1 τm g ∗ f  X ≤ g1 ω( f, 2−m , X ). 2 In the proofs of our Theorems 2.27 and 2.28, we will apply summation by parts. As a result of it, Dirichlet kernels will be replaced by Fejér kernels. The nth Walsh–Fejér kernel K n is defined as n 1 Dk (n ∈ N). Kn = n j=1 The L p norm (0

1 2

i 

p 2

k

≤ 2p

k=0

p

(X 1 + X 2 ).

l−1 

1 0

p

X i ≤ C p (2l )2 p−1 . By

(22i ) p χ[0,2−i ) .

i=0

implies 

1

0

p

X1 ≤ 2p

l−1  (2i )2 p−1 ≤ C p (2l )2 p−1 . i=0

We show that the same estimate holds for X 2 . Let us start by noting that l−1 

 l−1   m  l−1    i τ2−k D2i = 2 χ[2−k ,2−k +2−l+1 ) + 2i χ[2−k +2−m−1 ,2−k +2−m )

i=k

i=k

m=k

≤ 2l χ[2−k ,2−k +2−l+1 ) +

l−2 

i=k

2m+1 χ[2−k +2−m−1 ,2−k +2−m ) .

m=k

This, in particular, means that the support of the various terms with respect to k in X 2 are pairwise disjoint. Therefore, we have 

1 0

p X2

 l−1 p l−1 l−1 l−1     k p p k p −k i = (2 ) τ2 D2 ≤2 (2 ) (2m ) p−1 . k=0

i=k

k=0

m=k

2.3 Approximation by Transforms of Walsh–Fourier Series

Then

1 2

< p < 1 implies

1 0

63

p

X 2 ≤ C p (2l )2 p−1 .

H p is not homogeneous Banach space for 0 < p < 1, and the norm of convolution operators cannot be estimated by (2.61) within H p . Let Pn be the set of linear combinations of wk s, k = 0, . . . , n − 1. In case of Walsh polynomials, the following lemma will serve as a substitute for (2.61) in H p . Lemma 2.11 Let h, g ∈ P2l (i ∈ N). Then h ∗ g p ≤ (2l )1/ p−1 h p g p . Proof Set xk = k2−l , and t j = j2−l (k, j = 0, . . . , 2l − 1). Since h and g are Walsh polynomial of order less than 2l , they are constants on dyadic intervals of length 2−l . Consequently,  h ∗ g pp =

1 0

   

0

1

p  l l  p 2 2 −1  −1    1  1 h(x + t)g(t)dt  d x = l h(x + t )g(t ) k j j   l 2 k=0  2 k=0 

2 −1   2l −1 1  1 p = l |h(xk + t j )| p |g(t j )| p 2 k=0 2l j=0 l

 =

1 2l

l  p+1 2 −1

j=0

|g(t j )|

p

l −1 2

|h(xk + t j )| p = (2l )1− p g pp h pp .

k=0

Proofs Proof of Theorem 2.27. Let n = 2m + k(n, m ∈ N, 0 ≤ k < 2m ), and f ε X In case (qk ) is increasing we will use the following decomposition: 1  qn− j S j f Q n j=1 ⎞ ⎛ m−1 2l+1 −1 n  1 ⎝  qn− j ( f − S j f ) + qn− j ( f − S j f )⎠ = Q n l=0 m l j=2 j=2 ⎞ ⎛ l+1 m−1 2 −1 n  1 ⎝  = qn− j ( f − S2l f ) + qn− j ( f − S2m f )⎠ (2.68) Q n l=0 m l j=2 j=2 ⎞ ⎛ m−1 2l+1 −1 n  1 ⎝  − qn− j (S j f − S2l f ) + qn− j (S j f − S2m f )⎠ Q n l=0 m l j=2 +1 n

f − tn f = f −

j=2

= A1 + A2 . By (2.63), we have

64

2 Walsh–Fourier Series

⎞ ⎛ m−1 2l+1 −1 n  1 1 ⎝  A1  X ≤ qn− j ω( f, 2−l , X ) + qn− j ω( f, 2−m , X )⎠ . 2 Q n l=0 m l j=2 j=2

Let us consider A2 . We can write S2l + j f − S2l f = (τl D j ) ∗ dl f ( j = 1, . . . , 2l , l ∈ N). Then for the first term in A2 , we have 2l+1 −1 

⎛ qn− j (S j f − S2l f ) = ⎝τl

j=2l +1

l −1 2

⎞ qn−2l − j D j ⎠ ∗ dl f.

(2.69)

j=1

In the same way, we obtain for the second term that m 2 +k

⎛ qn− j (S j f − S2m f ) = ⎝τm

j=2m +1

k 

⎞ qn−2m − j D j ⎠ ∗ dm f.

(2.70)

j=1

Let us use summation by parts to obtain l 2 −1

qn−2l − j D j =

j=1

l 2 −2

jqn −2l − j K j + (2l − 1)qn−2l+1 +1 K 2l −1 .

(2.71)

j=1

Recall q j = q j − q j+1 , and K j is the jth ( j ∈ N) Walsh–Fejér kernel. Since K 1 ≤ 2 (see [2]), we have   ⎛l ⎞ l  2 2 −2   −1  ⎝ qn−2l − j D j  jqn −2l − j K j + (2l − 1)qn−2l+1 +1 ⎠  ≤2    j=1 j=1 1

=2

l 2 −1

qn−2l − j .

j=1

In a similar way, we have   ⎛ ⎞   k−1 k    k    ⎝ ⎠ m m q D ≤ 2 jq + kq qn−2m − j . = 2 n−2 − j j  n−2 − j 0   j=1  j=1 j=1 1

Then, it follows that (2.61) ⎛ ⎞ m−1 2l+1 −1 n  2 ⎝  A X ≤ qn− j dl f  X + qn− j dm f  X ⎠ . Q n l=0 m l j=2 +1 j=2 +1

(2.72)

2.3 Approximation by Transforms of Walsh–Fourier Series

65

Since dl f = S2l+1 ( f − S2l f ), we have by (2.61) and (2.63) that dl f  X ≤ ω ( f, 2−l , X ). Consequently, ⎞ ⎛ m−1 2l+1 −1 n  2 ⎝  A X ≤ qn− j ω( f, 2−l , X ) + qn− j ω( f, 2−l , X )⎠ . Q n l=0 l j=2m +1 j=2 +1

The proof of part (i) of the theorem is completed. In case (qk ) is depressing, we slightly modify the decomposition in (2.68) by using S2l+1 f instead of S2l f . Then, we will have the following three terms: f − tn f 1  qn− j S j f Q n j=1 ⎞ ⎛ m−1 2l+1 −1 n  1 ⎝  qn− j ( f − S2l+1 f )⎠ + qn− j ( f − S2m f ) + (2.73) = Q n l=0 j=2m j=2l ⎞ ⎛ n m−1 2l+1 −1 1 ⎝  1  + qn− j (S2l+1 f − S j f )⎠ − qn− j (S j f − S2m f ) Q n l=0 Q n j=2m l n

= f −

j=2

= B1 + B2 + B3 .

(2.74)

In the same way as for A1 , we have ⎞ ⎛ m−1 2l+1 −1 n  1 1 ⎝  B1  ≤ qn− j ω( f, 2−(l+1) , X ) + qn− j ω( f, 2−(m) , X )⎠ . 2 Q n l=0 m l j=2 j=2

In order to estimate B2  X , let us consider the corresponding kernel 2l+1 −1 

qn− j (D2l+1 − D j f ).

j=2l

By use summation of Walsh functions, it is easy to see that w2n −1 ws = w2n −1−s holds for any 0 ≤ s < 2v , v ∈ N. Hence D2l+1 − D j f = w2l+1 −1 − D2l+1 − j . This implies

66

2 Walsh–Fourier Series



2l+1 −1 

qn− j (S2l+1 f − S j f ) X = 

2l+1 −1 

j=2l

qn− j (w2l+1 −1 − D2l+1 − j ) ∗ dl f 

j=2l 2  l

≤

qn−2l+1 + j D j 1 dl f  X .

j=1

In the last sum, the coefficients qn−2l+1 + j decrease by j. Therefore, similarly to (2.72), we can obtain 2l 2l   qn−2l+1 + j D j 1 ≤ 2 qn−2l+1 + j .  j=1

j=1

Consequently, m−1 2 −1 2   B2  ≤ qn− j ω( f, 2−l , X ). Q n l=0 l l+1

j=2

Recall that n = 2m + k, 0 ≤ k < 2m . Then, for the last term B3 , we can write      n  k     1  1    m m (w q (S f − S f ) = qn−2m − j D j ) ∗ dm B3  X = n− j j 2 2  Qn  Qn    j=2m  j=1 X     k  1   ≤ qn−2m − j D j    dm f  X . Q n  j=1 

   f  

X

1

At the point, we will use the Sidon-type inequality in Theorem A to obtain B3  X = C

  n (n − 2m )qn− j 1  ω( f, 2−m , X ). qn− j 1 + log n Q n j=2m +1 q m s=2 +1 n−s

Summarizing the estimate for B1  X (i = 1, 2, 3), we conclude ⎞ ⎛ ⎛ m−1 2l+1 −1 1 ⎝ 5 ⎝   f − tn  X ≤ qn− j ω( f, 2−l , X )⎠ + Q n 2 l=0 l j=2

⎞   n  (n − 2m )qn− j + C ω( f, 2−m , X )⎠ . qn− j 1 + log n m +1 qn−s m s=2 j=2

Proof of Theorem 2.28 Let us first consider the case when (q j ) is decreasing. Since the maximal function operator is sublinear and p < 1, we have a decomposition similar to (2.68)

2.3 Approximation by Transforms of Walsh–Fourier Series

67

⎞∗ ⎞ p n  1 ⎝⎝ f − qn− j S j f ⎠ ⎠ ≤ A∗1, p + A∗2, p , Q n j=1 ⎛⎛

where A∗1, p

⎛ ⎛ ⎞p ⎞p ⎞ ⎛ m−1 2l+1 −1 n      1 ⎝ ⎝  p p = p qn− j ⎠ ( f − S2l f )∗ ) + ⎝ qn− j ⎠ ( f − S2m f )∗ ) ⎠ , Q n l=0 l j=2m j=2

and ⎛ A∗2, p =

m−1 

⎛⎛

⎞∗ ⎞ p

2l+1 −1 

1 ⎝ ⎝⎝ qn− j (S j f − S2l f )⎠ ⎠ + p Q n l=0 l j=2 +1 ⎛⎛ ⎞∗ ⎞ p ⎞ n  + ⎝⎝ qn− j (S j f − S2m f )⎠ ⎠ ⎠ . j=2m +1

It follows from (2.64) that ⎛



1 0

A∗1, p ≤

m−1 

⎛

1 ⎝ ⎝ p Q n l=0

2l+1 −1 

⎛

⎞p

qn− j ⎠ E 2l ( f, H p ) + ⎝ p

n 

⎞

⎞p

qn− j ⎠ E 2m ( f, H p )⎠ . p

j=2m

j=2l

In connection with A∗2, p , let us first note that    2l+1 −1      ⎝ ⎠ qn− j (S j f − S2l f ) =  qn− j (S j f − S2l f ) .  j=2l +1  j=2l +1 ⎛

2l+1 −1 

⎞∗

Then by (2.69),(2.70), and (2.71), we obtain ⎛ ⎞ l −2 2  1 ⎝τl = p⎝ jqn−2l − j K j + τl (2l − 1)qn−2l+1 +1 K 2l −1 ⎠ ∗ dl n l=0  j=1 ⎛ p⎞ ⎞   k−1    + ⎝τm jqn−2m − j K j + τm kq0 K k ⎠ ∗ dm f  ⎠ .   j=1 ⎛

A∗2, p

m−1 

It follows from (2.64) and Lemmas 2.9, 2.10 that

p   f  + 

68

 0

2 Walsh–Fourier Series 1

A∗2, p

1 ≤ p Qn × 

m−1 

l 2 −2

(2l )1− p

l=0

jqn−2l − j K j + (2l − 1)qn−2l+1 +1 K 2l −1  pp dl f  pp +

j=1

+ (2m )1− p  ⎛

k−1 

⎞ jqn−2m − j K j + τm kq0 K k  pp dl f  pp ⎠

j=1

⎛ ⎞p m−1 2l −2 1 ⎝ ⎝  p ≤ Cp p | jqn−2l − j | + (2l − 1)qn−2l+1 +1 ⎠ E 2l ( f, H p )+ Q n l=0 j=1 ⎛ ⎞ ⎞p k−1  p + ⎝ j| jqn−2m − j | + kq0 ⎠ E 2m ( f, H p )⎠ j=1

= Cp

⎛

m−1 

⎛

2l+1 −1 

1 ⎝ ⎝ p Q n l=0

⎛

⎞p

qn− j ⎠ E 2l ( f, H p ) + ⎝ p

n 

⎞

⎞p

qn− j ⎠ E 2m ( f, H p )⎠ , p

j=2m

j=2l

which is the desired estimate. Finally, let us prove part (ii), that is, when (qk ) is decreasing and condition (2.65) holds for the sequence. Then, similarly to (2.73), we have ⎛ ⎛ l+1 ⎞p m−1 2 −1     p 1 p ⎝ ( f − t n f )∗ ≤ p ⎝ qn− j ⎠ ( f − S2l+1 f )∗ Q n l=0 j=2l ⎛ ⎞p ⎞ n    p + ⎝ qn− j ⎠ ( f − S2m f )∗ ⎠ j=2m

+

1 p Qn

m−1 

2l+1 −1 

⎝

l=0

⎛

+

⎛

n 

j=2l

⎞p qn− j ⎠

(2.75)

 p (S2l+1 f − S j f )∗ +

⎞p

 p 1 ⎝ qn− j ⎠ (S j f − S2m f )∗ p Q n j=2m

∗ ∗ ∗ = B1, p + B2, p + B3, p .

(2.76)

Following the reasoning used for Ai ∗ and for Bi (i = 1, 2) in the proof part (ii) of Theorem (2.27), we receive that the proper estimates for B1∗ and B2∗ . Let us ∗ start the case of B3, p by noting the spectrum nj=2m qn− j (S j f − S2m f ) is part of {2m , . . . , n − 1}. Recall that n = 2m + k. Set

2.3 Approximation by Transforms of Walsh–Fourier Series

 ϕs =

1 Qn

69

k− j−1

qi , if s = 2m+1 ± j, j = 0, . . . , k − 1; otherwise.

i=0

0,

Then, it is easy to see that n k−1   qn−1 τm+1 (S j f − S2m f ) = Qn j=2m j=0



k− j−1 1  qi Q n i=0

 fˆ(2m + j)w2m+1 + j

= Tϕ (τm+1 (dm f )). Hence



1

0

∗ B3, p =



1 0

  Tϕ (τm+1 (dm f )) p = Tϕ (τm+1 (dm f )) p p . H

We will show that condition (2.67) in Theorem B holds for ϕ. Since |uϕ | =

 qk−i−1 Qn

0,

, if u = 2m+1 ± i, i = 0, . . . , k − 1; otherwise,

we have that there are only two dyadic blocks for which the left side of (2.67) is not 0. For those two blocks, we obtain ⎛ 2m+1 ⎝

j=2m+2 −1



j=2m+1

⎞1/r ⎛ ⎞1/r n−1 k−1 r  q | j ϕ|r ⎠ n 1−1/r  r j ⎠ m+1 1−1/r ⎝ = (2 ) ≤ 2 q , 2m+1 Q rn Q n j=0 j j=0 (2.77)

and ⎛ 2m ⎝

j=2m+1 −1



j=2m

⎞1/r ⎛ ⎞1/r n−1 k−1 r 1−1/r   q | j ϕ|r ⎠ j m 1−1/r ⎝ ⎠ ≤n = (2 ) q rj . r 2m Q Q n n j=0 j=0

(2.78)

Then (2.78), (2.77), and our assumption (2.65) on the q j s implies that condition (2.67) holds for ϕ (We note that in (2.65) we may suppose 1 < r ≤ 2). Consequently, Tϕ (τm+1 (dm f )) H p ≤ C(τm+1 (dm f )) H p = Cdm f  p ≤ C f − S2m f  H p ≤ C p E 2m ( f, H p ). This means that

 0

This proves Theorem 2.28(ii).

1

∗ p B3, p ≤ C p E 2m ( f, H ). p

70

2 Walsh–Fourier Series

2.4 Applications to Signal and Image Processing A signal can be treated as a function of one variable while an image can be represented by a function of two variables. A digital signal is nothing but a discrete function or a sequence, and a digital image can be represented by a matrix. The problem of digital signal processing is in general deals with the representation of signals by sequences of numbers or symbols, and the processing by a digital computer of these numbers. The goal of such processing is to find certain characteristic parameters of the signal, or it may be to obtain a simpler form of handling the signal as in the case of digital transmission. Walsh–Fourier series are ideally suited for this purpose. Communication, storing, improving, or refining images are challenging problems. Applications of Walsh–Fourier analysis may provide superior results in many situations.

2.4.1 Image Representation and Transmission Let f (t) be a periodic signal with period 1. f (t) can be written as f (t) =  ∞ n=0 cn wn (t), where  1 f (t)wn (t) dt cn = 0

By Parseval’s formula for Walsh analysis, we have 

1

f 2 (t) dt =

0

∞ 

cn2 .

n=0

In view of these results, the signal energy is distributed in the Walsh domain over the Walsh coefficients. Therefore, instead of transmitting the signal in the spatial domain (for example, time domain), we may encode it by its Walsh–Fourier coefficients and transmit these coefficients instead. In case of rapid convergence of the Walsh series, many higher order coefficients are of low magnitude and become possible to transmit fewer coefficients without sacrificing a big loss in the total signal energy. A data reduction in Walsh coefficients entails bandwidth reduction in the sequency Walsh domain. Let us consider images f (x, y) and g(x, y) in the spatial domain and a Walsh function and a Walsh function of D = {(x, y)/0 ≤ x < P, 0 ≤ y < ∞} two variables. In the Walsh or sequency domain, a representation of the image made possible by the application of a double Walsh transform, that is,  F(σ, α) = 0

∞



∞ 0

f (x, y)wσ (x)wα (y) d x d y.

(2.79)

2.4 Applications to Signal and Image Processing

71

By inverse Walsh transform, we have 



∞

f (x, y) = 0

∞

F(σ, α)wσ (x)wα (y) dσ dα.

(2.80)

0

Analogue of Parseval’s formula for two-dimensional case is  0

∞



∞

 (| f (x, y)|)2 d x d y =

0

∞



0

∞

(|F(σ, α)|2 ) dσ dα.

(2.81)

0

This indicates conservation of energy under double Walsh transform. Let f (x, y) be an image defined over a finite domain R = {(x, y)/0 ≤ x < a, 0 ≤ y < b} and   [ f (x, y)]2 d x d y < ∞.

(2.82)

R

It is possible to expand f (x, y) in the form of double Walsh–Fourier series f (x, y) =

∞  ∞ 

amn wm (x/a)wn (y/b),

(2.83)

f (x, y)wm (x/a)wn (y/b) d x d y.

(2.84)

m=0 n=0

where x, y ∈ R and amn =

1 ab



a 0



b

0

Thus, it is possible to encode a continuous image by its corresponding Walsh–Fourier coefficients. The application of digital image transmission is more profitable than analogue or analytic methods. There are three main problems associated with image representation: (a) Image digitization and coding, (b) Image enhancement and restoration, and (c) Image segmentation and description. Image Digitization and Coding This comprise conversion of continuous image (Continuous function of two variables) into a discrete form, using double Walsh– Fourier series. Coding the digitized picture is a succeeding scheme which aims at efficient and a more simplified representation of the digitized data. Digitization often leads to data compression. Once the image has been sampled and quantized within a tolerable quantized error, the result is a discrete image made of say N × N = N 2 samples. The sampling theorem is a very fundamental concept in signal processing with many applications. It is a notion linking continuous and discrete signals. We present

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2 Walsh–Fourier Series

below a Dyadic Sampling Theorem without proof, for details, we refer to Butzer and Splettstöber in [2]. Walsh–Fourier Analysis x ∈ R+ = [0, ∞), x=

∞ 

xi 2−i , xi ∈ {0, 1}, N (x) ∈ Z = {0, ±1, ±2, . . .}

i=−N (x)

D+ = {x ∈ R+ | x = p/2q , p ∈ P = {0, 1, 2, . . .}, q ∈ Z } dyadiic rational numbers.

If x ∈ D+ , select finite expansion. x∈ / D+ , unique representation. ψ(y, x), x, y ∈ R+ -generalized Walsh functions, y-sequency. r J (x, r ) = 0 wx,s ds-Walsh–Dirichlet kernel. r = 2n ,  n  2 0 ≤ x < 2−n , 1 x ∈ [2−n s, 2−n (s + 1)), n n J (x, 2 ) = , J (1, 2 x ⊕ s) = 0 other wise 0 other wise.

Theorem 2.29 Dyadic Sampling Theorem If f and its Walsh–Fourier spectrum S j (w) belong to L 1 (R+ ), d , f is continuous on R+ f is continuous from right on D+ . S f (w) = 0 for w ≥ 2n , n ∈ Z , Z—the set of integers, i.e., f is sequency limited, then f (x) =

∞ 

f (s/2n )J (1, 2n x ⊕ s), x ∈ R+ .

(2.85)

s=0

(b) Image Enhancement and Restoration This process deals essentially with improvement in blurred and noisy images. Our main objective is to minimize the expression   [ f (x, y) − f γ (x, y)]2 d x d y,

(2.86)

R

where f (x, y) and f γ (x, y) denote, respectively, the original and reconstructed images. These functions can be represented by Walsh–Fourier series. (c) Image Segmentation and Description Image segmentation is devoted to division of given image into segments for measurements and studies on a particular segment, or classification or the description of the image in terms of these segments or parts. Different series with orthonormal systems have used to study this theme. However, Walsh–Fourier series gives better results in many cases. Figure 2.1 presents a block diagram of a basic transform image transmission system.

2.4 Applications to Signal and Image Processing

73

Fig. 2.1 Diagram of a basic image transmission system

In Fig. 2.1, the original is first sampled to get a two-dimensional array of elements, also called pixel or pels. This is followed by two-dimensional Walsh transform. Thresholding the transformed image data results in the rejection of those transform coefficients that fall below the threshold. The remaining Walsh transform components are then quantized and coded. At the receiver, an inverse process takes place. The received data is first decoded and then inverse transformed to give two-dimensional data from which an image is reconstructed. The digital image is usually represented by an m × n matrix. Elements of such an image are called pixels, pels, or simply points. The digitized image can be represented by a two-dimensional discrete Walsh transform, that is, an image point denoted by g(n, m) yields discrete Walsh transform G(x, s) =

N −1  N −1 

g(n, m)wn (r )wm (s).

(2.87)

n=0 m=0

The inverse two-dimensional Walsh is given by g(n, m) =

N −1 N −1 1  q(r, s)wn (r )wm (s). N 2 r =0 s=0

Conservation of energy between the spatial and sequency domain exists via Parsevaltype formula N −1 N −1 N −1  N −1  1  2 2 g (n, m) = 2 G (r, s). (2.88) N n=0 m=0 n=0 m=0

2.4.2 Data Compression Data compression is devoted to reduction of data size according to appropriate criteria. In transform image transmission, data compression is applied to the transform points. A decision criterion must be applied for discarding certain transformed points. A popular criterion is thresholding. Hence, the transmitted data are specified by

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2 Walsh–Fourier Series

 G (n, m) =

f (n, m) | f (n, m)| ≥ threshold 0 | f (n, m)| < threshold.

According to this thresholding scheme, the normalized mean-squared error arising in the reconstruction of the image from the compressed data is given by r eemn =

  2 n m [ f (n, m) − G (n, m)]   2 . n m f (n, m)

(2.89)

2.4.3 Quantization of Walsh Coefficients The quantized image is operated on by a coder which converts quantized transformed image samples into appropriate digital code words, one code word for each transformed point; for more details, we refer to [8].

2.4.4 Signal Processing Signal process deals with the representation of signals by sequence of numbers or symbols and processing of these numbers. The goal may be to estimate characteristic parameters of the signal, or it may be to yield a simpler form of treating a signal. Signal processing via Walsh–Fourier series have been used to signals like speech and electrocardiogram (ECG). For details, we refer to [8] and references therein.

2.4.5 ECG Analysis An electrocardiogram (ECG) heart wave represents a graphic recording of the electrical activity associated with the heartbeat. By applying electrodes to various positions on the human body and connecting those electrodes to an electrocardiographic apparatus (monitor or recorder), an ECG signal is obtained. By properly sampling an ECG graph, a discrete signal is obtained. Let f T = [ f 0 , f (1), . . . , f N −1 ] be such a signal vector, where T denotes the vector or matrix transposition. Using Walsh function system {wn (x)}, we have F(r ) = Fr =

N −1 

f n wr (n), r = 0, 1, 2, . . . , N − 1.

n=0

By inverse discrete Walsh transform, we obtain

(2.90)

2.4 Applications to Signal and Image Processing

f (n) =

N −1 1 F(r )wr (n), n = 0, 1, 2, 3, . . . n r =0

75

(2.91)

For detailed ECG analysis using Walsh analysis, we cite [8, pp. 227–230]. It has been noted that there are several advantages of using Walsh transform.

2.4.6 EEG Analysis Brain is an extremely complex system. The cerebral cortex of the human brain contains roughly 15−33 billion neurons, perhaps more, depending on gender and age, linked with up to 10,000 synaptic connections each. Each cubic millimeter of cerebral cortex contains roughly one billion synapses. These neurons communicate with one another by means of long protoplasmic fibers called axons, which carry trains of signal pulses called action potentials to distant parts of the brain or body and target them to specific recipient cells. Methods of observation such as EEG recording and functional brain imaging tell us that brain operations are highly organized, while single-unit recording can resolve the activity of single neurons, but how individual cells give rise to complex operations is unknown. Electroencephalography (EEG) is the recording of electrical activity along the scalp produced by the firing of neurons within the brain. In clinical contexts, EEG refers to the recording of the brain’s spontaneous electrical activity over a short period of time, usually 20–40 mins, as recorded from multiple electrodes placed on the scalp. In neurology, the main diagnostic application of EEG is in the case of epilepsy, as epileptic activity can create clear abnormalities on a standard EEG study. A secondary clinical use of EEG is in the diagnosis of coma, encephalopathies, and brain death. EEG used to be a first-line method for the diagnosis of tumors, stroke, and other focal brain disorders, but this use has decreased with the advent of anatomical imaging techniques such as MRI and CT. A routine clinical EEG recording typically lasts 20–30 mins (plus preparation time) and usually involves recording from scalp electrodes. Routine EEG is typically used in the following clinical circumstances: to distinguish epileptic seizures from other types of spells, such as psychogenic seizures, syncope (fainting), sub-cortical movement disorders, and migraine variants; to differentiate “organic” encephalopathy or delirium from primary psychiatric syndromes such as catatonia; to serve as an adjunct test of brain death; to prognosticate, in certain instances, in patients with coma; and to determine whether to wean anti-epileptic medications. Both effects are independent and additive.

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2 Walsh–Fourier Series

Walsh–Fourier methods applied in Sect. 2.4.5 could be applied for EEG analysis. For an interesting application of Walsh–Fourier analysis to EEG, we cite references 62–65 in [8].

2.4.7 Speech Processing Man communicates through speech, and the theme of transmission of speech from a transmitter to a listener has been investigated using orthonormal systems including Walsh systems [8]. The use of Walsh functions to serve as a basis for the transform processing aspect yields several advantages, for example, bandwidth reduction for transmitted speech, efficient speech synthesis, and automatic speech or word recognition. For applying the Walsh transform, the number of speech in a window is often of the form N = 2n , for n positive integer. This yields fast computational algorithms. If a signal is sampled at a frequency fs, then considering N sampled values for a sampling window, the sampling interval is T = 1fs , and the analysis window has a time duration of N T . The frequency resolution of this window is given by N1T . Out of the N block of transformed samples, we may select the most significant M Walsh components of speech synthesis. The fundamental theorem of speech synthesis is to maximize the efficiency of voice transmission while preserving reasonable distortion levels for voice characteristics . Now let f (t) be a speech signal. Sampling the signal properly, and choosing a block of N samples, we got a set of sampled values, { f (nT )/n = 0, 1, 2, . . . , N − 1}, where T signifies the sampling interval. The discrete Walsh transform of this block of speech samples can be specified by F(k) =

N −1 

f (n)wn (k).

(2.92)

n=0

As in the image transmission problem, one can encode the speech by its Walsh transform coefficients and transmit these coefficients instead of the original speech samples.

2.4.8 Pattern Recognition Pattern space P comprises a large amount of data (digital or analogue signals). One may be interested in finding special features and the pattern and combining correlated data (redundancy) to obtain bandwidth reduction. The resulting data gives a feature space F. Significant information of F is used to form a classification space C, in which the original pattern will have been properly indicated. The main objective of applying discrete Walsh transform in pattern recognition is to reduce computation time for the transformed data from the pattern space P. A more general approach

2.5 Exercises

77

in pattern recognition is to carry out a discrete Walsh transform of the pattern and its model, and to cross-correlate the two transformed sets of values to determine the degree of recognition, rather than to attempt cross-correlation of the original signal. Use of discrete Walsh Transform is advantageous for computation. For a detailed discussion on pattern recognition, we cite Kamath [9].

2.5 Exercises   2.1 Let f be continuous on [0,1] and fˆ(k) = o k(log1 k)α as k → ∞ for some α > 1, then show that f is constant. Is this result true for α = 1? 2.2 How can one characterize those functions f ∈ L p such that  σn ( f ) − f  p = o(n −1 ) for some 1 ≤ p ≤ ∞? 2.3 Let f ∈ L p , and that sup

n −1 2

w(2) ( f, I (k, n)) < ∞.

n∈N k=0

2.4 2.5 2.6 2.7

Prove that fˆ(k) = o( k1 ) as k → ∞. Prove that Sn ( f ) → f uniformly as n → ∞ for f ∈ C W which is of bounded variation. Let f (x) = χ[0, 13 ] (x). Show that Sn f ( 13 ) not converge as n → ∞. Let f ∈ C W . Then show that σn f → f uniformly as n → ∞, where σn f is arithmetic mean of Walsh–Fourier series of f . Write an essay on application of Walsh analysis to ECG and EEG analyses.

References 1. 2. 3. 4.

Siddiqi, A. H. (1978). Walsh function. Aligarh: AMU. Schipp, F., Wade, W.R., & Simon, P. (1990) Walsh series. Bristol: Adam Hilger Siddiqi, A. H. (2018). Functional analysis with applications. Berlin: Springer Nature. Fridli, S., Manchanda, P., & Siddiqi, A. H. (2008). Approximation by Walsh Norlund means. Acta Scientiarum Mathematicarum (Szeged), 74, 593–608. 5. Moricz, F., & Siddiqi, A. H. (1992). Approximation by Norlund means of Walsh-Fourier. Journal of Approximation Theory, 70(3–6), 375–389. 6. Golubov, B. I., Efimov, A. V., & Skvortsov, V. A. (2008). Walsh series and transforms (English Translation of 1st ed.). Moscow: Urss; Dordrecht: Kluwer (1991). 7. Malozemov, V. N., & Masharskii, S. M. (2002). Generalized wavelet bases related to the Discrete Vilenkin-Chrestenson transform. St. Petersburg Mathematical Journal, 13, 75–106.

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8. Maqusi, M. (1981). Applied Walsh analysis. London: Heyden. 9. Kamath, C. (2009). Scientific data mining: A practical perspective. PhiladelPhia: SIAM. 10. Skvorcov, V. A. (1981). Certain estimates of approximation of function by Cesaro means of Walsh-Fourier series. Matematicheskie Zametki, 539–547. [Russian] 11. Watari, Sh. (1963). Best approximation by Walsh polynomials. Tôhoku Mathematical Journal, 15, 1–5. 12. Yano, Sh. (1951). On Walsh series. Tohoku Mathematical Journal, 3, 223–242. 13. Yano, Sh. (1951). On approximation by Walsh function. Proceedings of the American Mathematical Society, 2, 962–967. 14. Jastrebova, M.A. (1968). On the approximation of function satisfying a Lipshitz condition by the arithmetic means of their Walsh Fourier Series. American Mathematical Society Translations: Series 2, 77, 149–162.

Chapter 3

Haar–Fourier Analysis

3.1 Haar System and Its Generalization It is well known that the history of Walsh series began with Haar’s(Hungarian Mathematician Alfred Haar) dissertation of 1909 Zur Theories Orthogonal Function system in which Haar system was introduced. Supervisor of Haar, David Hilbert at Göttingen university asked him to find an orthonormal system on the interval whose Fourier series of continuous functions converges uniformly. He constructed the system which is under discussion in this chapter, now known as Haar system provided answer to the problem posed by Hilbert. Haar system {h k }∞ k=0 is defined as h 0 (x) = 1, h 1 (x) = χ[0,1/2) (x) − χ(1/2,1] (x)

(3.1)

In general writing k1 = 2n + k, 0 ≤ k < 2n where n is the largest power of 2 which is less than or equal to k, one defines 2 1 < x < 2k − n+1 2n+1 2 1 2k = −2n/2 , for 2k − n+1 < x < n+1 2 2 = (2n /4)1/2 , for x = k/2n , = 0, otherwise.

h k (x) = h kn (x) = 2n/2 , for 2k −

The Haar orthonormal sequences of functions labeled by two indices are designated by 2k−1 1 1 1 (x), . . . , } {h m n (x)} = {h 0 (x), h 1 (x), h 2 (x), . . . , h k (x), . . . , h k ,. . . are given by where h 0 (x), h 11 (x), . . ., h 2k−1 k h 0 (x) = 1, 0 ≤ x ≤ 1. © Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2_3

79

80

3 Haar–Fourier Analysis

 h 11 (x) =

1, 0 ≤ x < 1/2 −1, 1/2 < x ≤ 1

⎧√ 2, 0 ≤ x < 1/4 ⎨ √ h 12 (x) = − 2, 1/4 < x < 1/2 ⎩ 0, 1/2 < x ≤ 1

h 22 (x) =

⎧ 0, ⎨√

0 ≤ x < 1/2 2 1/2 < x < 3/4 ⎩ √ − 2, 3/4 < x ≤ 1.

(3.2)

and in general

hm n (x) =

⎧√ 2n−1 , ⎪ ⎪ √ ⎪ ⎪ ⎪ − 2n−1 , ⎪ ⎨ ⎪ 0, ⎪ ⎪ ⎪ ⎪ 0, ⎪ ⎩

m−1 ≤x 2n−1 m−1/2 < 2n−1

< m−1/2 2n−1 m x < 2n−1

... 0 < x < m−1 2n−1 m a f (a, b) = [ f (b) − f (a)]/(b − a).

(3.13)

From (3.12), it is clear that anm is proportional to the difference of two adjacent steps of Sn (x), namely the steps on either side of x = (2m − 1)/2n . This mean-value property of the coefficient {anm } is responsible for the modest computational requirements of Haar transforms. Relations Between Haar and Walsh Functions Relations between Haar and Walsh functions may be developed by expanding one set in terms of the other, and vice versa. Kremer, see for Example [1] developed a convenient method for establishing conversion relations between Haar and Walsh functions. This method utilizes the block orthogonal functions as an intermediate step. Block orthogonal function may be defined on a unit interval by  b2n (r, x) =

1 2rn ≤ x < r2+1 n 0 otherwise

(3.14)

where r = 0, 1, 2, . . . and n = 0, 1, . . . , 2r − 1. In terms of Walsh functions, these block functions can be expressed by b1 (0, x) = wal(0, x) = 1.

(3.15)

2 −1 1  b2n (r, x) = n wr (k/2n )wk (x). 2 k=0

(3.16)

n

Now the Haar functions may be expressed in terms of block functions by h 0 (x) = b1 (0, x)

(3.17)

84

3 Haar–Fourier Analysis

and hm n (x) =

√ 2n−1 [b2n (m, x) − b2n (m + 1, x)].

(3.18)

Comparison of (3.15) and (3.17), and (3.16) and (3.18), respectively, gives φ0 (x) = w0 (x) = 1 and

2  1 = √ [wm (k/2n ) − wm+1 (k/2n )]wk (x). 2 2n−1 k=0

(3.19)

n−1

φnm (x)

(3.20)

Relation (3.20) express Haar functions in terms of Walsh functions. The converse may be established in a similar way. Discrete Haar Transform As in the case the discrete Walsh transform, a discrete Haar transform (DHT) may be developed for the representation of discrete signals. Thus for a discrete signal sequence { f (i) : i = 0, 1, 2, . . . , N − 1} a corresponding discrete Haar transform sequence may be defined by N −1  2  i

F(n) =

f (i)φnm (i), n = 0, 1, . . . N − 1.

(3.21)

i=0 m=1

where φnm (i) denotes the nth value of the Haar function φnm (x) in the i-th subinterval. In this case, the unit interval is divided into N subintervals of length 1/N each. The inverse discrete transform is given by N −1 2 1  F(n)φnm (i), i = 0, 1, . . . , N − 1. f (i) = N n=0 m=1 n

(3.22)

The sequence of functions h m n (i) : i, n = 0, 1, . . . , N − 1 obviously defines a set of discrete Haar functions. If the values of these discrete functions are arranged properly in a matrix, we refer to this matrix as a Haar matrix. This matrix constitutes the kernel of the discrete Haar transformations. In matrix form, (3.21) and (3.22) can then be written as

and

F = Hf

(3.23)

f = H −1 F,

(3.24)

where the matrices H and H −1 denote direct and inverse Haar matrices of order N .

3.2 Haar Fourier Series

85

Extension of the one-dimensional discrete Haar transform to a two-dimensional case is rather straightforward. Assume a two-dimensional discrete signal specified by { f (i, j) : i, j = 0, 1, 2, . . . , N − 1}. Its discrete Haar transform components are then specified by N −1  2 N −1  2   n

F(n, r ) =

r

m2 1 f (i, j)h m n (i)h r ( j), n, r = 0, 1, 2, . . . , N − 1.

i=0 j=0 m 1 =1 m 2 =1

(3.25) N −1  N −1  2n  2r  1 m2 1 f (i, j) = 2 f (i, j)h m n (i)h r ( j), n, r = 0, 1, 2, . . . , N − 1. N n=0 r =0 m =1 m =1 1 2 (3.26) Applications of Haar Functions. To a large extent the utility of Walsh functions is based on two primary aspects. The first concerns the ease with which these functions may be generated on digital computers. The second aspect lies in the simplicity of implementing fast computational techniques based on these functions. Both features rely on the fact that a Walsh function attains a constant value of either +1 and −1 on each subinterval of length, say 1/2n , where the unit interval of operation is divided into 2n equal subintervals. Ignoring scaling constants, a Haar function assumes one of three possible values on each such subinterval. The three values are +1, −1 and 0. Thus, generation of Haar functions, their binary representations, and other involved operations are not likely to be as convenient as with Walsh functions. However, certain other features of Haar functions make them attractive for certain applications. For instance, the convergence aspect of a Haar series expansion provides desirable approximations, particularly in some cases of discontinuous signals. Another feature is noted if we consider the Haar expansion of function f (x). The resulting first two Haar coefficients are effected by all values of f (x). But the rest of the coefficients are affected by only small sections of f (x). Thus the first two Haar functions help to represent f (x) globally, and are thus called global functions. The remaining Haar functions describe f (x) on local sections, and are thus called local functions. In this sense, Walsh functions are all global functions. The Haar function feature of localization may be very desirable if interest is in describing certain local portions of a s signal as in television images where the center of a picture is the significant part. Practical applications of Haar functions and discrete Haar transforms have been made in a number of other areas. In data coding, for instance, a signal may be encoded by its Haar coefficients. Information transmission is then based on these coefficients instead of the original time domain signal. In this case, the rapid convergence aspect may be put to advantage in achieving good bandwidth reduction. In addition, certain immunity to channel errors results. Multiplexing with Haar carries is another area of application. In general, Haar multiplexing provides an operation with features between time division and sequency division multiplexing. Cross-talk problems may be less severe, and signal recovery at the demultiplexing and may be performed rather easily and accurately.

86

3 Haar–Fourier Analysis

Further application is made in some areas of pattern recognition. The property of a Haar coefficient being easily related to the mean value of the function is made useful in studying certain aspects of pattern recognition such as edge detection. Applications of discrete Haar transforms have been made in digital signal processing of one-dimensional and two-dimensional data. For instance, Haar transforms are found useful in the image transmission of slowly varying images, or images which are rather constant over large areas.

3.3 Haar System as Basis in Function Spaces In this section we introduce C W , L p , 1 < p < ∞, H p , BMO and VMO spaces indicating that Haar system is basis in these spaces. We will prove that Haar system is a complete orthonormal system of L 2 [0, 1]. Let C W represent the set of functions g : [0, 1) → R which are continuous at every dyadic irrational, continuous from the right on [0, 1) and have a finite limit from the left on (0, 1], all this in the usual topology. Let L p denote the set of all almost everywhere finite, Lebesgue measurable functions from [0, 1) into [−∞, ∞] 1 such that ( 0 | f | p d x)1/ p < ∞. 1  f  p = ( 0 | f | p d x)1/ p is a norm and so L p , 1 ≤ p < ∞, is a Banach space. Dyadic Hardy spaces H p , 1 < p < ∞, BMO and VMO are discussed in detail in [2] and these spaces are special cases of Homogeneous Banach spaces discussed in [2, 3], see also Chap. 2. Let (X, .) be a Banach space. {h n } is called a (Schauder) basis in X if to each x ∈ X there exists a unique sequence {xk } such that x=

∞ 

xk h k

(3.27)

k=0

in X , that is, the sequence of partial sums sn (x) =

n−1 

xk h k

k=0

converges to x in the norm of X as n → ∞. Remark 3.1 (i) The Haar system {h n } is a basis in L 1 . Infact, it is a basis in L p for 1 < p < ∞. (ii) The Haar system {h n } is a basis in the dyadic Hardy space H p . (iii) The Haar system {h n } is a basis in C W . (iv) The shifted Haar system {h n } is a basis in VMO. (v) The famous problem posed by Stefan Banach whether every separable Banach space has a basis is known as the “Basis problem”. It has been shown [2,

3.3 Haar System as Basis in Function Spaces

87

Sect. 5.6] that answer is no for the dyadic Hardy spaces as there is a dyadic Hardy space(separable) having no basis. Remark 3.2 (i) Let

⎧ ⎨ 1 0 ≤ t < 1/2, ψ(t) = −1 1/2 ≤ t < 1, ⎩ 0, otherwise

Haar mother wavelet and  φ(t) =

0≤t = 0 and < g, χ[0,1] >= 0 for all (i, j) ∈ A. We show that g = 0 a.e. Let I lj,k = [2− j , 2− j k + 2− j−1 ], I rj,k = [2− j k + 2− j−1 , 2− j (k + 1)] then ψi, j = 2− j (χ Ii,l j − χ Ii,r j ), so

l Ii, j

ψ=

r Ii, j

ψ for all (i, j) ∈ A.

88

3 Haar–Fourier Analysis

 [0,1]

ψ d x = 0.

Hence ψ = 0 a.e. Alternative Approach. To show that every continuous function on [0, 1] is a uniform limit of linear combination of Haar functions. Suppose f is continuous, n is a positive integer and let  g= < f, ψ j,k > ψ j,k . j≤n,k

By construction of piecewise constant function: it is constant on subintervals of the form [2−n−1 k, 2−n−1 (k + 1)), k = 0, 1, . . . , 2n+1 . We claim that on each subinterval, the value of g is equal to the average of f on that subinterval: this implies uniform convergence. Thanks to the uniform continuity of f . The proof is by induction. The base case is 

1



1

ψ=

0

 < f, χ[0,1] > χ[0,1] =

0

1

f. 0

which holds because all functions except χ[0,1] have zero mean on [0, 1]. Once it has been established that on some dyadic interval I j,k the averages of f and g are equal, consider its halves I− and I+ : then 



g = 2− j/2

g− I+



I−

1

ψ j,k g = 2− j/2



0

1





ψ j,k f =

f − I+

0

f I−

which together with 





g+ I+

imply that

I+

g=

I+



g= I−

f and

g= I

I−



g=

f = I

I−

 f + I+

f. I−

f.

3.4 Non-uniform Haar Wavelets In the first place we discuss uniform Haar wavelets or classical wavelets. Let I j,k = [2− j k, 2− j (k + 1)) be a dyadic interval. A dyadic step function with scaling j is a function which is constant on each interval I j,k (with j fixed). Haar scaling function of order j are given by φ j,k (x) = 2 j/2 φ(2 j x − k), where

3.4 Non-uniform Haar Wavelets

89

 φ(x) =

10≤x φ j,k , where φ j,k are the Haar scaling functions of order j. P j is called the approximation operator of order j. (ii) The Haar detail operator Q j of order j is defined as ∞ 

Q j( f ) =

< f, ψ j,k > ψ j,k .

k=−∞

For more details, see Walnut [4]. Non-uniform Haar wavelets. Now we present investigation of Dubeau et al. [5] on a frame work for non-uniform wavelets based on non-uniform partition of R. In this paper Haar scaling function is used to define the associated basic non-uniform Haar wavelet. Non-uniform multiresolution associated to these functions. Basic Haar Scaling Function and Non-uniform Haar Wavelet The characteristic function of the interval [0, 1[, noted χ[0,1[ will be considered as the basic Haar scaling function ϕ

90

3 Haar–Fourier Analysis

 ϕ(x) =

1 i f x ∈ [0, 1[, 0 elsewher e.

(3.28)

For any α1 , α2 ∈ [0, 1] such that α1 < α2 we have  ϕ

x − α1 α2 − α1

= χ[α1 ,α2 [ (x),

(3.29)

hence, for any α ∈]0, 1[, we obtain ϕ(x) = χ[0,α[ (x) + χ[α,1[ (x) = ϕ

x  α

 +ϕ

x −α 1−α

(3.30)

We define the basic non-uniform Haar wavelet, noted ψα , by ψα (x) = −(1 − α)χ[0,α[ (x) + αχ[α,1[ (x) 

x  x −α + αϕ = −(1 − α)ϕ α 1−α

(3.31)

Proposition 3.2 The functions ϕ and ψα have the following properties: 

∞

ϕ(x)d x = 1,

(3.32)

ψ(x)d x = 0

(3.33)

|ϕ(x)|n d x = 1,

(3.34)

|ψα (x)|n d x = α(1 − α)[(1 − α)n−1 + α n−1 ]

(3.35)

−∞  ∞ −∞

and for any n ∈ N



∞

−∞



∞ −∞

We can solve for ϕ

x α

and ϕ

 x−α  1−α

and obtain

x 

= αϕ(x) − ψα (x). α  x −α = (1 − α)ϕ(x) + ψα (x) ϕ 1−α ϕ

Now for any A, B ∈ R we have from (3.36) and (3.37)

(3.36)

(3.37)

3.4 Non-uniform Haar Wavelets

91

Fig. 3.2 Graph of φ(x) and ψα (x)

Aϕ

x  α

 + Bϕ

x −α 1−α

= [Aα + B(1 − α)]ϕ(x) + [B − A]ψα (x),

(3.38)

and from (3.30) and (3.31) Aϕ(x) + Bψα (x) = [A − B(1 − α)]ϕ

x  α

 + [A + Bα]ϕ

x −α 1−α

.

(3.39)

Remark 3.5 In (3.38) the value Aα + B(1 − α) is the average of the left-hand side expansion on [0, 1[. The value B − A is the step size of the discontinuity at x = α. In (3.39), the value B of the step size at x = α is splitted in two parts −B(1 − α) on [0, α[ and Bα on [α, 1[, and added to the average A on [0, 1[. Non-uniform Multiresolution Analysis The multiresolution analysis is based on a family {m }m∈Z of partitions of R such that the partition {m+ } is finer than {m } for any m ∈ Z. To be more precise, {m } = {xk(m) }k∈Z (m ∈ Z) are such that for any k ∈ Z (Fig. 3.2). (m+1) (m+1) (m+1) (m) < x2k+1 < x2k+2 = xk+1 xk(m) = x2k

and

lim xk(m) = −∞,

k→−∞

For each m ∈ Z and k ∈ Z we set  ϕk(m) (x)

=ϕ

(3.40)

lim xk(m) = ∞.

k→∞



x − xk(m) (m) xk+1 − xk(m)

= χ[x (m) ,x (m) [ (x) k

k+1

(3.41)

and, for α = αk(m) ∈]0, 1[ defined by αk(m) = And using (3.31), we set

(m+1) (m+1) − x2k x2k+1 (m+1) (m+1) x2k+2 − x2k

=

(m+1) − xk(m+1) x2k+1 (m) xk+1 − xk(m)

(3.42)

92

3 Haar–Fourier Analysis

 ψk(m) (x)

=

ψ (m) αk(m)

x − xk(m)

 (3.43)

(m) xk+1 − xk(m)

= −(1 − αk(m) )χ[x (m+1) ,x (m+1) [ (x) + αk(m) χ[x (m+1) ,x (m+1) [ (x). 2k

2k+1

2k+1

2k+2

In the next proposition, properties of ϕ and ψα are extended to ϕk(m) and ψk(m) . Proposition 3.3 The functions ϕk(m) and ψk(m) are such that 

∞

−∞  ∞ −∞

(m) ϕk(m) (x)d x = xk+1 − xk(m) ,

(3.44)

ψk(m) (x)d x = 0,

(3.45)

and for any n ∈ N 

∞

−∞  ∞ −∞

(m) |ϕk(m) (x)|n d x = xk+1 − xk(m) ,

(3.46)

   (m) |ψk(m) (x)|d x = αk(m) (1 − αk(m) ) (1 − αk(m) )n−1 + (αk(m) )n−1 xk+1 − xk(m) (3.47)

Moreover, orthogonality conditions and useful integrals are given in the proposition 4, 5 [5] with proof. We present these results in Remark 3.6. Remark 3.6 (a) For any m, n ∈ Z and k, l ∈ Z we have  (i) (ii)

∞

−∞  ∞ −∞  ∞ −∞

ϕk(m) (x)ψl(n) (x)d x = 0

(3.48)

 (m) ϕk(m) (x)ϕl(n) (x)d x = δklmn xk+1 − xk(m) ,

(3.49)

  (m) ψl(m) (x)ψl(n) (x)d x = δklmn αk(m) 1 − αk(m) xk+1 − xk(m) 

where δklmn =

1 i f m = n and k = 1, 0 elsewher e

(m) (m) , ϕ2k+1 are related as follows (b) The functions ϕk(m−1) , ψk(m−1) , ϕ2k (m) (m) (x) + ϕ2k+1 (x) ϕk(m−1) (x) = ϕ2k

(3.50)

(m) (m) (x) + αk(m) φ2k+1 (x) φk(m−1) (x) = −(1 − αk(m−1) )φ2k

(3.51)

3.4 Non-uniform Haar Wavelets

and

93

(m) (x) = αk(m−1) φk(m−1) (x) − ψk(m−1) (x), φ2k

(3.52)

(m) (x) = (1 − αk(m−1) φk(m−1) (x)) + ψk(m−1) (x). φ2k+1

(3.53)

Let us define the vector spaces Vm = Lin{φk(m) /k ∈ Z} and

Wm = Lin{ψk(m) /k ∈ Z}

for any m ∈ Z. Remark 3.6 yields Vm = Vm−1 ⊕ Wm−1 .

(3.54)

{Vm } is a multiresolution.

3.5 Generalized Haar Wavelets and Frames Definition 3.3 A generalized Haar wavelet of degree M is a wavelet with the property that there exists M ∈ R and si ∈ R such that ψ/[si , si+1 ] = ci ∈ C, Msi ∈ Z for all i ∈ Z. Definition 3.4 A family of functions {φi }i∈I in a Hilbert space H is a frame if there exists A > 0 and B < ∞ such that for all f ∈ H A f 2H ≤



| < f, ψi > |2 ≤ B f 2H

(3.55)

i∈I

The result given in Remark 3.7 and its generalization are proved in G. E. Pfander [6]. Remark 3.7 For generalized Haar wavelet ψ ∈ l2 (Z) the following statements are equivalent (i) The family {ψn,m }m∈Z+ , n ∈ Z is a frame for l2 (Z). (ii) There exists A > 0 and B < ∞ such ≤ ψ s (r ) ≤ B for almost all r ∈ T

that A −s s + ˆ )2 a.e. where ψ : T → R ∪ {∞}, r → m∈Z+ m |dm (r )ψ(mr dm (r ) =

m−1 

e−2πilr .

l=0

P. E. Pfander and J. J. Benedetto [6] have studied applications of generalized Haar wavelet to predict occurrence of seizure to an epileptic patient. The theory of pe-

94

3 Haar–Fourier Analysis

riodic wavelet transforms presented in [6] was developed to deal with the problem of epileptic seizure prediction. The main result is the characterization of wavelets having time and scale periodic transforms. In reality it has been proved that such wavelets are nothing but generalized Haar wavelets plus a logarithmic term. In the cited paper an algorithm for periodicity detection based on the periodicity of wavelet transforms defined by generalized Haar wavelets and implemented by wavelet averaging methods. The algorithm detects periodicities imbedded in significant noise.

3.6 Applications of Haar Wavelets Applications of Haar functions are discussed in Sect. 3.2. Haar wavelets have similar applications but we explain here applications to initial value, boundary value problems and integral equations. This section is based on Lepik and Hein [7] which contains important results of 19 reputed research papers.

3.6.1 Applications to Solutions of Initial and Boundary Value Problems Problems Consider the nth order linear differential equation n 

Av (x)y (v) (x) = f (x), x ∈ [α, β]

(3.56)

y (v) (A) = y0(v) , v = 0, 1, . . . , n − 1.

(3.57)

v=0

with the initial conditions

Here Av (x), f (x) are prescribed functions, y0(v) given constants. The Haar wavelet solution is sought in the form y (n) (x) =

2M 

ai h i (x),

(3.58)

ai pn−v,i (x) + Z v (x),

(3.59)

i=1

where ai are the wavelet coefficients. By integrating (3.58) n-v times we obtain y (v) (x) =

2M  i=1

3.6 Applications of Haar Wavelets

where Z v (x) =

95

n−v−1  σ =0

1 (x − A)α y0(v+σ ) . σ!

(3.60)

We shall satisfy (3.56), (3.58)–(3.60) in the collocation points x(l). If we substitute x → x(l) and replace (3.58)–(3.59) into Eq. (3.56) we get a system of linear equations for calculating the wavelet coefficient ai . After solving this system, the wanted function y = y(x) is calculated from (3.59) by assuming v = 0. It is essential to estimate the exactness of the achieved results. For this purpose we introduce the following error estimates: (i) If the exact solution y = yex (x) is known, we define the error estimates as     y(xl )  − 1 (local estimate) = max l  y (x )

(3.61)

σex = y − yex /(2M) (global estimate).

(3.62)

δex

ex

l

and

(ii) If the exact solution is unknown, we define the error vector ε(l) =

n 

Av (xl )y (v) (xl ) − f (xl ).

(3.63)

v=0

The smaller the s(l)/(2M) the more exact the solution is (ε = 0 in the case of exact solution). If ε(l)/(2M) is not small enough, we have to go to a higher level of resolution J and repeat the calculations. Boundary Value Problems. This topic is explained nicely in [7]. We mention here main steps. Let us consider (3.56). Let us assume that μ < m are given the initial conditions specified at x = A, but the remaining n − μ conditions are prescribed in some interval point x1 < B or at the boundary x = B. Furthermore assume boundary condition y (ν) (x) = y (ν) , where y∗(ν) is a given number. It follows from (3.59) and (3.60) 2M ai pn−v,i (x∗ ) + Z v (x∗) = y∗(v) Σi=1

Z v (x∗) =

n−v−1  σ =0

1 (x∗ − A)σ y0v+σ σ!

96

3 Haar–Fourier Analysis

3.6.2 Applications to Solutions of Integral Equations The integral equations can be classified in the following way: (i) Fredholm equation 

B

u(x) =

K [x, t, u(t)] dt + f (x), (x, t) ∈ [A, B].

(3.64)

K [x, t, u(t)] dt + f (x), (x, t) ∈ [A, B].

(3.65)

A

(ii) Volterra Equation 

x

u(x) =

A

(iii) Integro-differential equation αu  (x) + βu(x) =

 x A

K [x, t, u(t), u  (t)] dt + f (x), for u  (A) = u 0 , x ∈ [A, B].

(3.66)

Here α, β, A, B denote prescribed constants, f and K are given functions, the quantity K is called the kernel. The problem is to find the unknown function u(x). If f (x) = 0 the integral equation is called first kind equation. If f (x) = 0 we have the second kind equation. If the kernel has the form K [x, t, u(t)] = A(x, t)u(t).

(3.67)

the Eqs. (3.64)–(3.66) are linear integral equations, if (3.67) does not hold, these equations are nonlinear. In the case K [x, t, u(t)] = A(x, t)(x − t)−γ , where γ ∈ (0, 1)

(3.68)

the corresponding integral equation is weakly singular. Fredholm Integral Equation. Consider this linear Fredholm integral equation 

B

u(x) =

K (x, t)u(t) dt + f (x), x ∈ [A, B].

(3.69)

A

The solution is sought in the form u(x) =

2M 

ai h i (x),

(3.70)

i=1

where ai are the wavelet coefficients and h i (x) is ( ith) computed Haar wavelet according to (2.1) of [7]. Substituting (3.70) into (3.69) we obtain

3.6 Applications of Haar Wavelets 2M 

97

ai h i (x) −

2M 

i=1

ai G i (x) = f (x),

(3.71)

i=1

where



B

G i (x) =

K (x, t)h i (t) dt.

(3.72)

A

(i) Collocation method. Satisfying (3.71) only at the collocation points [7], we get a system of linear equations 2M 

ai [h i (xl ) − G i (xl )] = f (xl ), l = 1, 2, . . . , 2M.

(3.73)

i=1

The matrix form of this system is a(H − G) = F

(3.74)

where G il = G i (xl ), Fl = f (xl ). (ii) Galerkin Method. For realizing this approach each term of (3.71) is multiplied by h l (x) and the result is integrated over x ∈ [A, B]. Due to the orthogonality condition (2.4) [7] we obtain  al − ai Γil = ml i=1 2M



B

f (x)h l (x) d x.

(3.75)

A

Here l = m + k + 1, m = 2 j , j = 0, 1, . . . , J , k = 0, 1, . . . , m − 1 and 

B

Γil =

G i (x)h l (x) d x.

(3.76)

A

3.7 Exercises 3.1 Write down a Haar series expansion of the following function 3.2 Find expansion of the function given in Exercise 3.1 with respect to non-uniform wavelet given by (3.31). 3.3 What is the relationship between Haar and non-uniform wavelet? 1 3.4 Sketch graph of the Haar Fourier series of f (x) = x 2 for x ∈ (0, 1). 3.5 Discuss the evaluation of integrals 

β α

f (x)h i (x)d x

98

3 Haar–Fourier Analysis

where h i (x), i = 0, 1, 2, ..., m − 1 are Haar functions. 3.6 Using Haar wavelets solve the boundary problems: x2

d3 y d2 y dy + 6x +6 = 6, x ∈ [4, 8] 3 2 dx dx dx

for y(4) = y(8) = 0, , y  (3) = 0. 3.7 Discuss the numerical results obtained by the Haar wavelet of the equation  u(x) − Π λ

t

cos Π (x + t)u(t)dt = 0

0

References 1. Maqusi, M. (1981). Applied Walsh analysis. London, Philadelphia, Rheine: Heyden. 2. Schipp, F., Wade, W. R., & Simon, P. (1990). Walsh series. Bristol: Adam Hilger. 3. Fridli, S., Manchanda, P., & Siddiqi, A. H. (2008). Approximation by Walsh Norlund means. Acta Scientiarum Mathematicarum (Szeged), 74, 593–608. 4. Walnut, D. F. (2002). Wavelet analysis. Basel: Birkhauser. 5. Dubeau, F., Elmejdani, S., & Krantini, R. (2004). Nonuniform Haar wavelets. Applied Mathematics and Computation, 159, 675–693. 6. Pfander, G. E., & Benedetto, J. J. (2000). Periodic wavelet transforms and periodicity detection. SIAM Journal of Applied Mathematics, 62(4), 1329–1368. 7. Lepik, U., & Hein, H. (2014). Haar wavelets with applications. Berlin: Springer.

Chapter 4

Construction of Dyadic Wavelets and Frames Through Walsh Functions

4.1 Preliminary Walsh system of functions {wl : l ∈ Z+ } on half line R+ = [0, ∞) is determined by the equations w0 (x) ≡ 1, wl (x) =

k 

(w1 (2 j x))ν j , l ∈ N, x ∈ R+ ,

j=0

where the numbers k and ν j are taken from the binary expansion l=

k 

ν j 2 j , ν j ∈ {0, 1}, νk = 1, k = k(l),

j=0

but w1 (x)-function given on [0, 1) by the formula  1, if x ∈ [0, 21 ), w1 (x) = −1, if x ∈ [ 21 , 1), and continued on R+ such that w1 (x + 1) = w1 (x) for all x ∈ R+ . The integer and fractional parts of number x ∈ R+ are denoted by [x] and {x}, respectively. For each x ∈ R+ and any j ∈ N, the numbers x j , x− j ∈ {0, 1} are determined as follows x j = [2 j x](mod 2), x− j = [21− j x](mod 2).

(4.1)

These numbers are digits of binary expansion x=

 j0

© Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2_4

99

100

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

(in case of dyadic rational x, we get disintegration with finite number of nonzero terms). For x, y ∈ R+ , we get x⊕y=



|x j − y j |2− j−1 +

j0

where x j , y j are calculated by the formula (4.1). From definition, we assume x  y = x ⊕ y (since x ⊕ x = 0). It is well known that wm (x)wn (x) = wm⊕n (x), m, n ∈ Z+ , but if the sum x ⊕ y is double irradiational, then wn (x ⊕ y) = wn (x)wn (y), n ∈ Z+ ,

(4.2)

(in such a way that in case of fixed y, the Eq. (4.2) takes place for all x from R+ , in except countable set). The Walsh functions are 1-periodic, these are examined in the interval Δ = [0, 1). For a given integer n, the numerical intervals Ik(n) = [k2−n , (k + 1)2−n ], k ∈ Z+ are called dyadic interval of rank n. It is easy to see that Ik(n) ∩ Il(n) = φ and k = l, Δ =

n 2 −1

Ik(n) .

k=0

As Walsh functions wl (x), 0 ≤ l ≤ 2n − 1, are either equal to 1 or −1 in each of the dyadic interval of rank n and wl (x) = 1 at x ∈ I0(n) . Any 1-periodic function w(x), constant in each dyadic interval Ik(n) , 0 ≤ k ≤ 2n − 1 is representable in the form w(x) =

n 2 −1

cl wl (x),

(4.3)

l=0

i.e., is the Walsh polynomial of order not more than 2n − 1. Such a representation of the function w(x) is unique. The coefficients cl in the formula (4.3) coincide with the values bk = w(k2−n ), 0 ≤ k ≤ 2n − 1, discrete Walsh transformation: 2 −1 1  cl = n bk wk (l2−n ), 0 ≤ l ≤ 2n − 1. 2 k=0 n

(4.4)

Theorem 4.1 The Walsh system {wl : l ∈ Z+ } is the basis in L p (Δ) for 1 < p < ∞ (orthonormal basis in L 2 (Δ)) and not the basis in L 1 (Δ).

4.1 Preliminary

101

The function f : R+ → C is called W -continuous at the point x ∈ R+ , if sup | f (x ⊕ h) − f (x)| → 0 as n → ∞. 0≤h< 21n

Each Walsh function W -continuous at each point of R+ . Any function f , continuous in the classical sense at the point x, is W -continuous at this point. Let E ⊂ R+ . If the function f is W -continuous at each point of the set E, then it is called W -continuous on E. If the function f : R+ → C continuous at each irrational point of R+ and has a finite limit to the left at each point of R+ \{0}, then the function f is W -continuous on R+ . The function f is called uniformly W -continuous on E, if for any ε > 0 there exists the number n ∈ Z+ such that the inequality | f (x) − f (y)| < ε follows from the conditions x ∈ E and x ⊕ y < 2−n . If the function f : R+ → C is constant in the dyadic interval of rank n, i.e., f (x) = ck , where x ∈ I0(n) , where {ck } certain numerical sequence, then f is uniformly W -continuous on R+ . For any x, y ∈ R+ , we write χ (x, y) = (−1)σ (x,y) , σ (x, y) =

∞ 

x j y− j + x− j y j ,

j=1

where the numbers x j , y j are calculated by the formulae (4.1). The function χ (x, y) possesses the properties as given below: χ (y, x) = χ (x, y), χ (x, 0) = 1; χ (x, y) = χ (x, [y])χ ([x], y); χ (x, k) = wk (x) = wk ({x}) and any k ∈ Z+ ; if x, y, z ∈ R+ and x ⊕ y are dyadic irrational, then χ (x, z)χ (y, z) = χ (x ⊕ y, z); (5) in case of fixed y ∈ [0, 2n ), where n ∈ Z+ , the function χ (x, y) is constant for j , (2j+1) j ∈ Z+ ; x on each dyadic interval I j(n+1) = [ 2n+1 n+1 ], (6) for any k ∈ Z+ , we have (1) (2) (3) (4)



k+1

 χ (x, y)dy =

k

wk (x), 0,

0 ≤ x < 1, x ≥ 1.

The Fourier–Walsh transform of function f ∈ L 1 (R+ ) ∩ L 2 (R+ ) is defined by the formula  ˆ f (x)χ (x, ω)d x, ω ∈ R+ , f (ω) = R+

102

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

and valid for all functions from L 2 (R+ ). Let us remember that

f L p (R+ ) =



| f (t)| p

R+

 1p

, 1 ≤ p < ∞.

Proposition 4.1 The following properties hold: (a) if f ∈ L 1 (R+ ), then fˆ is W -continuous function on R+ and fˆ(ω) → 0 when ω → ∞; (b) if f, fˆ ∈ L 1 (R+ ), and f are the W -continuous function, then at each point x ∈ R+ there takes place the inversion formula  f (x) =

R+

fˆ(ω)χ (x, ω)dω;

(c) if f ∈ L 2 (R+ ), then fˆ ∈ L 2 (R+ ) and fˆ L 2 (R+ ) = f L 2 (R+ ) . The following criterion of orthonormality of the system of shifts of the function from L 2 (R+ ). Proposition 4.2 Let f ∈ L 2 (R+ ). For orthonormality of the system { f (· ⊕ k) : k ∈ Z+ } in L 2 (R+ ), it is necessary and sufficient that 

| fˆ(ω ⊕ l)|2 = 1, for almost all ω ∈ R+ .

l∈Z+

Let f k ∈ L 2 (R+ ), k ∈ Z+ . The family { f k } is Riesz system in L 2 (R+ ), if there exist the positive constants A and B such that for any set of coefficients a = {ak } ∈ l 2 is fulfilled the inequality  ak f k A a l 2 ≤ ≤ B a l 2 . k∈Z+

By A and B, it is usually their best values. Then, the maximum value of A is the least Riesz constant and the minimum value of B is the upper Riesz constant. According to Parseval’s theorem, the constants A and B are equal to 1 for the orthonormal system { f k }. Proposition 4.3 Let { f k } be a Riesz system in L 2 (R+ ) with the constants A and B. Then

∞ 2 (a) { f k } is Riesz basis in space V := { f = ∞ k=0 ck f k : k=0 |ck | }; (b) V = span{ f k : k ∈ Z+ }; (c) for any f ∈ L 2 (R+ ), the following inequality is fulfilled A f L 2 (R+ ) ≤

∞   f, f k  ≤ f L 2 (R+ ) ; k=0

4.1 Preliminary

103

The Proposition 4.3 follows from the well-known property of Riesz system in Hilbert spaces (see, e.g., Novikov et al. [1, Theorem 1.1.2]). Proposition 4.4 Let f ∈ L 2 (R+ ). For the system { f (· ⊕ k) : k ∈ Z+ } in L 2 (R+ ) to be Riesz system in L 2 (R+ ) with the constant A and B, it is necessary and sufficient that  | fˆ(ω ⊕ l)|2 ≤ B, ω ∈ R+ . A≤ l∈Z+

Let En (R+ ) be the set of functions given on R+ and constant on dyadic intervals of rank n. Clearly, each Walsh polynomial of order 2n − 1 to the set En (R+ ). The characteristic function of the set M ⊂ R+ is denoted by 1 M . If f ∈ En (R+ ), then f (x) =

∞ 

f (2−n k)1 I (n) (x), x ∈ R+ . k

k=0

The set E (R+ ) is defined as the collection of all sets En (R+ ). Let us indicate by Ec (R+ ), the set of functions from E (R+ ) having compact support. Proposition 4.5 The following properties are true: (a) (b) (c) (d)

each function f ∈ E (R+ ) is uniformly W -continuous on R+ ; the sets E (R+ ) and Ec (R+ ) are compact in the spaces L p (R+ ), 1 ≤ p ≤ ∞; if f ∈ L 1 (R+ ) ∩ En , then m 0 supp f ⊂ [0, 2n ]; if f ∈ L 1 (R+ ) and supp f ⊂ [0, 2n ], then fˆ ∈ En .

The last two properties show that the dyadic entire functions on R+ are analogous to the usual entire functions on R+ (compare with the Paley-Wiener theorem on Fourier transform of entire functions).

4.2 Orthogonal Wavelets and MRA in L 2 (R+ ) As on the real line R+ , orthogonal wavelets on positive half line R+ are constructed using the notion of multiresolution analysis. For an arbitrary function f ∈ R+ , we set f j,k (x) := 2 j/2 ψ j,k (2 j x ⊕ k), j ∈ Z, k ∈ Z+ . Definition 4.1 An orthogonal wavelet in L 2 (R+ ) is a function ψ so that the system {ψ j,k : j ∈ Z, k ∈ Z+ .} is an orthonormal basis in L 2 (R+ ). In other words, ψ is an orthogonal wavelet in L 2 (R+ ) if the system {ψ j,k } is orthonormal and every function f ∈ L 2 (R+ ) can be expanded in a Fourier series in this system:

104

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

f =

  f, ψ j,k ψ j,k . j,k

The simplest example of an orthogonal wavelet in L 2 (R+ ) is the Haar wavelet ⎧ ⎪ ⎨+1, if x ∈ [0, 1/2) ψ H (x) = −1, if x ∈ [1/2, 1) ⎪ ⎩ 0, otherwise. Definition 4.2 MRA in L 2 (R+ ) A multiresolution analysis in L 2 (R+ ) is a family of closed subspaces V j ⊂ L 2 (R+ ) satisfying the following conditions: (i) (ii) (iii) (iv)

V ⊂ V j+1 for all j ∈ Z; j  V j = L 2 (R+ ) and V j = {0}; f (·) ∈ V j if and only if f ∈ ψ(2· j ) ∈ V0 ; there exists a function φ ∈ L 2 (R+ ) so that the system {φ(· ⊕ k) : k ∈ Z} is an orthonormal basis in V0 .

According to (i), the family {V j } is a sequence of embedded subspaces. Let W j be the orthogonal complement of V j in V j+1 , that is, V j+1 = V j ⊕ W j+1 , j ∈ Z.

(4.5)

From properties (i) and (ii), it follows the equalities L 2 (R+ ) = ⊕ j∈Z W j = V0 ⊕ (⊕ j≥0 W j ).

(4.6)

Property (iii) allows us to reproduce the entire family {V j } from one subspace V0 . From properties (iii) and (iv), it follows that the system of functions φ1,k (t) =

√ 2φ(2t ⊕ k), k ∈ Z+

is an orthonormal basis of the subspace V1 . Since φ ∈ V0 ⊂ V1 function φ can be expanded in a Fourier series as per this system as shown below: φ=



φ, φ1,k φi,k .

(4.7)

k∈Z+

Definition 4.3 A scaling function in L 2 (R+ ) is a function φ from L 2 (R+ ) such that φ(t) =

 k∈Z+

where {ck } is some sequence from l 2 .

ck φ(2t ⊕ k),

(4.8)

4.2 Orthogonal Wavelets and MRA in L 2 (R+ )

105

The function φ from condition (iv) of Definition 4.2 is called the scaling function of the multiresolution analysis {V j }. According to (4.7), for this function of Eq. (4.8) is satisfied with constants  √  ck = 2 φ, φi,k  = 2 φ(t)φ(2t ⊕ k)dt, k ∈ Z+ . (4.9) R+

From (4.8) and (4.9), by virtue of the orthonormality of the system {φ(· ⊕ k) : k ∈ Z+ }, by means of Parseval’s identity, the equality is derived as 

ck ck⊕2l ¯ = 2δ0,l and, in particular,

k∈Z+



|ck |2 = 2.

(4.10)

k∈Z+

For an arbitrary sequence {ck } from l 2 , equality (4.8) can be considered as a functional equation with respect to φ. This equation is called the scaling function for φ. We say that a function φ generates an MRA in L 2 (R+ ), if, first, the system {φ(· ⊕ k) : k ∈ Z+ } is orthonormal in L 2 (R+ ) and, second, the family of subspaces V j = span{φ j,k : k ∈ Z+ }, j ∈ Z+

(4.11)

satisfies the conditions V j ⊂ V j+1 ,



V j = L 2 (R)



V j = {0}.

Note that if the function φ generates MRA, then equality (4.7) is true for it, and hence φ is a scaling function in L 2 (R+ ). The function m 0 (ω) =

1  ck wk (ω) 2 k∈Z

(4.12)

+

is called the mask of the scaling equation (4.8) (or its solution φ). By the Parseval’s theorem, from (4.10) and (4.12), we have 

1

|m 0 (ω)|2 dω =

0

1  1 |ck |2 = . 4 2

(4.13)

k∈Z+

Applying the Walsh–Fourier transform, we can write Eq. (4.8) in the form ˆ φ(ω) = mˆ 0

ω ω φˆ , ω ∈ R+ . 2 2

(4.14)

106

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

Proposition 4.6 Suppose that φ ∈ L 2 (R+ ) satisfies the scaling equation (4.8) and ˆ φ(0) = 0. If the system {φ(· ⊕ k) : k ∈ Z+ } is orthonormal in L 2 (R+ ), then m 0 (0) = 1, and|m 0 (ω)|2 + |m 0 (ω ⊕ 1/2)|2 = 1, a.e. ω ∈ R+ .

(4.15)

Proof The first equality in (4.15) follows from (4.14). According to Proposition 4.2, the orthonormality of the system {φ(· ⊕ k) : k ∈ Z+ } is equivalent to the condition 

ˆ + l)|2 = 1, almost everywhere ω ∈ R+ . |φ(ω

(4.16)

l∈Z+

By virtue of the 1-periodicity of the mask m 0 , the second equality in (4.15) is obtained after substituting (4.14) into (4.16) and grouping the summands with even and odd indices. The proposition is proved. Recall that for integer m, we denote by Em (R+ ), the collection of all functions f on R+ which are constant on [2−m s, (s + 1)2−m ) (cf. Proposition 4.5). Further, we set  E˜m (R+ ). E˜m (R+ ) := { f : f is W -continuous and fˆ ∈ E (R+ )} and E˜ (R+ ) := m

It is known (see Golubov et al. [2, §6.2 and §10.5]) that (1) E˜ (R+ ) is dense in L p (R+ ) for 1 ≤ p ≤ ∞; (2) if f ∈ L 1 (R+ ) ∩ E˜m (R+ ), then supp f ∈ [0, 2m ]. Proposition 4.7 Let {V j } be a family of subspaces defined by (4.1) with a fixed function ϕ ∈ L 2 (R+ ) φ ∈ L 2 (R+ ) with the help of formula (4.11).  If the system {φ(· ⊕ k) : k ∈ Z+ } is orthonormal basis of the subspace V0 , then V j = {0}.  Proof Let P j be the orthogonal projection of L 2 (R+ ) to V j , let f ∈ V j . Given an ε > 0, we choose u ∈ L 2 (R+ ) ∩ E˜m (R+ ) such that f − u < ε. Then

f − P j u ≤ P j ( f − u) ≤ f − u < ε, and so

f ≤ P j u + ε for every j ∈ Z. Now choose R > 0 so that suppu ⊂ [0, R). Then  (P j u, φ j,k ) = (u, φ j,k ) = 2 j/2 0

Hence, by Cauchy–Schwarz inequality

R

u(x)φ(2 j x ⊕ k)d x.

(4.17)

4.2 Orthogonal Wavelets and MRA in L 2 (R+ )

P j u = 2



107

|(P j (u), φ j,k )| ≤ u 2

2

k∈Z+





2

R

j

|φ(2 j x ⊕ k)|2 d x.

0

k∈Z+

Therefore, if j is chosen enough so that Rp j < 1, then 



P j u ≤ u 2

|φ(x)| d x = u

2

2

SR j

where S R j :=



k∈Z+ {y

2 R+

1 SR, j (x)|φ(x)|2 d x,

(4.18)

⊕ k : y ∈ [0, 2 j R)}. It is easy to check that / Z+ . lim 1 SR j = 0 for all x ∈

j→−∞

Thus, by the dominated convergence theorem from (4.18), we get lim P j u = 0.

j→−∞

In view of (4.17), this implies that f < ε, and thus



V j = {0}.

Theorem 4.2 Let the solution φ of the scaling equation (4.8) generates MRA in ˆ L 2 (R+ ) and satisfies the condition φ(0) = 1. If the function ψ is defined by the formula  (−1)k ck⊕l φ(2x ⊕ k), x ∈ R+ . (4.19) ψ(x) = k∈Z+

then, ψ is an orthogonal wavelet in L 2 (R+ ). Proof According to (4.6), for the function ψ to be an orthogonal wavelet, it is sufficient that the system {ψ0,k : k ∈ Z+ } to be an orthonormal basis in W0 . Applying the Walsh–Fourier transform to (4.19), we obtain ˆ ˆ ψ(ω) = m 1 (ω/2) · φ(ω/2),

where m 1 (ω) = 21 k∈Z+ (−1)k ck⊕l wk (ω) = −w1 (ω)m 0 (ω ⊕ 21 ). From this and from (4.15), we see that the matrix 

m 0 (ω) m 0 (ω ⊕ 21 ) m 1 (ω) m 1 (ω ⊕ 21 )



is unitary for almost all ω ∈ R+ . Further, using (4.15) and (4.16), we have

(4.20)

108

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions



ˆ ⊕ l)|2 = |ψ(ω

l∈Z+



ˆ ⊕ 2l)|2 + |ψ(ω

l∈Z+



ˆ ⊕ 2l ⊕ 1)|2 |ψ(ω

l∈Z+

   ω 2    ω 2   ω 1 2   ˆ  ⊕ l  + m 0 ⊕ = m 0  + φ 2 2 2 2  l∈Z+     ω 1 2  ˆ + φ 2 ⊕ l ⊕ 2  l∈Z+     ω 1 2   ω 2  ⊕ = m 0 + m 0  . 2 2  2

Hence, by Proposition 4.2, the system {ψ0,k : k ∈ Z+ } is orthonormal. Besides,  φ(t ⊕ k)ψ(t)dt = 0,

R+

In fact, 

 R+

φ(t ⊕ k)ψ(t)dt = =

ˆ ˆ φ(ω) ψ(ω)χ (k, ω)dω

R+  1 0

 ˆ ⊕ l)ψ(ω ˆ ⊕ l) χ (k, ω)dω φ(ω

l

= 0.  l

ˆ ⊕ 2l)ψ(ω ˆ ⊕ 2l) + φ(ω

 l

ˆ ⊕ 2l ⊕ 1)ψ(ω ˆ ⊕ 2l ⊕ 1) φ(ω

2 ω   ω  ˆ ⊕l  φ 2 2 2 l∈Z+        ω ω ω 1 1   1 2 ˆ φ ⊕ ⊕ ⊕ l ⊕ + m0 m1 2 2 2 2 l∈Z  2 2  +     ω ω ω ω 1 1 + m0 ⊕ ⊕ = m0 m1 m1 2 2 2 2 2 2 = 0.

= m0

ω

m1

Thus, ψ ∈ W0 . We now prove that W0 = span{ψ0,k : k ∈ Z+ }. By virtue of relations V1 = V0 ⊕ W0 and V1 = span{φ0,l : l ∈ Z+ }. it follows from the fact that for each l ∈ Z+

4.2 Orthogonal Wavelets and MRA in L 2 (R+ )

φ(2t ⊕ l) =



109

αk φ(t ⊕ k) +

k∈Z+



βk ψ(t ⊕ k),

(4.21)

k∈Z+

where αk , βk are the Fourier coefficients of the  function φ(2t ⊕ l) with respect to the orthonormal system {φ(t ⊕ k) : k ∈ Z+ } {ψ(t ⊕ k) : k ∈ Z+ }. According to Parseval’s theorem, the expansion (4.21) holds the place in that and only in that case, when    2 2 |αk | + |βk | = |φ(2t ⊕ l)|2 . (4.22) R+

k∈Z+

Obviously, the right side in (4.22) is equal to 21 . Comparing (4.9) with expression  αk =

 R+

φ(2t ⊕ l)φ(t ⊕ l)dt =

in view of (4.12), we note that αk = coefficients βk , we have 

1 0

R+

φ(2x ⊕ 2k ⊕ l)φ(x)d x,

m 0 (ω)w2k⊕l (ω)dω =

c2k⊕l . Similarly, 2

for the

1

βk =

m 1 (ω)w2k⊕l (ω)dω   1 w1 (ω)w2k⊕l (ω)dω m0 ω ⊕ =− 2 R+    1 1 dω =− m 0 (ω)w2k⊕l⊕1 ω + 2 0  1 l = (−1) m 0 (ω)w2k⊕l⊕1 (ω)dω 0



0

(−1)l c2k⊕l⊕1 . = 2 Consequently,

 1   1 |αk |2 + |βk |2 = |ck |2 = . 4 2 k∈Z k∈Z +

+

Therefore, (4.22) is true and the theorem is proved. Under the conditions of the Theorem 4.2 for each fixed j, the system {ψ j,k : k ∈ Z+ } is orthonormal basis of space W j . Orthogonal projectors P j : L 2 (R+ ) → V j and Q j : L 2 (R+ ) → W j for each j ∈ Z are determined by the formulae Pj f =

 k

and

a jk φ jk , a jk =  f, φ jk 

110

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

Qj f =



d jk ψ jk , d jk =  f, ψ jk .

k

For any f ∈ L 2 (R+ ) in accordance with (4.5), we have P j+1 f = P j f + Q j f, P j+1 f 2 = P j f 2 + Q j f 2 , j ∈ Z.

(4.23)

Sometimes it is said that Q j f contains “the details” necessary for the transition from the j −th level of the approximation of the function f to the more accurate ( j + 1) level. Accordingly, the subspaces {V j } (and the coefficients {a jk }) are said to be approximating, and the subspaces {W j } (and coefficients {d jk }) are detailed. For a fixed j and any s ∈ N from equalities (4.23), we obtain P j f = P j−1 + Q j−1 = P j−s f + Q j−s f + · · · + Q j−1 f. From the relations



V j = L 2 (R+ ) and



(4.24)

V j = {0}.

It follows that for each f ∈ L 2 (R+ ), we have the equalities lim f − P j f = 0, and

j→+∞

lim P j f = 0.

j→−∞

Thus, as j increases, the approximation error f ≈ P j f decreases to zero, and if j → −∞, then the projections of P j f tend to the zero element of the space L 2 (R+ ). Theorem 4.3 Let f ∈ L 2 (R+ ) and let h k = φ, φ1,k , gk = ψ, φ1,k , k ∈ Z+ , where the scaling function φ satisfies the conditions of the Theorem 4.2 and the wavelet ψ is given by the formula 4.19. If the coefficients a jk of the expansion Pj f =



a jk φ jk

k∈Z+

are known, then the coefficients of the expansions of P j−1 f =





a j−1,k φ j−1,k , Q j f =

k∈Z+

d jk ψ j−1,k

k∈Z+

are calculated from formulas a j−1,k =

 l

h l⊕2k a jl , d j−1,k =



gl⊕2k a jl .

(4.25)

l

Conversely, if the coefficients a j−1,k and d j−1,k are known, then the coefficients a jl are recovered from the formula

4.2 Orthogonal Wavelets and MRA in L 2 (R+ )

a jl =



111

 h l⊕2k a j−1,l + gl⊕2k d j−1,k .

(4.26)

k

Proof According to (4.9) and (4.19), for the coefficients in formulas (4.25) and (4.26) there occur equality ck h k = √ , gk = (−1)k h k⊕l . 2 Taking into account (4.7), we have expansions φ(t) =

√  2 h l φ(2t ⊕ l)

(4.27)

l∈Z+

and ψ(t) =

√  2 gl φ(2t ⊕ l).

(4.28)

l∈Z+

Using (4.27), for any j ∈ Z, k ∈ Z+ , we have 2( j−1)/2 φ(2( j−1) t ⊕ k) = 2 j/2



φ(2 j t ⊕ (2k ⊕ l))

l∈Z

and, consequently φ j−1,l =



h l φ j,2k⊕l ,

l∈Z

That is, φ j−1,l =



h l⊕2k φ jl .

(4.29)

l∈Z

Similarly from (4.28), we obtain the equality ψ j−1,k =



gl⊕2k φ jl .

(4.30)

l∈Z

According to (4.29), we have a j−1,k =  f, φ j−1,k  =



h l⊕2k  f, φ j,l  =

l∈Z



h l⊕2k a jl .

l∈Z

Similarly from (4.30), we deduce d j−1,k =  f, φ j−1,k  =

 l∈Z

gl⊕2k  f, φ jl  =

 l∈Z

gl⊕2k a jl .

112

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

Thus, the formulas (4.25) are proved. We now prove (4.26). According to (4.29) and (4.30), we have (4.31) h l⊕2k = φ j−1,k φ jk , gl⊕2k ψ j−1,k , φ jk . Using the fact that the difference f − P j f is orthogonal to the subspace V j , and applying (4.23), we have a jl =  f, φ jl  = P j f, φ jl  = P j−1 f, φ jl  + Q j−1 f, φ jl . Taking into account (4.31), hence, by the definition of the operators P j and Q j , we obtain   a j−1,k φ j−1,k φ jl  + ψ j−1,k φ jl  a jl = k

=



a j−1,k h l⊕2k +

k



k

d j−1,k gl⊕2k ,

k

i.e., the formula (4.26) is valid. The theorem is proved. Further, we shall consider the case when the scaling function φ has a support on the segment. It is easy to see that then in Eq. (4.8) the coefficients ck can be non-zero only when 0 ≤ k ≤ 2n − 1. Indeed, let us assume that for the function φ from Definition 4.3, the condition supp φ ⊂ [0, 2n − 1] is fulfilled. According to (4.8), we have φ(x) = S1 (x) + S2 (x) =

n 2 −1

ck φ(2x ⊕ k) +

∞ 

ck φ(2x ⊕ k).

k=2n

k=0

If x ≥ 2n−1 and k < 2n , then 2x ⊕ k ≥ 2n , and hence φ(x) = S1 (x) = 0. If x ≥ 2n−1 , then S2 (x) = 0 (since 2x ⊕ k ≥ 2n for all k ≥ 2n ). Thus, if supp φ ⊂ [0, 2n−1 ], then φ(x) = S1 (x) for all x ∈ R+ and Eq. (4.8) can be written in the form of φ(x) =

n 2 −1

ck φ(2x ⊕ k), x ∈ R+ .

(4.32)

k=0

The mask of this scaling equation is the Walsh polynomial 2 −1 1 ck wk (ω). m 0 (ω) = 2 k=0 n

(4.33)

We denote by L 2c (R+ ) the class of all functions from L 2 (R+ ) with compact supports. The following theorem contains the properties that solution of Eq. (4.32) possesses in the general case.

4.2 Orthogonal Wavelets and MRA in L 2 (R+ )

113

Theorem 4.4 If the scaling equation (4.32) has a solution φ ∈ L 2c (R+ ) such that ˆ φ(0) = 1, then n 2 −1 ck = 2, suppφ ⊂ [0, 2n−1 ]. (4.34) k=0

This solution is unique and is given by the formula ˆ φ(ω) =

∞ 

m 0 (2− j ω)

(4.35)

j=0

and possess the following properties: ˆ (1) φ(r

) = 0, for all r ∈ N; (2) k∈Z+ φ(x ⊕ k) = 1 for x ∈ R+ . Proof Let φ be a solution of Eq. (4.32), belonging to the space L 2c (R+ ) and such, ˆ ˆ ˆ ω )m 0 ( ω ) at ω = 0. We obtain m 0 (0) = that φ(0) = 1. From the equality φ(ω) = φ( 2 2

2n −1 1, and hence k=0 ck = 2. Further, let j be the largest integer so that function φ on the section [ j − 1, j] does not turn into zero on a set of positive measure. Let us assume that j ≥ 2n−1 + 1. Take an arbitrary binary irrational x ∈ [ j − 1, j]. If [2x] = l(mod 2n ), where l ∈ {1, 2, 3, . . . , 2n − 1}, then 2x > 2n + l and for any k ∈ {0, 1, 2, 3, . . . , 2n − 1} 2x ⊕ k ≥ 2x − l ≥ 2n , since {2x} > 0, then 2x ⊕ k > 2n . Hence, from (4.32) it follows that j ≥ 2n + 1. Indeed, otherwise φ(2x ⊕ k) = 0 for almost all x ∈ [ j − 1, j] for any k.It means, by (4.32) and φ(x) = 0 almost everywhere on [ j − 1, j], which contradicts the election of j. Further, since x ≥ j − 1 and 2x is not an integer, for any k ∈ {0, 1, 2, 3, . . . , 2n − 1} 2x ⊕ k ≥ 2x − (2n − 1) > (2 j − 2) − (2n − 1) ≥ j. (The inequality j ≥ 2n + 1 is used). Hence, as above, we obtain that φ(x) = 0 almost everywhere on [ j − 1, j]. Thus, j ≤ 2n−1 and therefore supp φ ⊂ [0, 2n−1 ]. Thus, the relations (4.34) are proved. Further, since φ belongs to L 2c (R+ ), then it belongs to L 1 (R+ ). Thus, from (4.34) by Proposition 4.5 it follows that φˆ ∈ En−1 . Using the ˆ ˆ condition φ(0) = 1, we get that φ(ω) = 1 for all ω ∈ [0, 21−n ). On the other hand, 1−n m 0 (ω) = 1 for ω ∈ [0, 2 ). Therefore, for any ω ∈ [0, 2r ) (r is a natural number), we have r +n ∞  ˆ ˆ −r −n ω) φ(ω) = φ(2 m 0 (2−k ω) = m 0 (2−k ω), k=1

k=1

which proves (4.35) and uniqueness φ. Note that for any r ∈ N

114

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

ˆ ) = φ(r ˆ ) φ(r

j−1 

ˆ j r ) → 0. m 0 (2s r ) = φ(2

s=1

when j → ∞ (since φˆ ∈ L 1 (R+ ) and m 0 (2s r )) by virtue of the equality m 0 (0) = 1 ˆ ) = 0. Applying now and the periodicity of the polynomial (4.33). It follows that φ(r the Poisson summation formula, we get ∞ 

φ(x ⊕ k) =

∞ 

ˆ )wr (x), φ(r

r =0

k=0

(the equality is understood almost everywhere

with respect to Lebesgue measure) ˆ ˆ ) = δ0r , we obtain ∞ and using the equality φ(r k=0 φ(x ⊕ k) = φ(0)w0 (x) = 1. The theorem is proved.

4.3 Orthogonal Wavelets with Compact Support on R+ For every natural number n, we shall characterize all the solutions φ of the scaling equation n 2 −1 φ(x) = φ(2x ⊕ k), x ∈ R+ (4.36) k=0

generating multiresolution analysis in L 2 (R+ ) and we indicate the algorithm for constructing corresponding orthogonal wavelets. The mask of the scaling equation (4.36) is the Walsh polynomial 2 −1 1 ck wk (ω), m 0 (ω) = 2 k=0 n

(4.37)

and the Eq. (4.36) in terms of the Fourier–Walsh transform is written in the form ˆ φ(ω) = m0

ω ω φˆ , ω ∈ R+ . 2 2

(4.38)

By Theorem 4.4, the scaling equation (4.36), under rather general conditions, has a unique solution represented by (4.35). It follows from the properties of Walsh functions that the mask (4.33) assumes constant values at each of the intervals Il(n) = [l2−n , (l + 1)2−n ], 0 ≤ l ≤ 2−n . The coefficients of the scaling equation (4.36) are calculated from the values bl = m 0 (l2−n ), 0 ≤ l ≤ 2−n using the discrete Walsh transform

4.3 Orthogonal Wavelets with Compact Support on R+

ck =

n 2 −1

1 2n−1

bl wl (k2−n ), 0 ≤ k ≤ 2n − 1.

115

(4.39)

l=0

Thus, the choice of mask values m 0 on dyadic intervals of rank n completely determines the coefficients of Eq. (4.36), to which satisfies the corresponding function φ. ˆ = 0, satisfies the scaling equation (4.36). Let us assume that φ ∈ L 2c (R+ ), φ(0) Then, it follows from Propositions 4.2 and 4.5 that the orthonormality of the system {φ(· ⊕ k) : k ∈ Z+ } is equivalent to the condition 

ˆ ⊕ l)|2 = 1, for all ω ∈ R+ , |φ(ω

(4.40)

l∈Z+

and by Proposition 4.6, in this case, we have     1 2 m 0 (0) = 1 and |m 0 (ω)|2 + m 0 ω ⊕ = 1, for all ω ∈ R+ . 2 

(4.41)

Obviously, condition (4.41) means that for the values bl = m 0 (l2−n ) the following equalities are fulfilled b0 = 1, |bl |2 + |bl+2n−1 |2 = 1, 0 ≤ l ≤ 2n−1 − 1.

(4.42)

According to Theorem 4.2, if φ generates MRA in L 2 (R+ ), then corresponding orthogonal wavelet ψ is determined by the formula ψ(x) =

n 2 −1

(−1)k c¯k⊕l φ(2x ⊕ k), x ∈ R+ .

(4.43)

k=0

Example 4.1 If n = 1 and c0 = c1 = 1, then the solution of Eq. (4.36) is the Haar function φ = 1[0,1) (recall that 1 E is the characteristic function of the set E). In this case, the mask is determined by the formula  m 0 (ω) =

1, ω ∈ [0, 1/2), 0, ω ∈ [1/2, 1).

and the wavelet ψ coincides with the Haar wavelet which is given below: ⎧ ⎪ x ∈ [0, 1/2), ⎨1, ψ H (x) = −1, x ∈ [1/2, 1), ⎪ ⎩ 0, x ∈ R+ \[0, 1).

116

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

The corresponding system of wavelets {ψ jk } is the classical Haar basis. Example 4.2 The Lang scaling function is obtained if for n = 2, we set ⎧ 1, ⎪ ⎪ ⎪ ⎨a, m(ω) = ⎪0, ⎪ ⎪ ⎩ b, where 0 < |a| < 1, |b|
2n−1 , then f (x ⊕ k) = 0 for almost all x ∈ [0, 2n−1 ]. Therefore, the linear independence of the system { f (· ⊕ k) : k ∈ Z+ } in L 2 (R+ ) is equivalent to that for the finite system { f (· ⊕ k) : k = 0, 1, . . . , 2n−1 − 1}. Further, if some vector (a0 , a1 , . . . , a2n−1 −1 ) satisfies conditions 2n−1 −1

ak f (· ⊕ k) = 0, and |a0 | + · · · + |a2n−1 −1 | > 0,

(4.50)

k=0

then using the Walsh–Fourier transform, we obtain fˆ(ω)

2n−1 −1

ak wk (ω) = 0, for almost everywhere ω ∈ R+ .

k=0

n−1 The Walsh polynomial w(ω) = 2k=0 −1 ak wk (ω) is not identically equal to zero. Hence, among Is(n−1) , 0 ≤ s ≤ 2n−1 − 1, there exists an interval (denote it by I ) for which w(I ⊕ k) = 0, k ∈ Z+ . Since fˆ ∈ En−1 (R+ ), it follows that (4.50) holds if and only if there exists a dyadic interval I of range n − 1 such that f (I ⊕ k) = 0 for all k ∈ Z+ . Thus, (b) ⇔ (c). It remains to prove that (c) ⇒ (a). Suppose that fˆ does not have periodic zeros. Then F(ω) :=



| fˆ(ω ⊕ k)|2 , ω ∈ R+ ,

k∈Z+

is positive and 1-periodic function. Moreover, since fˆ ∈ En−1 (R+ ) we see that F is constant on each Is(n−1) , 0 ≤ s ≤ p n−1 − 1. The application of Proposition 4.4 completes the proof of the theorem. Remark 4.1 In the proof of Theorem 4.6, it was established that if the function f ∈ L 2 (R+ ) has compact support and the system of integer translates of this function is linearly dependent, then on the interval [0, 1) there exists a dyadic interval I , whose all points are periodic zeros for fˆ. Moreover, if supp f ⊂ [0, 2n−1 ], then the interval I has rank n − 1 and any periodic zero ω0 ∈ [0, 1) for f is located in such interval I . For an arbitrary M ⊂ [0, 1), we set T M :=

  1  1 1 M + M , 2 2 2

where α + β M := {α + βx : x ∈ M}. For the mask m 0 of the scaling equation (4.36), we introduce notations N (m 0 ) := {ω ∈ [0, 1) : m 0 (ω) = 0}. Suppose that the set M ⊂ [0, 1) is representable as a union of dyadic intervals of rank n − 1 or coincides with one of these intervals. A set M is called blocking set (for mask  m 0 ) if it does not contain the interval [0, 2−n+1 ) and has the property T M ⊂ M N (m 0 )

120

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

(that is, the mask m 0 turns into zero at each point of the set T M\M). Obviously, each mask can have only a finite number of blocking sets. ( j) If the dyadic interval Ik = [2− j k, 2− j (k + 1)] lies on [0, 1), then we sometimes ( j) write Ik = Id1 d2 ···d j where 0d˙1 d˙2 · · · d j = 2− j k. If M is the blocking set for the mask m 0 , then for any dyadic interval Id1 d2 ···dn−1 ⊂ M each of the intervals I0d1 d2 ···d j and I1d1 d2 ···d j is located either in M, or in N (m 0 ). It is easy to see that for the mask from Example 4.2 the interval [ 21 , 1), in the case, a = 0 is a blocking set. Theorem 4.7 Let φ be the solution of the scaling equation (4.36) such that φ ∈ ˆ L 2c (R+ ) and φ(0) = 1. The system {φ(· ⊕ k) : k ∈ Z+ } is linearly dependent if and only if the mask m 0 has a blocking set. Proof By Theorem 4.4, we have suppφ ⊂ [0, 2n−1 ], and therefore φˆ ∈ En−1 . If the system {φ(· ⊕ k) : k ∈ Z+ } is linearly dependent, then the set of all periodic zeros of the function φˆ on [0, 1) is the blocking set for m 0 . Indeed, let ˆ + k) = 0, for almost all k ∈ Z+ }. M0 = {ω ∈ [0, 1) : φ(ω According to Remark 4.1, the set M0 is the union of dyadic intervals of rank n − 1. ˆ Since φ(0) = 1,then M0 does not contain [0, 2−n+1 ). In addition, if ω ∈ M0 , then according to (4.38) we have  m0

   ω k ω k φ = 0, for all k ∈ Z+ , + + 2 2 2 2

and, consequently, the numbers ω2 , ω2 + 21 belong to either M0 or N (m 0 ). Conversely, ˆ + k) = if the mask m 0 has a blocking set M, then each ω ∈ M has the property: φ(ω 0, k ∈ Z+ , hence, φˆ has periodic zeros and (by Theorem 4.6) the system {φ(· ⊕ k) : k ∈ Z+ } is linearly dependent. In fact, presuppose that there is ω ∈ M such that ˆ + k) = 0 for some k. Then, we take so high j that 2− j (ω + k) < 21−n and for φ(ω each r ∈ {0, 1, . . . , j} denote the fractional part of the number 2−r (ω + k) through xr . Obviously, x0 = ω and x j = 2− j (ω + k). We have, ˆ + k) = φ(2 ˆ − j (ω + k)) φ(ω

j  r =1

ˆ j) m 0 (2−r (ω + k)) = φ(x

j 

m 0 (xr ).

(4.51)

r =1

It is easy to see that if xr ∈ M, then xr +1 ∈ T M and, therefore, xr +1 belongs to either / N (m 0 ) (otherwise, from (4.51) we would have N (m 0 ), or M. We also note that xr ∈ ˆ + k) = 0). Thus, if xr ∈ M, then xr +1 ∈ Mr and since x0 = ω ∈ M, the equality φ(ω then xr ∈ M for all 1 ≤ r ≤ j. However, this is impossible since x j = M. Indeed, x j = 2− j (ω + k) < 21−n , but M does not contain points of the interval [0, 21−n ). This completes the proof of the theorem. The question of the existence of a blocking set by Theorem 4.7 leads to the verification of some to the verification of some combinatorial fact, which can be verified, at least

4.3 Orthogonal Wavelets with Compact Support on R+

121

theoretically, in finite time by simple brute force. In practice, however, for large n n−1 this search can take considerable time to perform the order of 22 operations. In connection with this, we formulate the following two corollaries from Theorem 4.7. Let us call a number ω symmetric zero of the mask m 0 , if m 0 (ω) = m 0 (ω + 21 ) = 0. Corollary 4.1 If the mask m 0 of Eq. (4.36) has a symmetric zero, then it has a blocking set. Indeed, if ω = 0 · d1 · d2 · · · dn · · · is a symmetric zero, then the dyadic interval Id2 ···dn is a blocking set. Corollary 4.2 If m 0 ( 21 − 21n ) = 0, then the system of entire shift of the solution φ of the scaling equation (4.36) is linearly dependent. In fact, if m 0 ( 21 − 21n ) = 0, then the interval [1 − 21−n , 1) is a blocking set. According to the following proposition and Theorem 4.7, the condition m 0 ( 21 ) is necessary so that the mask of the scaling equation (4.36) does not have blocking sets. Proposition 4.8 If the system of entire shifts L 2 of the solution φ of the scaling equation (4.36) is linearly independent, then   2n−1 2n−1 −1 −1 1 = 0, m0 c2k = c2k+1 = 1. 2 k=0 k=0

(4.52)

Proof Suppose that m 0 ( 21 ) = 0. Using property 1 of Theorem 4.4, for any k ∈ Z+ , we have         1 1 1 1 ˆ ˆ ˆ = m0 . 0 = φ(2k + 1) = m 0 k + φ k+ φ k+ 2 2 2 2 ˆ + 1 ) = 0. Thus, the number 1 is the periodic zero of the function φˆ And, hence, φ(k 2 2 which contradicts Theorem 4.6. Therefore, m 0 ( 21 ) = 0 and, by virtue of (4.34), the equalities (4.52) are true. The proposition is proved. Theorem 4.8 Let φ be the solution of the scaling equation (4.36) such that φ ∈ ˆ L 2c (R+ ) and φ(0) = 1. The system {φ(· ⊕ k) : k ∈ Z+ } is an orthonormal family if and only if the mask m 0 satisfies condition (4.41) and does not have blocking sets. Proof We set F(ω) :=



ˆ ⊕ l)|2 . |φ(ω

l∈Z+

As noted above (see (4.40)), the orthonormality in L 2 (R+ ) of the system {φ(· ⊕ k) : k ∈ Z+ } is equivalent to the condition F(ω) ≡ 1. Moreover, as in the proof of Proposition 4.6, we have

122

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

       2 1 2 1 ω ω  + |m 0 ω ⊕ ⊕ . F(ω) = |m 0 (ω)  F F 2 2  2 2

(4.53)

1. Necessity. The absence of blocking sets follows from Theorem 4.7 and (4.41) from the condition F(ω) ≡ 1 by (4.53), (4.41) is obtained. 2. Adequacy. We set δ = in f F(ω)|ω ∈ [0, 1). By Theorem 4.8, we have F(0) = 1 and, hence, δ ≤ 1: The function F is nonnegative, has Period 1, and is constant on dyadic intervals of rank n − 1: Therefore, either δ > 0; or F converts into zero at one of these intervals (and then has a periodic zero). The latter is impossible by Theorem 4.6 since the system {φ(· ⊕ k) : k ∈ Z+ } is linearly independent. Thus, 0 < δ ≤ 1. We set Mδ = {F(ω) = δ : ω ∈ [0, 1)}. If 0 < δ < 1, then it follows from (4.41) and (4.53) that for any ω ∈ Mδ numbers ω2 and ω2 + 21 belong to either Mδ , or N (m 0 ). Hence, Mδ is a blocking set, which contradicts the condition. Thus, F(ω) ≥ 1 for all ω ∈ [0, 1). From this equation 

1

F(ω)dω =

0

 l∈Z+

By Theorem 4.4, we obtain



1

l

l+1

2 ˆ ˆ 22 |φ(ω)| dω = φ L (R+ ) .

F(ω)dω = 1.

0

Repeating the inequality F(ω) ≥ 1 and using the fact that the function F is constant on dyadic intervals of rank n − 1, we conclude that F(ω) ≡ 1 The theorem is proved. We recall that the family {[2− j )} is a fundamental system of neighborhoods of zero of the dyadic topology on R+ . The set E on R+ that is compact with respect to the dyadic topology is said to be W -compact. It is easy to see that the union of a finite number of dyadic intervals is W compact. Suppose that the set E is W compact. The set E is said to be congruent [0, 1) as per module Z+ (the notation: E ≡ [0; 1)(modZ+ )) if the Lebesgue measure of E is 1 and for every x ∈ [0, 1), there exists k ∈ Z+ , such that x ⊕ k ∈ E. We say that the Walsh polynomial m 0 satisfies the modified Cohen condition if there exists a W − compact set, E ⊂ R+ congruent [0, 1), modulo Z+ , containing a neighborhood of zero and such that inf inf |m 0 (2− j ω)| > 0. j∈Z ω∈E

The following theorem is an analogue of Cohen’s well-known theorem.

4.3 Orthogonal Wavelets with Compact Support on R+

123

Theorem 4.9 Let the Walsh polynomial 1 m 0 (ω) = ck wk (ω) 2 k=0 2n−1

(4.54)

satisfies the conditions     1 2 1  m 0 (0) = 1 |m 0 (ω)| + m 0 ω + = 1 for all ω ∈ [0, ),  2 2 2

(4.55)

and the function φ ∈ L 2 (R+ ) is defined with the help of the formula ˆ φ(ω) =

∞ 

m 0 (2− j ω).

(4.56)

j=1

The system {φ(· ⊕ k) : k ∈ Z+ } is an orthonormal family in L 2 (R+ ) then and only then, when polynomial m 0 is satisfied the modified condition of Coehen. Proof Note that φ belongs to class L 2 (R+ ) by Theorem 4.5. According to (4.56), ˆ ˆ ω )m 0 (( ω )) equivalent (4.36). It means that the given we have equality of φ(ω) = φ( 2 2 function φ satisfies scaling equation (4.36), mask of which is the polynomial (4.54). For each ω, all the multipliers of the work (4.56), starting with some numbers, are equal to 1 as a mask m 0 is equal to 1 on I0(n) as 2− j ω → 0 when j → ∞. From here it follows that φˆ is W -continuous function (noted in the Proposition 4.1 W -continuity of the function φˆ follows from the Theorem 4.4). Let the system {φ(· ⊕ k) : k ∈ Z+ } be orthonormal. As per (4.40), for every ω ∈ [0, 1), there exists a number lω such that, lω  1 ˆ ⊕ l)|2 > . |φ(ω 2 l=0 Since φˆ is a W -continuous function, for every ω ∈ [0, 1), such a dyadic interval Iω can be found, that lω  1 ˆ ⊕ l)|2 ≥ . |φ(t 4 l=0 For every t ∈ Iω . By the W compactness of the interval [0, 1) from the covering {Iω : ω ∈ [0, 1)}, a finite sub-covering {Iω1 , . . . , IωL } can be distinguished. Assuming that l0 = max{Iω1 , . . . , IωL }, then the inequality l0  l=0

ˆ ⊕ l)|2 ≥ |φ(ω

1 4

(4.57)

124

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

is true for all ω ∈ [0, 1). Let c0 =

According to (4.57), for each ω ∈ [0, 1), a ˆ ⊕ l)| ≥ c0 . As φ(0) ˆ = 1 and number l ∈ {0, 1, . . . , l0 } can be found such that |φ(ω function φˆ is W -continuous, the set √1 . 2 l0 +1

ˆ ≥ c0 } S0 := {ω ∈ [0, 1) : |φ(ω)| contains a neighborhood of zero. Let us assume ˆ ⊕ 1)| ≥ c0 } S1 := {ω ∈ [0, 1)\S0 : |φ(ω ˆ ⊕ 2)| ≥ c0 } S2 := {ω ∈ [0, 1)\(S0 ∪ S1 ) : |φ(ω ·······································   l0  ˆ ⊕ l0 )| ≥ c0 . Sl : |φ(ω St0 := ω ∈ [0, 1)\ l=0

0 The set E 0 = ll=0 (Sl ⊕ l) contains neighborhood of zero, which is W -continuous and congruent [0, 1), as per modulo Z+ . Using the fact that the polynomial m 0 (ω) is equal to 1 in neighborhood of zero, we choose j0 in such a way that, m 0 (2− j ω) = 1 for all j > j0 , ω ∈ E 0 .

(4.58)

In view of (4.56) ˆ |φ(ω)| =

j0 

ˆ − j0 ω)|, |m 0 (2− j ω)| · |φ(2

(4.59)

j=1

ˆ where |φ(ω)| ≥ c0 for ω ∈ E 0 . As |m 0 (ω)| ≤ 1. For any ω, it follows from (4.58) and (4.59) that |m 0 (2− j ω)| ≥

j0 

|m 0 (2−l ω)| ≥ c0 > 0, for all 1 ≤ j ≤ j0 , ω ∈ E 0 .

(4.60)

l=1

From (4.58) and (4.60), we acquire, that inf inf |m 0 (2− j ω)| > 0. j∈N ω∈E 0

(4.61)

Conversely, we assume that the polynomial m 0 (ω) in formula (4.56) satisfies conditions (4.61) and (4.56). We show that the system {φ(· ⊕ k) : k ∈ Z+ } is an orthonormal family in L 2 (R+ ). For every l ∈ N, we assume μ[l] (ω) =

l  j=1

m0

ω 2j

1E

ω 2j

, ω ∈ L 2 (R+ ),

4.3 Orthogonal Wavelets with Compact Support on R+

125

where E is the set from (4.61). As per the condition, E contains a neighborhood of zero and m 0 (ω) = 1 for all ω ∈ [0, 2−n ). Therefore, from (4.56) it follows that ˆ ω ∈ R+ . lim μ[l] (ω) = φ(ω),

(4.62)

l→∞

Also by (4.61) and (4.55), there exists a number j0 such that m 0 ( 2ωj ) = 1, j > j0 , ω ∈ E. Thus, j0  ˆ φ(ω) = m 0 (2− j ω), ω ∈ E. j=1

By (4.61), there is a constant c1 > 0 such that, |m 0 ( 2ωj )| ≥ c1 , j ∈ N, ω ∈ E. And so −j

ˆ c1 0 |φ(ω)| ≥ 1 E (ω), ω ∈ R+ . Therefore, |μ[l] (ω)| =

l  l   ω   ω   ω    ω     ˆ    − j0 1 ≤ c   φ l  , m 0 m E 0 1 j l j 2 2 2 2 j=1 j=1

which by (4.56) yields −j

ˆ |μ[l] (ω)| ≤ c1 0 |φ(ω)| for all l ∈ N, ω ∈ R+ . Now, for l ∈ N, we define



Il (s) :=

R+

(4.63)

|μ[l] (ω)|2 χ (s, ω)dω, s ∈ Z+ .

Setting El := {ω ∈ R+ : 2−l ω ∈ E} and ξ = p −l ω, we have Il (s) =

  l El

ω |m 0 ( j )|2 χ (s, ω)dω = 2l 2 j=1

 |m 0 (ξ )| E

2

l−1 

|m 0 (2 j ξ )|2 χ (s, 2l ξ )dξ.

j=1

(4.64) Where the last integrand is 1-periodic. Using the assumption E ≡ [0, 1)(modZ+ ) and changing the variable we get from (4.64). 

1

Il (s) = 2l−1 0

l  0

 i ξ |m 0 ( + )|2 |m 0 (2 j−1 ξ )|2 χ (s, 2l−1 ξ )dξ. 2 2 j=1 l−1

Therefore, in view of (4.55), 

l−2 1

Il (s) = 2l−1 0

j=0

|m 0 (2 j ξ )|2 χ (s, 2l−1 ξ )dξ

126

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

which by (4.64) becomes Il (s) = Il−1 (s). Since



1

Il (s) = p

 |m 0 (ξ )|2 χ (s, pξ )dξ =

0

1

χ (s, ξ )dξ = δ0,s .

0

We get Il (s) = δ0,s l ∈ N, s ∈ Z+ .

(4.65)

In particular, for all l ∈ N  Il (0) =

R+

|μ[l] (ω)|2 dω = 1.

By (4.62) and Fatou’s lemma, we then obtain  R+

2 ˆ |φ(ω)| dω ≤ 1.

Thus, from (4.62) and (4.63) by Lebesgue’s dominated convergence theorem it follows that  2 ˆ |φ(ω)| χ (s, ω)dω = lim Il (s). l→∞

R+

Hence, by (4.65) and 

 R+

φ(x)φ(x ⊕ k)d x =

we have

R+

2 ˆ |φ(ω)| χ (k, ω)dω, k ∈ Z+

 R+

φ(x)x ⊕ sd x = δ0,s , s ∈ Z+ .

The theorem is proved. The necessary and sufficient conditions, under which the solutions of the scaling equation (4.36) generate MRA, are contained in the following theorem. Theorem 4.10 Let φ be the solution of the scaling equation (4.36) such that φ ∈ ˆ L 2c (R+ ) and φ(0) = 1. Let us suppose that the mask m 0 of this solution is φ which satisfies the conditions (4.41). Then, the following three statements are equivalent: (a) The function φ generates MRA in L 2c (R+ ). (b) The mask m 0 does not have blocking sets. (c) The mask m 0 satisfies the modified Cohen condition.

4.3 Orthogonal Wavelets with Compact Support on R+

127

Proof Implications (a) ⇒ (b) and (a) ⇒ (c) follow from Theorems 4.8 and 4.9. To prove the converse implications, let us suppose that for m 0 any of the conditions (b) or (c) is completed. From Theorems 4.8 and 4.9, it follows that the system {φ(· ⊕ k) : k ∈ Z+ } is orthonormalized.  We define {V j } by the formula (4.11). According to Proposition 4.6, we have V j = {0}. The embedding V0 ⊂ V1 follows from the fact that φ satisfies Eq. (4.36). Taking (4.11) into account, we therefore have V j ⊂ V j+1 for all j ∈ Z. It remains to prove that  Or, equivalently,

V j = L 2 (R+ ).



Vj

⊥

= {0}.

(4.66)

 Let f ∈ ( V j )⊥ . For the given ε > 0, we choose a dyadic-integer function u ∈ 1 L (R+ ) ∩ L 2 (R+ ) such that f − u < ε (here and further through · signifies the norm of space L 2 (R+ )). Then, for any j ∈ Z+ for any orthogonal projection P j f of the function f on V j , we have

P j f 2 = (P j f, P j f ) = ( f, P j f ) = 0 and consequently,

P j u = P j ( f − u) ≤ f − u < ε.

(4.67)

Next, take a natural j so large that supp uˆ ⊂ [0, 2 j ) and 2− j ω ∈ [0, 2−n+1 ) for all ω ∈ ˆ − j ω) (such j exists in virtue of Proposition supp uˆ and then put g(ω) = supp uˆ φ(2 j 2 j 4.5). Since the system {2− 2 χ (2− j k, ·)}∞ k=0 is orthonormal basis in L [0, 2 ], we have 

|ck (g)| = 2 2

−j

 R+

 2j 0

2j

|g(ω)|2 dω,

0

k∈Z+

where ck (g) = 2− j



g(ω)χ (2− j k, ω)dω. Noting that

ˆ − j ω)χ (2− j k, ω). φ(2 j x ⊕ k)χ (x, ω)d x = 2− j φ(2

By the Plancherel formula, we obtain −j

2 2 (u, φ j,k ) = 2− j

 0

Therefore, in view of (4.68),

2j

g(ω)χ (2− j k, ω)dω.

(4.68)

128

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions



P j u = 2

 |(u, φ j,k )| = 2

2j

ˆ − j ω)|2 dω. |u(ω) ˆ φ(2

(4.69)

0

k∈Z+

Since m(ω) = 1 on I0(n) and since 2− j ω ∈ [0, 2−n+1 ) for ω ∈ supp uˆ from formula ˆ φ(ω) =

∞ 

m 0 (2− j ω)

j=1

ˆ − j ω) = 1 for all ω ∈ supp u. it follows that φ(2 ˆ Considering that supp uˆ ⊂ [0, 2 j ), from (4.67) and (4.68) by the Parseval theorem, we obtain ˆ = u . ε > P j u = u It means f < ε + u < 2ε and therefore it is true (4.66). The theorem is proved. Applying Theorems 4.2 and 4.10, we obtain the following algorithm for constructing orthogonal wavelets in L 2 (R+ ). Algorithm A0 • Step-1 Select the numbers b0 , b1 , . . . , b2n −1 , satisfying the condition b0 = 1, |bl |2 + |bl+2n−1 |2 = 1, 0 ≤ l ≤ 2n−1 − 1.

(4.70)

• Step-2 Calculate the coefficients ck by formulas ck =

1 2n−1

n 2 −1

bl wl (k2−n ), 0 ≤ k ≤ 2n − 1.

l=0

• Step-3 For the Walsh polynomial 2 −1 1 m 0 (ω) = ck wk (ω), ω ∈ R+ , 2 k=0 n

verify the feasibility of conditions (b) or (c) of Theorem 4.10. • Step-4 Find the function φ ∈ L 2 (R+ ) such that ˆ φ(ω) =

∞ 

m 0 (2− j ω), ω ∈ R+ .

j=1

• Step-5 Determine the function ψ by the formula

(4.71)

4.3 Orthogonal Wavelets with Compact Support on R+

ψ(x) =

n 2 −1

(−1)k c¯k⊕1 φ(2x ⊕ k), x ∈ R+ .

129

(4.72)

k=0

Step-2 of the algorithm A0 is realized using the fast Walsh transform. Step-3 will be implemented if the numbers bl in Step-1 are chosen so that bl = 0 for 0 ≤ l ≤ 2n−1 − 1. In this case, m 0 (ω) = 0 on [0, 21 ) and condition (c) of Theorem 4.10, that is, the modified Cohen condition, is satisfied with E = [0, 1). Note that if there are blocking sets for the polynomial m 0 found at Step-2, then the function ψ defined by the algorithm A0 generates the Parseval Frame in L 2 (R+ ) (for more details, see Chap. 5). Let us explain how to implement Step-4. For the values of bs selected in Step-1, we set γ (i 1 , i 2 , . . . , i n ) = bs , if s = i 1 20 + i 2 21 + · · · + i n 2n−1 , i j ∈ {0, 1}, and the numerical sequence {dl : l ∈ N} as follows. Let l=

k 

μ j 2 j , μk = 1, k = k(l) ∈ Z+ ,

(4.73)

j=0

then dl = γ (μ0 , 0, 0, . . . , 0, 0), i f k(l) = 0; dl = γ (μ1 , 0, 0, . . . , 0, 0)γ (μ0 , μ1 , 0, . . . , 0, 0), i f k(l) = 1; ··········································· dl = γ (μk , 0, 0, . . . , 0, 0)γ (μk−1 , μk , 0, . . . , 0, 0) · · · γ (μ0 , μ1 , μ2 , . . . , μn−2 , μn−1 ).

If k = k(l) ≥ n − 1. In the last product, the indices of each factor, starting from the second one, are obtained by “shifting” the indices of the previous factor by one position to the right and adding one new digit to the vacant first place from the binary expansion of the number l. We set M = M0

 {1, 2, . . . , 2n−1 − 1},

where M0 is the set of integers l ≥ 2n−1 , for which in the above binary expansion among the sets {μ j , μ j+1 , μ j+2 , . . . , μ j+n−1 }, there is no set (0, 0, . . . , 0, 1). Step-4 is justified by the following proposition. Proposition 4.9 Suppose that the polynomial m 0 and the numbers dl are defined by the values of bs under the condition (4.70) as indicated above. If the function φ is given by its Fourier transform according to formula (4.71), then it belongs to the

130

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

class L 2 (R+ ) and

⎧ 1 ⎪ ⎨1, ω ∈ [0, 2n−1 ), l ˆ φ(ω) = dl , ω ∈ [ 2n−1 , (l+1) ), l ∈ M, 2n−1 ⎪ ⎩ (l+1) l / M. 0, ω ∈ [ 2n−1 , 2n−1 ), l ∈

(4.74)

Proof By Parseval’s theorem and Theorem 4.5, we obtain that it belongs to the class L 2 (R+ ). If ω ∈ I0(n−1) , then 2− j ω ∈ I0(n) for all j ∈ N. But m 0 (ω) ≡ 1 on I0(n) . This and (4.71) imply the first equality in (4.74). We take l ∈ N with the expansion (4.73) and find the smallest number j0 such that 2 j0 > l + 1. Then, for any ω ∈ I0(n−1) for j ≤ j0 , we have 2 j0 ω ∈ I0(n) , and hence m 0 (2− j ω) = 1. Hence, according to (4.71), we obtain j0 −1  ˆ φ(ω) = m 0 (2− j ω), ω ∈ Il(n−1) . (4.75) j=1

In the case, l = μ0 (that is, when k(l) = 0), in formula (4.75) there will be j0 = and 2− j ω ∈ I0(n) . Therefore, if l ∈ {1, 2, . . . , p − 1} and ω ∈ I0(n−1) , 2, 2−1 ω ∈ Iμ(n) 0 then ˆ φ(ω) = b1 = γ (μ0 , 0, 0, . . . , 0, 0) = dl . Similarly, in the case k(l) = 1, we have m 0 (2−1 ω) = γ (μ1 , 0, 0, . . . , 0, 0), m 0 (2−2 ω) = γ (μ0 , μ1 , 0, . . . , 0, 0), . . . , ˆ and, hence, φ(ω) = dl . In the general case, from formula (4.73) for any j ∈ N, it follows that l 2n= j−1

=

 μ j−2 1  μ0 k−n + μ . + · · · + + μ 2 + · · · + μ 2 j−1 j k 2n 2 j−1 2 (n+ j−1)

In addition, if ω ∈ Il(n−1) then 2− j ω ∈ Il . If l ∈ / M then among the factors in (4.71) there is a zero (since by virtue of (4.70)). If, however, l ∈ M then in formula (4.71) there will be , 2−2 ω ∈ Iμ(n) , . . . , 2 j0 ω ∈ Iν(n) , 2−1 ω ∈ Iμ(n) k 0 k−1 +2μk where ν0 = μ j0 −1 + μ j0 2 + · · · + μk 2k−n . Hence, taking into account the periodicity of the polynomial m 0 , formula (4.74) follows. The proposition is proved. In connection with step-5, we note that under the conditions of Theorem 4.10 the orthogonal wavelet with respect to the scaling function φ is not uniquely determined. Namely, if for the Walsh polynomial 2 −1 1  (1) c wk (ω) m 1 (ω) = 2 k=0 k n

4.3 Orthogonal Wavelets with Compact Support on R+

matrix

131

m (ω) m (ω ⊕ 1 ) 0 0 2 m 1 (ω) m 1 (ω ⊕ 21 )

is unitary for all ω ∈ R+ , then the burst ψ can be given by formula ψ(x) =

n 2 −1

ck(1) φ(2x ⊕ k), x ∈ R+ .

k=0

We write formula (4.74) in the form ˆ φ(ω) = 1[0,

1 2n−1

) (ω)

+



dl 1[0,

1 2n−1

) (ω

⊕ 2−n+1 l).

(4.76)

l∈M

Because  R+

1[0,

1 2n−1

−n+1 l)χ (x, ω)dμ(ω) = χ (x, 2−n+1 l) ) (ω ⊕ 2

= 2−n+1 1[0,1) (2



1 2n−1

0 −n+1

χ (x, ω)dμ(ω)

x)χ (2−n+1 x, l)

= 2−n+1 1[0,1) (2−n+1 x)wl (2−n+1 x), the formal application of the inverse Fourier transform to (4.76) leads to the decomposition  φ(x) =

1 2n−1



   x   x   1[0,1) n−1 dl wl n−1 1+ , x ∈ R+ . 2 2 l∈M

(4.77)

We denote C A(n) 0 , the class of all scaling functions φ defined by steps 1–4 of algorithm A0 for which the series (4.77) converges absolutely and uniformly on the interval [0, 2n−1 ). Every function φ ∈ C A(n) 0 satisfies a scalable equation of the form (4.36) ˆ ω ). The Lang’s scaling function from ˆ since by (4.71), we have φ(ω) = m 0 ( ω2 )φ( 2 Example 4.2 belongs to the class C A(2) (R ) since for n = 2 the expansion (4.77) + 0 coincides with (4.46). Any binary-integer scalable function belongs to the class C A(2) 0 (R+ ) (in this case, the expansion (4.77) contains a finite number of nonzero terms). The scaling function φ from the following example does not belong to the class C A(3) 0 (R+ ). Example 4.3 Let n = 3 and the function φ is determined by the algorithm A0 with the parameters 1 b0 = 1, b4 = 0, bl = √ , for all the remaining l. 2

132

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

Grouping the coefficients of the series (4.77) with respect to the number of factors γ (μ j , μ j+1 , μ j+2 ) included in the definition dl , l ∈ M, we note that the number of coefficients in the k-th group is equal to Fk+1 from the Fibonacci sequence F0 = 0, F1 = F2 = 1, Fk+1 = Fk + Fk−1 , k = 2, 3, . . . . In addition, dl ≥ 2

(k+1) 2

for 2k ≤ l < 2k + 1. Hence, since Fk+2 lim = k→∞ Fk+1

√ 5+1 , 2

on the basis of Dalember’s theorem, it is easy to conclude that for the chosen values of the parameters bl , the series (4.77) diverges at the point x = 0. Therefore, the series (4.77) does not converge absolutely. The following example shows that for any n ≥ 3 there are Walsh series of the form (4.77) with a finite number of nonzero summands. Example 4.4 For an arbitrary n ≥ 3, we set ⎧ 1, ⎪ ⎪ ⎪ ⎨α, m 0 (ω) = ⎪ 0, ⎪ ⎪ ⎩ β, where 0 ≤ |α| < 1, |β| =



1 ω ∈ [0, 21 − 2n−1 ) ∪ [ 21 − 21n , 21 ), 1 , 21 − 21n ), ω ∈ [ 21 − 2n−1 1 1 1 ω ∈ [ 2 , 2 − 2n ) ∪ [1 − 21n , 1), 1 , 1 − 21n ), ω ∈ [1 − 2n−1

1 − |α|2 . For this mask with the help of (4.77), we obtain

2n−1  x   x  −3  x  1 + wl n−1 + αw2n−1 − 2 n−1 n−1 2 2 2 2 l=1  x   x  + w2n−1 −1 n−1 + βw2n −2 n . 2 2

φ(x) =

1

1 n−1 [0,1)

This function generates MRA in L 2 (R+ ) since condition (c) of Theorem 4.10 is satisfied for E = [0, 1 −

1 2n−2

) ∪ [1 −

1 2n−1

, 1) ∪ [2 −

1 2n−2

,2 −

1 2n−1

).

The corresponding orthogonal wavelet ψ is found from formula (4.72). The following exercise shows that the Lang surge ψ from Example 4.2 can be interpreted as a multiwavelet on R+ . Example 4.5 Let φ, ψ be defined s in Example 4.2 in such a way that

4.3 Orthogonal Wavelets with Compact Support on R+

133

φ(x) = c0 φ(2x) + c1 φ(2x ⊕ 1) + c2 φ(2x ⊕ 2) + c3 φ(2x ⊕ 3), ψ(x) = c¯1 φ(2x) − c¯0 φ(2x ⊕ 1) + c¯3 φ(2x ⊕ 2) − c¯4 φ(2x ⊕ 3), where c0 =

1+a+b 1+a−b 1−a−b 1−a+b , c1 = , c2 = , c3 = . 2 2 2 2

It is easy prove that the vector functions 

   φ(x) ψ(x) φ(x) = , ψ(x) = φ(x ⊕ l) ψ(x ⊕ 1) satisfy the equalities x x φ( ) = P0 φ(x) + P2 φ(x ⊕ 2), ψ( ) = Q 0 φ(x) + Q 2 φ(x ⊕ 2), 2 2 where

    1 c0 c1 1 c2 c3 , P2 = √ P0 = √ 2 c2 c3 2 c0 c1     1 c1 −c0 1 c3 −c2 , Q2 = √ . Q0 = √ 2 c3 −c2 2 c1 −c0

The dyadic derivative of the function f ∈ L 1 (R+ ) at the point x ∈ R+ is defined by the equality f [1] (x) = lim dl f (x), l→∞

where

l−1 

dl f (x) =

2 j−1 ( f (x) − f (x ⊕ 2− j−1 )).

j=−l+1

For any integer r ≥ 2, we put f [r ] (x) = ( f [r −1] (x))[1] .

4.4 Estimates of the Smoothness of the Scaling Functions In this section, we obtain estimates for the smoothness of the functions φ ∈ L 2c (R+ ), satisfying a scaling equation of the form

134

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

φ(x) =

n 2 −1

ck φ(2x ⊕ k), x ∈ R+ ,

(4.78)

k=0

and belonging to the class C A(n) 0 (R+ ). In this case, for the mask 2 −1 1 m 0 (ω) = ck wk (ω), 2 k=0 n

(4.79)

the conditions are fulfilled     1 2 = 1, for all ω ∈ R+ , m 0 (0) = 1, |m 0 (ω)|2 + m 0 ω + 2  and the coefficients of Eq. (4.78) are determined from the values bl = m 0 ( 2ln ), l = 0, 1, . . . , 2n , using the formula ck =

1 2n−1

n 2 −1

bl wl (k2−n ), 0 ≤ k ≤ 2n − 1.

l=0

We assume that the values of the parameters bl are real, although all estimates of smoothness given below are also preserved in the complex case. Since φ ∈ C A(n) 0 (R+ ) the expansion takes place    x   x   φ(x) = n−1 1[0,1) n−1 dl wl n−1 1+ , x ∈ R+ , 2 2 2 l∈M 1

(4.80)

where the coefficients are defined as in Proposition 4.9 and the series converges 1 absolutely and uniformly on [0, 2n−1 ). Recall that a dyadic metric on a positive half line is introduced by means of the distance ρ(x, y) = x ⊕ y, x, y ∈ R+ . Functions that are continuous with respect to this metric are said to be binary continuous or W -continuous. It is easy to show that the dyadic metric on the half line majorizes the Euclidean metric. Therefore, the topology generated by this metric is stronger than the Euclidean one. A sequence that converges in dyadic topology converges also in Euclidean topology, and functions that are continuous in the usual sense are also W -continuous. The Walsh functions wk (x) are dyadic continuous at each point x ∈ R+ . The dyadic modulus of continuity of a scaling function φ, satisfying Eq. (4.78) is defined by ω(φ, δ) := sup{|φ(x ⊕ y)| − φ(x) : x, y ∈ R+ , |y| < δ}, δ > 0. If the function φ is such that ω(φ, 2− j ) ≤ C2α j , j ∈ N for some α > 0, then there exists see [5] the constant C(φ, α), such that

4.4 Estimates of the Smoothness of the Scaling Functions

135

ω(φ, δ) ≤ C(φ, α)δ α .

(4.81)

We denote through αφ the upper bound of the set of all values α > 0, for which inequality (4.81) is satisfied. By Theorem 4.4 for a given function φ, we have suppφ ⊂ [0, 2n−1 ]. Therefore, an estimate of the quality αφ reduced to a study of a sequence ω j (φ) := sup{|φ(x) − φ(y)| : x, y ∈ [0, 2n−1 ), x ⊕ y ∈ [0, 2− j )} j ≥ 1 − n. This can be seen from formula (4.72) that the smoothness of the orthogonal wavelet ψ in L 2 (R+ ), corresponding to the scaling function φ coincides with αφ . We set N = 2n−1 . By Proposition 4.8, we have N −1 

N −1 

c2k =

k=0

c2k+1 = 1.

(4.82)

k=0

Recall that the left-side limit of function f at point a is denoted by f − (a), that is f − (a) := lim f (x). x→a −0

From formula (4.80) and easily verified equations N −1 

 wm

k=0

it follows that



k 2n−1 N −1  k=0

=

N −1 

wm−

k=0

φ(k) =

N −1 



k 2n−1

 = 0, m ∈ N,

φ − (k + 1) = 1.

(4.83)

k=0

For i, j, k ∈ {1, 2, . . . , 2N − 1} the equality k = i ⊕2 j by definition means that ks = i s + js (mod 2) for s ∈ {0, 1, . . . , n − 1}, where i s , js , and ks are the digits of binary expansions i=

n−1  s=0

i s 2s , j =

n−1  s=0

js 2s , k =

n−1 

k s 2s .

s=0

We define (N × N ) the matrices T0 and T1 by the formulas (T0 )i, j = c2(i−1)⊕2 ( j−1) and (T1 )i, j = c(2i−1)⊕2 ( j−1) ,

(4.84)

136

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

where i, j ∈ {1, 2, . . . , N }. In particular, for n = 2, we have    c0 c1 c1 c0 , T1 = , T0 = c2 c3 c3 c2 

where the coefficients ck are computed as in Example 4.2. Example 4.6 Suppose that n = 3 and b0 = 1, b1 = a, b2 = b, b3 = c, b4 = 0, b5 = α, b6 = β, b7 = γ . Then, for the coefficients ck , we have formulas c0 = c1 = c2 = c3 = c4 = c5 = c6 = c7 =

1 (1 + a + b + c + α + β + γ ), 16 1 (1 + a + b + c − α − β − γ ), 16 1 (1 + a − b − c + α − β − γ ), 16 1 (1 + a − b − c − α + β + γ ), 16 1 (1 − a + b − c − α + β − γ ), 16 1 (1 − a + b − c + α − β + γ ), 16 1 (1 − a − b + c − α − β + γ ), 16 1 (1 − a − b + c + α + β − γ ). 16

In the case, the matrices (4.84) have the form ⎛ c0 ⎜c2 T0 = ⎜ ⎝c4 c6

c1 c3 c5 c7

c2 c0 c6 c4

⎞ ⎛ c3 c1 c0 ⎜c3 c2 c1 ⎟ ⎟,T = ⎜ c7 ⎠ 1 ⎝c5 c4 c5 c7 c6

c3 c1 c7 c5

⎞ c2 c0 ⎟ ⎟. c6 ⎠ c4

Note that for a = 0 and c = 0, the blocking sets for the mask 1 ck wk (ω) 2 k=0 7

m 0 (ω) =

are the intervals [ 41 , 1) and [ 43 , 1), respectively. If a and c are nonzero, then the mask m 0 satisfies the modified Cohen condition (with interval [0, 1) for b = 0 and with the set [0, 21 ) ∪ [ 43 , 1) ∪ [ 23 , 74 ) for b = 0). Let e1 = (1, 1, . . . , 1) be an N-dimensional vector with all components equal to 1. According to (4.82), we have e1 T0 = e1 T1 = e1 .

(4.85)

4.4 Estimates of the Smoothness of the Scaling Functions

137

For any two N -dimensional vectors v = {v1 , . . . , v N } and w = {w1 , . . . , w N }, we set v · w := t

N 

v j w j , and v :=

√ v · vt ,

j=1

where vt and wt are the columns vector obtained from the row vectors v and w by transposition. We denote through E 1 the space of N -dimensional column vectors orthogonal to the vector e1 : E 1 := {u = (u 1 , . . . , u N )t : u 1 + · · · + u N = 0}. For an arbitrary real (N × N ) matrix M, we set

M := sup{

Mu : u ∈ R+ , uc = 0}

u

and

M| E1 := sup{

Mu : u ∈ E 1 , u = 0}.

u

It is well known that the value M coincides with the square root of the largest eigenvalue of the matrix M t M. The following Proposition holds: Proposition 4.10 Suppose that the function φ satisfies Eq. (4.78) and belongs to the class C A(n) 0 (R+ ). Suppose that the elements (N × N ) matrices T0 , T1 have the form (4.84). If for all m ∈ N max{ Td1 Td2 · · · Tdm | E1 : d j ∈ {0, 1}, 1 ≤ j ≤ m} ≤ Cq m ,

(4.86)

where 0 < q < 1 and C > 0, then for any integers j ≥ n the inequality ω j (φ) ≤ Cq j .

(4.87)

Proof For any x ∈ [0, 1) and {0, 1, . . . , N − 1}, we set φ0 (x ⊕ k) := (1 − x)φ(k) + xφ − (k).

(4.88)

The sequence of vector functions v j (x) for j ∈ Z+ , x ∈ [0, 1) we define the equalities v0 (x) := (φ0 (x), φ0 (x ⊕ 1), . . . , φ0 (x ⊕ (N − 1)))t ,  v j+1 (x) :=

T0 v j (2x), if x ∈ [0, 21 ), T1 v j (2x ⊕ 1), if x ∈ [ 21 , 1).

138

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

For each x ∈ [0, 1) having double expansion x=

∞ 

x j 2 j+1 , x j ∈ {0, 1}.

j=0

Set d j (x) = x j (if the number x is dyadic rational, then among the digits d j (x) only a finite number of nonzero ones). Let  τ x :=

2x, if x ∈ [0, 21 ), 2x ⊕ 1, if x ∈ [ 21 , 1).

Then v j+1 (x) = Td1 (x) v j (τ x), and, consequently, v j (x) = Td1 (x) Td2 (x) · · · Td j (x) v0 (τ j x).

(4.89)

From (4.83) and (4.88), we find e1 · v0 (x) =

N −1 

φ0 (x ⊕ k)

k=0

= (1 − x)

N −1 

φ(k) + x

k=0

N −1 

φ − (x)

k=0

= 1. From this and (4.85) and (4.89), we obtain e1 · v j (x) = e1 Td1 (x) Td2 (x) · · · Td j (x) v0 (τ j x) = e1 · v0 (τ j x) = 1. Consequently, for each l ∈ Z+ , we have e1 · (v j+1 (x) − vl (x)) = 0. Thus, for all x ∈ [0, 1) and for all j, l ∈ Z+ e1 · v j (x) = 1 and v j+1 (x) − vl (x) ∈ E 1

(4.90)

and according to (4.89), v j+1 (x) − v j (x) = Td1 (x) Td2 (x) · · · Td j (x) [vl (τ j x) − v0 (τ j x)]. For l = 1 from (4.86), (4.90), and (4.91), we have

(4.91)

4.4 Estimates of the Smoothness of the Scaling Functions

139

v j+1 (x) − v j (x) ≤ Cq j sup v1 (y) − v0 (y) . y∈[0,1)

Therefore,

v j (x) ≤ v0 (x) +



vl (x) − vl−1 (x)

l=1

≤ sup v0 (x) + C(1 − q)−1 sup v1 (y) − v0 (y) . y∈[0,1)

y∈[0,1)

Thus, the sequence {v j (·)} is uniformly bounded on [0, 1): sup{ v j (x) |x ∈ [0, 1), j ∈ Z+ } < ∞.

(4.92)

As above, for any x ∈ [0, 1) and l ∈ N in view of (4.86), (4.90), and (4.91), we have

v j+1 (x) − v j (x) ≤ Cq j sup v1 (y) − v0 (y) . y∈[0,1)

From this and (4.92) follow the inequality sup v j+1 (x) − v j (x) ≤ Cq j ,

(4.93)

y∈[0,1)

where C does not depend on l. Hence, the sequence {v j (·)} is fundamental in the space [C([0, 1))] N = C([0, 1)) × · · · × C([0, 1)). By (4.93), for the limit vector-valued function v˜ (·), we have sup ˜v(x) − v j (x) ≤ Cq j .

(4.94)

y∈[0,1)

Let v(x) := (φ(x), φ(x ⊕ h [1] ), . . . , φ(x ⊕ h [N −1] ))t . Then, for x ∈ [0, 1) v(x) = Td1 (x) v(τ x) and assuming j → ∞ in equality v j+1 (x) = Td1 (x) v j (τ x). We conclude that v˜ (x) = v(x). From this and (4.94) for all j ∈ Z+ , we have (4.95) sup φ(x) − φ j (x) ≤ Cq j . y∈[0,1)

We fix an integer j ≥ n and choose in [0, N ) of the numbers x=

∞  j=2−n

x j 2 j+1 and y =

∞  j=2−n

y j 2 j+1 .

140

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

So that xi = yi for 2 − n ≤ i ≤ j and x j+1 = y j+1 . It is seen that both the numbers x and y belong to the interval [ 2mj , (m+1) ), where 2j m=

j 

j 

xi 2 j−i =

i=2−n

yi 2 j−i .

i=2−n

According to (4.95), we have |φ(x) − φ(y)| ≤ |φ(x) − φ j (x)| + |φ j (x) − φ j (

m )| 2j

m ) − φ j (y)| + |φ j (y) − φ(y)| 2 j  m    m      + (y) − φ ≤ 2Cq j + φ j (x) − φ j φ  . j j 2j 2j + |φ j (

(4.96)

Suppose that x ∈ [0, 1) and x0 = x−1 = · · · = x2−n = 0. Then, m = x1 2 j−1 + x2 2 j−2 + · · · + x j 20 and, similarly to (4.91) v j (x) − v j (

m τ jm ) = Td1 (x) Td2 (x) · · · Td j (x) [v0 (τ j x) − v0 ( j )]. j 2 2

(4.97)

Taking into account that v0 (τ j ·) are uniformly bounded on [0, 1), and also that   m  m   ≤ (x) − v ( ) v  φ j (x) − φ j j j 2j 2j and

Td1 (x) Td2 (x) · · · Td j (x) ≤ Cq j from (4.97), we obtain the inequality   m     ≤ Cq j . φ j (x) − φ j 2j

(4.98)

Let now x ∈ [1, N ). We set x  = x ⊕ k and m  = m − k, where k = x0 2 j + x−1 2 j+1 + · · · + x2−n 2 j+n−2 . Then x  ∈ [0, 1) and v j (x  ) − v j (m  /2 j ) = Td1 (x  ) Td2 (x  ) · · · Td j (x  ) [v0 (τ j x  ) − v0 (

τ j m )]. 2j



Because |φ j (x) − φ j ( 2mj )| ≤ v j (x  ) − v j ( m2 j ) , we again come at the estimate (4.98). The last term in (4.98) is estimated similarly. Therefore, the inequality (4.87) is true and Proposition 4.10 is proved. We note that the vector-valued function v(x) is related to the matrices T0 and T1 by the formula

4.4 Estimates of the Smoothness of the Scaling Functions

 v(x) =

T0 v(2x), if x ∈ [0, 21 ), T1 v(2x ⊕ 1), if x ∈ [ 21 , 1).

141

(4.99)

From this formula for x = 0, it follows that the vector v(0) = (φ(0), φ(1), . . . , φ(N − 1))t is an eigenvector of the matrix T0 corresponding to the eigenvalue 1. Formula (4.99) is useful in calculating the values of the function φ. For example, for n = 2, the values φ(0) and φ(1) given in Example 4.2 are obtained from (4.83) and (4.99). Example 4.7 Let n = 2, 0 < |b| < 1, and let the function φ be given by (4.46). Then, we have the estimate ω j (φ) ≤ C|b| j , j ∈ N. (4.100) Indeed, in this case E 1 = {v ∈ R2 |v1 + v2 = 0} = {te10 |t ∈ R} and T0 e10 = be10 = −be10 , where e10 = (−1, 1)t . Therefore, inequality (4.100) follows from Proposition ab j 4.10. Because φ(0) − φ(2− j ) = 1−b , then the estimate (4.100) is exact in order. Example 4.8 Let φ ∈ C A(3) 0 (R+ ) and the coefficients ck be defined as in Example 4.6. Suppose that a = 1 and |γ | < 1. Then, using (4.83) and (4.99), we obtain the equations c(1 − β) b 1+c−γ − + , 2(1 − γ ) 4(1 − γ ) 4 1−c−γ c(1 − β) b φ(1) = + − , 2(1 − γ ) 4(1 − γ ) 4 b c(1 − β) φ(2) = − , 2 2(1 − γ ) b c(1 − β) φ(3) = − + . 2 2(1 − γ )

φ(0) =

We show that ω j (φ) ≤ C|γ | j , j ∈ N.

(4.101)

For n = 3, the basis of the space E 1 is formed by the vectors ⎛

⎛ ⎞ ⎛ ⎞ ⎞ 1 1 1 ⎜−1⎟ ⎜−1⎟ ⎜1⎟ ⎜ ⎟ ⎜ ⎟ ⎟ e1 = ⎜ ⎝ 1 ⎠ , e2 = ⎝−1⎠ , e3 = ⎝−1⎠ . −1 1 −1 Under the condition a = 1, we have

(4.102)

142

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

T0 e1 = T1 e1 = 0, T0 e2 = −T1 e2 = βe1 + γ e2 , T0 e3 = T1 e3 = be1 + ce2 . Using these equalities, for an arbitrary vector ν = ν1 e1 + ν2 e2 + ν3 e3 . Space E 1 , we obtain T02 w = −T1 T0 w = (ν2 γ + ν3 c)T0 e2 , T12 w = −T0 T1 w = (ν2 γ − ν3 c)T0 e2 , (4.103) and, in particular, T02 e2 = T12 e2 = −T0 T1 e2 = −T1 T0 e2 = γ T0 e2 . Therefore, for any d1 , d2 , . . . , dm ∈ {0, 1}, the following equality is correct: Td1 · Td2 · · · Tdm e2 = ±γ m−1 T0 e2 . Taking into account (4.103), we obtain

Td1 · Td2 · · · Tdm | E1 ≤ C|γ |m , and estimate (4.101) follows from Proposition 4.10. From formula (4.80) with a = 1, b = 0, |γ | < 1 for all j ∈ N, we derive the equality  φ(0) − φ

1 2j

 =

cγ j+3 . 4(1 − γ )

Thus, the estimate (4.101) is exact in order. The estimate is also preserved for γ = 0 since, in this case, φ(x) =

1 x 1[0,1) (y)(1 + w1 (y) + bw2 (y) + w3 (y) + βw6 (y)), y = 4 4

(See Example 4.4). We recall that the joint spectral radius of two complex matrices A0 and A1 of size N × N is determined by the formula 1

ρ(A ˆ 0 , A1 ) := lim max{ Ad1 Ad2 · · · Adk k : d j ∈ {0, 1}, 1 ≤ j ≤ k}, k→∞

where · is an arbitrary norm in C N ×N (a bibliographic reference on the joint spectral radius is given in the comments on Chap. 7 in see [1]). Obviously, if A0 = A1 , then value ρ(A ˆ 0 , A1 ) coincides with the spectral radius ρ(A0 ). The following two propositions take place: Proposition 4.11 For any (d × d) matrices A0 , A1 and for any q ≤ 0 the following conditions are equivalent: (1) there exists constants C > 0, p ≤ 0 such that for all m ∈ N, we have

4.4 Estimates of the Smoothness of the Scaling Functions

143

max{ Ad1 · · · Adm : d j ∈ {0, 1}, 1 ≤ j ≤ m} ≤ Cm p q m ; (2) ρ(A ˆ 0 , A1 ) ≤ q. Moreover, if any of these properties are fulfilled, then we can always take p ≤ d − 1, and if the matrices are irreducible (do not have nontrivial real common eigensubspaces), then p = 0. Proposition 4.12 Let the function φ and the matrices T0 , T1 be the same as in Proposition 4.10, and let ρˆ = ρ(L ˆ 0 , L 1 ) be the joint spectral radius of the linear operators L 0 L 1 given on R N by the matrices T0 , T1 and bounded on the subspaces E 1 = {u = (u 1 , . . . , u N )t : u 1 + · · · + u N = 0}. Then

1 αφ = log2 ( ). ρˆ

(4.104)

Analogues of Propositions 4.11 and 4.12 for wavelets on the line R have been proved by V. Yu. Protasov [4], and for dyadic wavelets on the half line R+ , the proofs are completely analogous. Formula (4.104) allows us to accurately find the Hölder exponent; however, its practical use is complicated by the need to calculate the joint spectral radius. It follows from this formula that the function φ is W -continuous, then, and only then ρˆ < 1. In the general case, if for the matrices T0 and T1 , the inequality is fulfilled max{ Td1 · · · Tdm | E1 : d j ∈ {0, 1}, 1 ≤ j ≤ m} ≤ Cm p q m , m ∈ N, where 0 ≤ q < 1, p ≥ 0 then ρ(L ˆ 0 , L 1 ) ≤ q. It follows from Proposition 4.12 and Example 4.7 that for the Lang scaling function 1 ) is correct (so that αφ → ∞ for b → 0 and αφ → 0 for the equality αφ = log2 ( |b| b → 1). We indicate a few more cases when the value αφ can be calculated exactly. Let n = 3 and the coefficients of the scaling equation (4.78) are defined as in Example 4.6 with respect to the parameters b0 = 1, b1 = a, b2 = b, b3 = c, b4 = 0, b5 = α, b6 = β, b7 = γ , where |a|2 + |α|2 = |b|2 + |β|2 = |c|2 + |γ |2 = 1. In this case, the matrices of the linear operators L 0 , L 1 defined on R4 by the matrices T0 and T1 and bounded on the subspaces E 1 , we have the following form in the basis (4.102): ⎛ ⎞ ⎛ ⎞ 0βb 0 −β b (4.105) A0 = ⎝ 0 γ c ⎠ , A1 = ⎝ 0 −γ c ⎠ α 00 −α 0 0 and in this case ρ(L ˆ 0 , L 1 ) = ρ(A ˆ 0 , A1 ). Theorem 4.11 Suppose that φ be the solution of (4.78) defined above for n = 3 and let ρˆ be the joint spectral radius of the matrices (4.105). Then

144

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

⎧√ 3 ⎪ if b = 0, |c| = 1, 0 ≤ |α| < 1, ⎨ |α|,√ ρˆ = max{ |α|, |γ |}, if |b| = 1, 0 ≤ |α| < 1, 0 ≤ |γ | < 1, ⎪ ⎩ |γ |, if |a| = 1, 0 ≤ |γ | < 1. and in this case αφ = log2 ( ρ1ˆ ). Proof Suppose that β = c = 1. Then, for the basis vectors (4.102), we have T0 e1 = −T1 e1 = αe3 , T0 e2 = −T1 e2 = e1 , T0 e3 = T1 e3 = e2 . We denote through mod (m, 3) the remainder when dividing m by 3. By induction, we verify m+2 T0m e1 = α [ 3 ] e3−mod (m+2,3) , m ∈ N, where [x] is the integer part of the number x. Similarly, T0m e2 = α [

m+1 3 ]

e3−mod (m+4,3) , T0m e3 = α [ 3 ] e3−mod (m+3,3) , m ∈ N. m

Hence, we obtain that m

T0m e j ≤ C|α| 3 , j = 1, 2, 3. Moreover, since T0 e j = ±T1 e j , similar estimates exist for the vectors Td1 · Td2 · · · Tdm e j . Therefore, for all m ∈ N m

Td1 · Td2 · · · Tdm | E1 ≤ C|α| 3 , d1 , d2 , . . . , dm ∈ {0, 1}, √ and the estimate ρˆ ≤ 3 |α| is valid (see Propositions 4.10 and 4.11). We prove the converse inequality. Since b = γ = 0, by the formula (4.80), we obtain the expansion φ(x) =

 " 1 1[0,1) (y) 1 + α w1 (y) + w3 (y) + w6 (y) + αw13 (y) + αw27 (y) + αw54 (y) 4 # + α 2 w109 (y) + α 2 w219 (y) + α 2 w438 (y) + α 3 w877 (y) + · · ·

,

where y = x4 . The absolute convergence of this series follows from the condition |α| < 1. It is also seen that φ(0) = In addition, if s = 3k, then

3a  1 1+ . 4 1−α

4.4 Estimates of the Smoothness of the Scaling Functions

φ(

145

" # 1 1 s−3 s s+3 3 − α3 − α 3 + ··· 1 + α 3 + 3α + · · · + 3α ) = 2s 4  1 s 1 + α(3 − 4α 3 )(1 − α)−1 . = 4

Further, if s = 3k + 1, then φ(

" # 1 1 s−4 s−1 s+2 s+5 3 + α 3 − α 3 − α 3 ··· 1 + a 3 + 3α + · · · + 3α ) = 2s 4  1 s−1 s+2 s−1 1 + a{(3(1 − α 3 ) − α 3 )(1 − α)−1 + α 3 } , = 4

and if s = 3k + 2, then φ(

" # 1 1 s−2 s+1 s+4 3 − α 3 − α 3 + ··· 1 + a 3 + 3α + · · · + 3α ) = 2s 4  1 s+1 1 + a{(3 − 4α 3 )(1 − α)−1 } . = 4

From the above expansions, we obtain      φ(0) − φ 1  = C|α| 3s , s ∈ N,  s 2  √ and, therefore, ρˆ ≥ 3 |α|. Cases β = −1, c = 1, β = 1, c = −1, and β = c = √ −1 are considered similarly. Therefore, if b = 0, |c| = 1, 0 ≤ |α| < 1, then ρˆ = 3 |α|. Let now b = 1. Then T0 e1 = −T1 e1 = αe3 , T0 e2 = −T1 e2 = γ e2 , T0 e3 = T1 e3 = e1 + ce2 . Therefore, for an even m T0m e1 = α 2 (e1 + ce2 ) + (γ 2 α 2 −1 + γ 4 α 2 −2 + · · · + γ m−2 α)ce2 , T0m e2 = γ m e2 , m

m

m

T0m e3 = α 2 e3 + (γ α 2 −1 + γ 3 α 2 −2 + · · · + γ m−1 α)ce2 m

and

m

m

146

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

 m  m m

T0m e1 ≤ C |α| 2 + |γ |2 |α| 2 −1 + |γ |4 |α| 2 −2 + · · · + |γ |m−2 |α|  Cm|γ |m , |α| ≤ |γ |2 , ≤ m Cm|γ | 2 , |α| ≥ |γ |2 ,

T0m em ≤ C|γ |m ,  Cm|γ |m , |α| ≤ |γ |2 , m

T0 em ≤ m Cm|α| 2 , |α| ≥ |γ |2 . Further, for odd m, we have  m−1  m+1 m−3 T0m e1 = α 2 e3 + γ α 2 + γ 3 α 2 + · · · + γ m−2 α ce2 , T0m e2 = γ m e2 , T0m e3 = α

m−1 2

  m−3 m−5 (e1 + ce2 ) + γ 2 α 2 + γ 4 α 2 + · · · + |γ |m−2 α ce2

and, as above,  m+1  m−1 m−3

T0m e1 ≤ C |α| 2 + |γ ||α| 2 + |γ |3 |α| 2 + · · · + |γ |m−2 |α|  Cm|γ |m , |α| ≤ |γ |2 , ≤ m Cm|γ | 2 , |α| ≥ |γ |2 ,

T0m em ≤ C|γ |m ,  Cm|γ |m , |α| ≤ |γ |2 , m

T0 em ≤ m Cm|α| 2 , |α| ≥ |γ |2 . Similar estimate also occurs for the vectors Td1 Td2 · · · Tdm e j . Therefore,  1 1

Td1 Td2 · · · Tdm | E1 m ≤ Cm m max{ |α|, |γ |}, m ∈ N, √ and, consequently, ρˆ ≤ max{ |α|, |γ |}. For the proof of the inverse inequality, we use the expansion  4φ(x) = 1 + α w1 (y) + w2 (y) + cw3 (y) + αw5 (y) + γ cw7 (y) + αw10 (y) + αcw11 (y) + γ 2 cw15 (y) + α 2 w21 (y) + αγ cw23 (y) + γ 3 cw31 (y) + α 2 w42 (y) + α 2 cw43 (y) + αγ 2 cw47 (y) + γ 4 cw63 (y) + α 3 w85 (y) + α 2 γ cw87 (y) + αγ 3 cw95 (y) + γ 5 cw127 (y) + α 3 w170 (y) + α 3 cw171 (y)  + α 2 γ 2 cw175 (y) + αγ 4 cw191 (y) + γ 6 cw255 (y) + α 4 w341 (y) + · · · , (4.106)

4.4 Estimates of the Smoothness of the Scaling Functions

where y =

x 4

147

and x ∈ [0, 4). Substitute in (4.106) the value of x = 0, we find φ(0) =

α  c  1 1+ 2+ . 4 1−α 1−γ

Further, it follows from (4.106) that if s is even, then  c(γ s − 1 ) #  s s s−2 2a " 1 1 2 1+ 1 − α 2 − c α 2 + γ 2 α 2 + · · · + αγ s−2 − , φ( s ) = 2 4 1−α 1−γ

and if s is odd, then φ(

 c(γ s − 1 ) #  s−1 s+1 s−3 2a " 1 1 2 1+ 1 − α 2 − cγ α 2 + γ 2 α 2 + · · · + αγ s−3 − . )= s 2 4 1−α 1−γ

Therefore,

    s    φ(0) − φ 1  = Cs max{ |α|, |γ |} , s ∈ N,   2s

√ √ and, ρˆ ≥ max{ |α|, |γ |}, Thus, ρˆ = max{ |α|, |γ |}.The case b = −1 is treated similarly. Derivation of necessary estimates for the case |a| = 1, 0 ≤ |γ | < 1 is contained in Example 4.8. The equality αφ = log2 ( ρ1ˆ ) follows from Proposition 4.12. The theorem is proved. For n = 4, the mask of Eq. (4.78) is written in the form 1 ck wk (ω). 2 k=0 15

m 0 (ω) =

(4.107)

Here, the coefficients ck are defined as in the algorithm A0 with respect to the parameters (4.108) b0 = 1, bl = βl , b8 = 0, bl+8 = γl , 1 ≤ l ≤ 7, where |βl |2 + |γl |2 = 1. We give a complete list of sets of parameter values, for which the mask (4.107) has blocking sets, and for each of these sets we indicate one blocking set as given below: (1) (2) (3) (4) (5)

β1 β6 β2 β2 β3

= 0, M = [ 18 , 1), = 0, M = [ 78 , 1), = β3 = 0, M = [ 41 , 21 ) ∪ [ 58 , 1), = γ5 = 0, M = [ 41 , 1) ∪ [ 58 , 34 ), = β5 = 0, M = [ 83 , 21 ) ∪ [ 85 , 1).

Also note that for n = 4 vectors

148

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

e1 = (1, −1, 1, −1, 1, −1, 1, −1)t , e2 = (1, 1, −1, −1, −1, −1, 1, 1)t , e3 = (1, 1, 1, 1, −1, −1, −1, −1)t , e4 = (1, −1, 1, −1, −1, 1, −1, 1)t , e5 = (1, 1, −1, −1, 1, 1, −1, −1)t , e6 = (1, −1, −1, 1, 1, −1, −1, 1)t , e7 = (1, −1, −1, 1, 1, −1, 1, −1)t , (4.109) for a basis of E 1 . We recall that T0 and T1 are matrices, determined by formulas (4.84) under the condition (4.108). Let us introduce the matrices ⎛

0 −β7 ⎜0 0 ⎜ ⎜γ1 0 ⎜ β7 B0 = ⎜ ⎜0 ⎜0 0 ⎜ ⎝ 0 β6 − β7 0 2β7

0 β3 0 0 β2 0 0

0 γ3 0 0 γ2 0 0

β4 0 0 β5 0 0 0

γ4 0 0 γ5 0 0 0

⎛ ⎞ 0 −β7 0 ⎜0 0⎟ 0 ⎜ ⎟ ⎜γ1 0⎟ 0 ⎜ ⎟ ⎜ 0⎟ β7 ⎟ B1 = ⎜ 0 ⎜0 0⎟ 0 ⎜ ⎟ ⎝ 0 β6 − β7 0⎠ 0 0 2β7

0 β3 0 0 β2 0 0

0 −γ3 0 0 −γ2 0 0

β4 0 0 β5 0 0 0

−γ4 0 0 −γ5 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

and denote by Γ matrix of size 7 × 7, for which the first six columns are zero, and the last column coincides with the vector (γ4 − γ7 , −γ3 , γ1 , γ5 + γ7 , −γ2 , γ6 − γ7 , γ7 )t . Then, the linear transformations y = T0 x, y = T1 x, x ∈ E 1 have in the basis (4.109) of matrices 1 1 A0 = B0 + Γ, A1 = B1 − Γ, 2 2 respectively. By Proposition 4.12, we have ρ(A ˆ 0 , A1 ) = 2−αφ . We indicate three cases when the values of αφ is calculated exactly. Theorem 4.12 Let the function φ ∈ C A(4) 0 (R+ ) satisfy Eq. (4.78), in which n = 4 and the coefficients are determined from the parameters (4.108). Then (1) αφ = ( 21 ) log2 ( |γ12 | ), if γ2 = 0, γ1 = γ5 = γ7 = 0; (2) αφ = ( 21 ) log2 ( |β51γ2 | ), if β5 γ2 = 0, γ1 = γ3 = γ7 = 0; (3) αφ = log2 ( |γ17 | ), if γ7 = 0, and one of the following conditions is fulfilled (a) γ1 = γ2 = γ3 = 0, (b) γ1 = γ2 = γ5 = 0, (c) β5 = γ1 = γ3 = 0. Proof Since φ ∈ C A(4) 0 (R+ ), the following expansion takes place   x  x   1 φ(x) = 1[0,1) dl wl 1+ , x ∈ R+ , 8 8 8 l∈M where the coefficients dl are defined as in Proposition 4.9. 10 : Let β1 = β5 = β7 = 1 and γ2 = 0. Then (4.110) is written as

(4.110)

4.4 Estimates of the Smoothness of the Scaling Functions

149

 8φ(x) = 1[0,1) (y) 1 + w1 (y) + β2 w2 (y) + β3 w3 (y) + β2 β4 w4 (y) + β2 w5 (y) + β3 β6 w6 (y) + β3 w7 (y) + β2 γ2 w10 (y) + β2 γ3 w11 (y) + β3 β6 γ4 w12 (y) + β3 γ6 w14 (y) + β2 β4 γ2 w20 (y) + β2 γ2 w21 (y) + β2 β6 γ3 w22 (y) + β2 γ3 w23 (y) + β3 γ4 γ6 w28 (y) + β2 γ22 w42 (y) + β2 γ2 γ3 w43 (y) + β2 β6 γ3 γ4 w44 (y) + β2 γ3 γ6 w46 (y) + β2 β4 γ22 w84 (y) + β2 γ22 w85 (y) + β2 β6 γ2 γ3 w86 (y)  + β2 γ2 γ3 w87 (y) + · · · ,

where y = x8 . We put G 1 := 2 + 2β2 + 2β3 + β2 β4 + β3 β6 + 2β2 γ2 + 2β2 γ3 + β3 β6 γ4 + β3 γ6 + β2 β4 γ3 + β2 β6 γ3 + β3 γ4 γ6 . Then d2  2 2γ2 + 2γ2 γ3 + β6 γ3 γ4 + γ3 γ6 1 − γ2  + β4 γ22 + β6 γ2 γ3 + γ3 γ4 γ6 .

8φ(0) = G 1 +

Moreover, for any odd s ≥ 5 8φ(

 s−2  1 β2  2 2 (γ ) = G + + γ γ + β γ γ + γ γ ) 1 − 2γ 1 2 3 6 3 4 3 6 2 2 2s 1 − γ2  + (γ22 + β4 γ22 + β6 γ2 γ3 + γ2 γ3 + γ3 γ4 γ6 ,

and for any given s ≥ 6 8φ(

β2  2 1 (γ2 + γ2 γ3 + β6 γ3 γ4 + γ3 γ6 ) ) = G + 1 2s 1 − γ2  s−4  + (γ22 + β4 γ22 + β6 γ2 γ3 + γ2 γ3 + γ3 γ4 γ6 ) 1 − 2γ2 2 . s

Therefore, |φ(0) − φ( 21s )| = C|γ2 | 2 , and hence αφ ≤ 21 log2 ( |γ12 | ). To prove the converse inequality, we note that for the vectors (4.109) and for any i, j, k, l ∈ {0, 1}. Ti e1 = Ti T j Tk Tl e2 = 0, Ti e4 = ±(γ3 e2 + γ2 e5 ) Ti T j = ±Ti e4 . m

Therefore, Td1 Td2 · · · Tdm ei ≤ C|γ2 | 2 for i = 1, 2, 4, 5. Further, for all i, j, k, l ∈ {0, 1}, the equalities take place

150

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

Ti T j e6 = 0, Ti e3 = β3 e2 + β2 e5 , Ti T j Tk Tl e4 = ±γ22 (β4 e1 + e4 ) + γ3 (Ti T j Tk e2 ± γ2 Ti e2 ), 1 Ti T j Tk e7 = ± Ti T j e4 . 2 Hence, for any d1 , d2 , . . . , dm ∈ {0, 1} and m ∈ N, we obtain the estimate m

Td1 Td2 · · · Tdm | E1 ≤ C|γ2 | 2 , from which, Proposition 4.12, we deduce αφ ≥ 21 log2 ( |γ12 | ). Thus, if β1 = β5 = β7 = 1 and γ2 = 0, then αφ = 21 log2 ( |γ12 | ). 20 Let β1 = β3 = β7 = 1 and β5 γ2 = 0. Then, according to (4.110)  8φ(x) = 1[0,1) (y) 1 + w1 (y) + β2 w2 (y) + w3 (y) + β2 d4 w4 (y) + β2 β5 w5 (y) + β6 w6 (y) + w7 (y) + β2 β5 γ2 w10 (y) + β6 γ4 w12 (y) + β6 γ5 w13 (y) + γ6 w14 (y) + β2 β4 β5 γ2 w20 (y) + β2 β52 γ2 w21 (y) + β6 γ2 γ5 w26 (y) + γ4 γ6 w28 (y) + γ5 γ6 w29 (y) + β2 β52 γ22 w42 (y) + β4 β6 γ2 γ5 w52 (y) + β5 β6 γ2 γ5 w53 (y) + γ2 γ5 γ6 w58 (y)  + β2 β4 β52 γ22 w84 (y) + · · · ,

where y = x8 . We set G 2 := 4 + β2 + β2 β4 + β2 β5 + β6 + β2 β5 γ2 + β6 γ4 + β6 γ5 + γ6 + β2 β4 β5 γ2 + β2 β52 γ2 + β6 γ2 γ5 + γ4 γ6 + γ5 γ6 . Then 8φ(

 1 γ2 2γ β2 β52 γ2 + β2 γ4 β5 2 + β4 β6 γ5 + β5 β6 γ5 ) = G2 + s 2 1 − β5 γ2  + γ5 γ6 + β5 β6 γ2 γ5 + β4 γ5 γ6 + β5 γ5 γ6 .

For any odd s ≥ 5 8φ(

s−3 s−3 1 γ2 ) = G2 + (β2 β52 γ2 + γ5 γ6 + β5 β6 γ2 γ5 )(1 − 2(β5 γ2 ) 2 + 2(β5 γ2 ) 2 ) 2s 1 − β5 γ2

+ (β4 β6 γ5 + β5 β6 γ5 + β5 γ5 γ6 + β5 γ5 γ6 )(1 − 2(β5 γ2 )

For any even s ≥ 6

s−1 2

+ β2 β4 β52 γ2 + β2 β52 γ2 ),

4.4 Estimates of the Smoothness of the Scaling Functions

 8φ

1 2s



151

   γ2 s−2 β2 β52 γ2 + (β4 β6 γ5 + β5 β6 γ5 ) 1 − 2(β5 γ2 ) 2 1 − β5 γ2   s−4 + (β4 β2 β52 γ2 + β2 β52 γ2 + β4 γ5 γ6 + d5 γ5 γ6 ) 1 − 2(β5 γ2 ) 2     s−2 s s−4 + γ5 γ6 1 − 2(β5 γ2 ) 2 + 2(β5 γ2 ) 2 + β5 β6 γ2 γ5 1 − 2(β5 γ2 ) 2  s−2 + 2(β5 γ2 ) 2 ) .

= G2 +

s

Therefore, |φ(0) − φ( 21s )| = C|β5 γ2 | 2 and αφ ≤ for vectors (4.109) and any i, j ∈ {0, 1}

1 2

log2 ( |β51γ2 | ). We now notice that

Ti e1 = 0, Ti T j e5 = ±γ2 β5 e5 , Ti e4 = ±γ2 e5 . m

Hence, for i = 1, 4, 5 estimate Td1 Td2 · · · Tdm ei ≤ C|β5 γ2 | 2 is correct. Moreover, for all i, j, k, l ∈ {0, 1} equalities take place Ti T j Tk Tl e2 = ±γ2 β5 Ti e4 ± (β6 − 1)γ2 γ5 Ti e5 + Ti T j Tk e7 , Ti e3 = e2 + β2 e5 , Ti T j e6 = ±γ5 Ti e4 , 1 Ti T j Tk e7 = ± (±γ2 γ5 Ti e5 + (±γ5 γ6 − β5 )Ti e4 ). 2 Using these equalities, for any m ∈ N, we obtain estimate m

Td1 Td2 · · · Tdm | E1 ≤ C|β5 γ2 | 2 . Consequently, according to Proposition 4.12 the equality αφ ≥ 30 Let β1 = β2 = β3 = 1 and γ7 = 0. Then

1 2

log2 ( |β51γ2 | ).

 8φ(x) = 1[0,1) (y) 1 + w1 (y) + w2 (y) + w3 (y) + β4 w4 (y) + β5 w5 (y) + β6 w6 (y) + β7 w7 (y) + β6 γ4 w12 (y) + β6 γ5 w13 (y) + β7 γ6 w14 (y) + β7 γ7 w15 (y) + β7 γ4 γ6 w28 (y) + β7 γ5 γ6 w29 (y) + β7 γ72 w31 (y) + β7 γ4 γ6 γ7 w60 (y) + β7 γ5 γ6 γ7 w61 (y) + β7 γ6 γ72 w62 (y) + β7 γ73 w63 (y) + β7 γ4 γ6 γ72 w124 (y)  + β7 γ5 γ6 γ72 w125 (y) + · · · , where y = x8 . We set G 3 := 4 + β4 + β5 + β6 + β6 γ4 + β6 γ5 . Then

152

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

8φ(0) = G 3 +

β7 (1 + γ6 (1 + γ4 + γ5 )). 1 − γ7

Moreover, for any integer s ≥ 2 8φ(

β7 1 ) = G3 + (1 − 2γ7s + (1 − 2γ7s−2 )(β6 γ5 + γ4 γ6 ) + γ6 (1 − 2γ7s−1 )). 2s 1 − γ7

Therefore, |φ(0) − φ( 21s )| = C|γ7 |s . For all i, j, k ∈ {0, 1} equalities are valid Ti T j Tk e1 = Ti T j Tk e4 = Ti T j Tk e5 = Ti T j Tk e6 = 0, Ti T j Tk e2 = ±β7 γ7 Ti T j e7 , Ti T j Tk e3 = (β6 − β7 )(γ4 e1 + γ5 e4 ) + 2β7 Ti e7 , 1 Ti T j Tk e7 = ±γ7 Ti T j e7 , Ti T j e7 = ± (±(γ6 − γ7 )(γ4 e1 + γ5 e4 ) + 2γ7 Ti e7 ). 2 Consequently, for any m ∈ N, we get

Td1 Td2 · · · Tdm | E1 ≤ C|γ7 |m . Hence, as above we conclude αφ ≥ log2 ( |γ17 | ). Thus, if β1 = β2 = β3 = 1 and γ7 = 0, then αφ = log2 ( |γ17 | ). 40 Let β1 = β2 = β5 = 1 and γ7 = 0. Then  8φ(x) = 1[0,1) (y) 1 + w1 (y) + w2 (y) + β3 w3 (y) + β4 w4 (y) + w5 (y) + β3 β6 w6 (y) + β3 β7 w7 (y) + γ3 w11 (y) + β3 β6 γ4 w12 (y) + β3 β7 γ6 w14 (y) + β3 β7 γ7 w15 (y) + β6 γ3 w22 (y) + β7 γ3 w23 (y) + β3 β7 γ4 γ6 w28 (y) + β3 β7 γ6 γ7 w30 (y) + β3 β7 γ72 w31 (y) + β6 γ3 γ4 w44 (y) + β7 γ3 γ6 w46 (y)  + β7 γ3 γ7 w47 (y) + · · · ,

where y = x8 . Let G 4 := 4 + β3 + β4 + β3 β6 + β3 β7 + γ3 + β3 β6 γ6 + β3 β7 γ7 + β6 γ3 + β7 γ3 + β3 β7 γ4 γ6 + β3 β7 γ72 + β6 γ3 γ4 + β7 γ3 γ6 + β7 γ3 γ7 + β3 β7 γ4 γ6 γ7 + β3 β7 γ6 γ72 . Then 8φ(0) = G 4 +

β7 (β3 γ72 + γ3 γ4 γ6 + γ3 γ6 γ7 + γ3 γ72 + d3 γ4 γ6 γ72 + β3 γ6 γ72 ). 1 − γ7

For any integer s ≥ 5

4.4 Estimates of the Smoothness of the Scaling Functions

8φ(B −s h [1] ) = G 4 +

153

β7  β3 γ72 (1 − 2γ7s−3 ) + γ3 (γ4 γ6 + γ6 γ7 + γ72 ) 1 − γ7

 (1 − 2γ7s−4 + 2γ7s−3 − 2γ7s−2 ) + d3 γ6 γ72 (γ4 + γ7 )(1 − 2γ7s−4 ) . From here it is seen that |φ(0) − φ(

1 )| = C|γ7 |s , 2s

and hence αφ ≤ log2 ( |γ17 | ). Further as above, for any i, j, k ∈ {0, 1} Ti e1 = Ti T j e6 = 0, Ti T j Tk e2 = ±β7 γ3 Ti e2 + 2β7 Ti T j e7 , Ti T j Tk e3 = β3 Ti T j e2 ± γ3 e2 , Ti e4 = Ti T j e5 = ±γ3 e2 , 1 Ti T j e7 = ± (γ3 Ti e2 + γ7 Ti e4 ± (γ6 − γ7 )γ4 e1 + γ7 Ti e7 ). 2 Consequently, for any m ∈ N the estimate is correct and according to the Proposition 4.12, αφ ≥ log2 ( |γ17 | ). Thus, if β1 = β2 = β5 = 1 and γ7 = 0 then αφ = log2 ( |γ17 | ). 50 Let β1 = β3 = β5 = 1 and γ7 = 0. Then  8φ(x) = 1[0,1) (y) 1 + w1 (y) + β2 w2 (y) + w3 (y) + β2 β4 w4 (y) + β6 w6 (y) + β7 w7 (y) + β6 γ4 w12 (y) + β6 w13 (y) + β7 γ6 w14 (y) + β7 γ7 w15 (y) + β6 γ2 w26 (y) + β7 γ4 γ6 w28 (y) + β7 γ6 w29 (y) + β7 γ6 γ7 w30 (y) + β7 γ72 w31 (y) + β4 d6 γ2 w52 (y) + β7 γ2 γ6 w58 (y) + β7 γ4 γ6 γ7 w60 (y)  + β7 γ6 γ7 w61 (y) + β7 γ6 γ72 w62 (y) + · · · , where y = x8 . We set G 5 := 3 + β2 + β2 β4 + 2β6 + β7 + β6 γ4 + 2β7 γ6 + β7 γ7 + β6 γ2 + β7 γ4 γ6 + β7 γ6 γ7 + 2β7 γ72 + β4 β6 γ2 + β2 γ2 γ6 + β7 γ4 γ6 γ7 + β7 γ6 γ72 . Then 8φ(0) = G 5 +

β7 (γ 3 + β4 γ2 γ6 + γ2 γ6 γ7 + γ4 γ6 γ72 + γ6 γ72 + γ6 γ73 ). 1 − γ7 7

For any s ≥ 5, we have

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4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

 β7  3  γ7 1 − γ7s−3 + γ6 (β4 γ2 + γ2 γ7 1 − γ7   + γ4 γ72 + γ72 + γ73 ) 1 − 2γ7s−4

8φ(B −s h [1] ) = G 5 +

and estimate |φ(0) − φ( 21s )| = C|γ7 |s is correct. Hence, αφ ≤ log2 ( |γ17 | ). Further, Ti e1 = Ti T j Tk e4 = Ti T j e5 = Ti T j Tk Tl e6 = 0, Ti e2 = −β7 e1 + β7 e4 + (β6 − β7 )e6 + 2β7 , Ti e3 = e2 + β2 e6 , 1 Ti e7 = ± ((γ4 − γ7 )e1 − (γ7 + 1)e4 − γ2 e5 + (γ6 − γ7 )e6 + γ7 e7 ). 2 Therefore, for any m ∈ N inequality is valid.

Td1 Td1 · · · Tdm | E1 ≤ C|γ7 |m . Using Proposition 4.12, from this we derive the estimate αφ ≥ log2 ( |γ17 | ). Thus, if γ5 = β1 = β3 = 1 and γ7 = 0 then αφ = log2 ( |γ17 | ). The theorem is proved. Recall that a binary-integer function of order r is a function f , defined on R+ and a constant on dyadic intervals of rank r . The space of such functions is denoted  by Er = Er (R+ ), and the set of all dyadic-integer functions is denoted by E = r Er . Walsh polynomial of order 2r − 1 belongs to the set Er . In the notations of Theorem 4.9, we indicate three cases, when ρ(A ˆ 0 , A1 ) = 0 and the solution of Eq. (4.78) for n = 4 is a binary-integer function. Suppose that x ∈ [0, 8) and y = x8 . Then, assuming W (y) := w1 (y) + β2 w2 (y) + β3 w3 (y) + β2 β4 w4 (y) + β3 β6 w6 (y) + β3 β7 w7 (y). We obtain form (4.110), the following three expansions: 1. If γ1 = γ2 = γ3 = γ7 = 0, then  8φ(x) = 1 + β1 W (y) + β2 β5 w5 (y) + β3 β6 γ4 w12 (y) + β3 β6 γ5 w13 (y)  + β3 β7 γ6 w14 (y) + β3 β7 γ4 γ6 w28 (y) + β3 β7 γ5 γ6 w29 (y) . 2. If γ1 = γ2 = γ5 = γ7 = 0, then

4.4 Estimates of the Smoothness of the Scaling Functions

155

 8φ(x) = 1 + β1 W (y) + β2 β5 w5 (y) + β2 β5 γ3 w11 (y) + β3 β6 γ4 w12 (y) + β3 β7 γ6 w14 (y) + β2 β5 β6 γ3 w22 (y) + β2 β5 β7 γ4 γ6 w28 (y) + β2 β5 β6 γ3 γ4 w44 (y) + β2 β5 β7 γ3 γ6 w46 (y)  + β2 β5 β7 γ3 γ4 γ6 w92 (y) . 3. If β5 = γ1 = γ3 = γ7 = 0, then  8φ(x) = 1 + β1 W (y) + β3 β6 γ4 w12 (y) + β3 β7 γ6 w14 (y) + d3 d6 γ2 γ5 w23 (y) + β3 β7 γ4 γ6 w28 (y) + β3 β7 γ5 γ6 w29 (y) + β3 β4 β6 γ2 γ5 w52 (y)  + β3 β7 γ2 γ6 w58 (y) + β3 β4 β7 γ2 γ5 γ6 w116 (y) . In Example 4.4, for an arbitrary n ≥ 3, there is a dyadic entire function of Eq. (4.78), generating MRA in L 2 (R+ ). The following theorem gives the conditions for the arrangement of the zeros of the mask of the scaling equation (4.78), for which its solution φ for a given natural number n is a dyadic entire function. Binary-rational numbers will sometimes be written in the binary system (for example, 0.01 = 41 ). Theorem 4.13 Let φ be a finite L 2 solution of a scaling equation (4.78) and m 0 is ˆ its mask. Suppose that φ(0) = 1. The function φ is not a dyadic entire if and only N , N ≤ j ≤ 2n−1 + n − 1, dk ∈ {0, 1}}, such if there exists a finite sequence {dk }k=1 that (a) d1 = · · · = dn−1 = 0, dn = 1; (b) There exists an integer j, n − 1 ≤ j ≤ N − 1, such, that d j−s = d N −s for s = 0, . . . , n − 2; (c) m 0 (0, ·dk+1 · · · dk+n ) = 0 for k = 0, . . . , N − n. N Proof Let {dk }k=1 be a sequence, satisfying conditions (a) − (c), and let

β = 0 · d1 · · · d j−n+1 (d j−n+2 · · · d N −n+1 ) is a binary-rational number with period (d j−n+2 · · · d N −n+1 ). Since m 0 ∈ En and m 0 (0) = 1, then for any r ∈ N. ˆ r β) = φ(2

∞ 

m 0 (2r −k β)

k=1

= =

∞  k=1 ∞  k=1

m 0 (d1 · · · dr −k , dr −k+1 · · · dr −k+n · · · ) m 0 (0 · dr −k+1 · · · dr −k+n ) = 0.

156

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

(Assume di = 0 for i ≤ 0, use the periodicity m 0 and the fact that each piece of length n of a binary entry of number β coincides with some piece of length n of N .). Hence, the Walsh–Fourier transform carrier φˆ is not compact the sequence {dk }k=1 and, therefore, by Proposition 4.5, the function φ is not binary integer. Conversely, suppose that a finite L 2 -solution φ of Eq. (4.78) is not a dyadic entire function. Then, φ ∈ L 1 (R+ ) and by Proposition 4.5 the support supp φˆ is not compact. Let us take n−1 ˆ = 0. The binary expansion of this number has the number ω > 22 such that φ(ω) the form ω=

l  j=0

ω− j−1 2 j +

∞ 

ω j 2 j , ω j ∈ {0, 1}, ω−l−1 = 1, l > 2n−1 .

j=1

It can be seen from 4.5 that m 0 (2−s ω) for all s ∈ N. Hence, as above using the fact that m 0 ∈ En , we get that at points γl = 0 · 0 · · · 0 · ω−l−1 , γl−1 = 0 · 0 · · · ω−l ω−l−1 , γ0 = 0 · ω−n · · · ω−2 ω−1 the mask m 0 does not vanish. In the binary notation of each binary-rational number γs (s = 0, . . . , l) there are exactly n significant digits. Since l > 2n−1 among the numbers γ0 , . . . , γl there are two, for which the first n − 1 digits coincide. Let r be the smallest number such that for some k ∈ N, the numbers γr and γr −k have the first n − 1 digits. Then, the sequence 0, . . . , 0, ω−l−1 , . . . , ω−r is required and the theorem is proved. Corollary 4.3 If the mask m 0 of the scaling equation (4.78) is equal to zero on the interval [2−r , 2−r +1 ) for some r ∈ N, then the solution φ of this equation is a dyadic entire function. Proof Since on the intervals [k, k + 2−n ) the mask m 0 is equal to 1, then 1 ≤ r ≤ n. Then, m 0 (ω) = 0 for any number ω, the binary record of which begins with the digits 0.0...01 (before the unit of r zeroes). It immediately follows that for this mask N satisfying the conditions (a) − −(c) of Theorem 4.13. there is no sequence {dk }k=1 In particular, if the mask is equal to zero in the interval [ 21 , 1), then the solution φ of Eq. (4.78) is a binary integer. Note also that if  m 0 (ω) =

1, ω ∈ [0, 2−r ), 0, ω ∈ [2−r , 2−r +1 ),

where r ∈ N, r ≤ n, then irrespective of the values taken by the mask m 0 on the remaining dyadic intervals, we have φ(x) = 21−r 1[0,2r −1 ) (x). For small n, the verification using Theorem 4.13, whether the solution of the scaling equation (4.78) is binary integer, reduces to a search of all sequences of zeros and units of length 2n−1 − n + 1. Example 4.9 For the case n = 3, the following step functions are solutions of Eq. (4.78):

4.4 Estimates of the Smoothness of the Scaling Functions

(1) (2) (3) (4) (5)

157

φ(x) = ( 41 )1[0,1) ( x4 ) (b1 = 0), φ(x) = ( 41 )1[0,1) ( x4 )(1 + b1 w1 ( x4 )) (b2 = b3 = 0), φ(x) = ( 41 )1[0,1) ( x4 )(1 + b1 w1 ( x4 ) + b1 b3 w3 ( x4 )) (b2 = b6 = b7 = 0), φ(x) = ( 41 )1[0,1) ( x4 )(1 + b1 w1 ( x4 ) + b1 b2 w2 ( x4 )) (b3 = b4 = b5 = 0), φ(x) = ( 41 )1[0,1) ( x4 )(1 + b1 w1 ( x4 ) + b1 b2 w2 ( x4 ) + b1 b3 w3 ( x4 ) + b1 b3 b6 w6 ( x4 )) (b4 = b5 = b7 = 0).

It is well known that for orthogonal wavelets with compact support on a line the smoothness cannot be infinite, it is always bounded by the length of the support of the scaling function. The exact analogue of this statement for dyadic wavelets is not true, if only because dyadic scaling functions can be dyadic entire finite functions. However, it turns out that for dyadic scaling functions there is an alternative, either the function is dyadic entire or it has finite smoothness. More precisely, by Theorem 4.14, if the solution φ of Eq. (4.78) is not binary integer, then its smoothness is finite. Moreover, the Holder exponent αφ is estimated above with the help of the smallest nonzero value of the mask module, and this estimate is unimprovable. Since the mask module |m 0 (ω)| is a piecewise constant function with step 2−n , it can take not more than 2n different values. One of these values is equal to zero, because of the modified Strang–Fix condition (Exercise 4.2.19), the mask must convert to zero on the period. Let am be the least positive value of the module of the mask m 0 , i.e., am = min{|m 0 (ω)| : ω ∈ R+ , m 0 (ω) = 0}. Since the mask has period 1 and is constant at dyadic intervals of rank n. Then, the value of am coincides with the minimal positive number from the set {|m 0 (2−n k)|, k = 0, . . . , 2n−1 }. Theorem 4.14 For any function φ ∈ L 2c (R+ ), that is a solution of Eq. (4.78) and ˆ satisfying the condition φ(0) = 1, the following alternative takes place: (1) either φ is binary integer; (2) either am < 1 and αφ ≤ − log2 am . Proof Suppose that φ is not a binary-integer function. Then, by Proposition 4.5, the support of the function φˆ is not compact. It means that there exists arbitrarily large ω ˆ for which φ(ω) = 0. Having determined the natural number k from the inequalities k 2 ≤ ω ≤ 2k+1 , and using the fact that m 0 (2− j ω) = 1 for all j ≥ n + k + 1, we obtain ∞ n+k   ˆ φ(ω) = m 0 (2− j ω) = m 0 (2− j ω). (4.111) j=1

j=1

ˆ Since φ(ω) then for each j = 1, . . . , n + k value of |m 0 (2− j ω)| is not equal to zero, and therefore not less than am . Having substituted the inequality |m 0 (2− j ω)| ≥ am in (4.111), we obtain ˆ (4.112) |φ(ω)| ≥ (am )n+k .

158

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

If we now assume that am ≥ 1, then, by (4.112), there exists arbitrarily large ω 2 2 ˆ ˆ ≥ 1. But, then, since the function |φ(ω)| is constant on binary for which |φ(ω)| ˆ ∈ / intervals of rank n − 1, it is not summable on the half line R+ , and hence φ(ω) / L 2 (R+ ) which contradicts the condition. Thus, am < 1. L 2 (R+ ). Consequently, φ ∈ n ˆ ≥ ω−(alogm 2) am . At Substituting in (4.112), the inequality k ≤ log2 ω, we have |φ(ω)| the same time, according to Shipp Wade Simon [5, Exercise 9.12], for any ω > 0 ˆ the inequality |φ(ω)| ≤ 21 ω(φ, ω2 ) is satisfied. Consequently, there exists a constant ˆ C > 0 such that |φ(ω)| ≤ ωCαφ for all ω ≤ 1. Therefore, αφ ≤ − log2 am . The theorem is proved. Remark 4.2 There exist infinitely smooth functions in R+ that are not dyadic integer. However, as from Theorem 4.14, this is impossible for scaling functions. The estimate αφ ≤ − log2 am is unimprovable for scaling functions. For example, for the Lang function (Example 4.2), it is achieved. In contrast to the scaling functions on the line R where for each n there exists only one scaling function of maximum smoothness (namely, a cardinal B-spline of order n − 1), the smoothness of the solutions of Eq. (4.78) for a given n is unbounded. However, it is limited for non-dyadic entire functions for a given value of am . This estimate is attained for the data n and am on the set of finite scaling dyadic functions. Thus, in constructing dyadic wavelets, we can check for a given mask m 0 in finite time whether the wavelets are dyadic entire and if not, then obtain a boundary from above for smoothness. Another useful consequence of Theorem 4.14 concerns the case where the mask module can take only two values: 0 and 1. This includes, in particular, the case when the mask is a characteristic function of the set obtained by combining some family of dyadic intervals of rank n. Corollary 4.4 If the mask module of the scaling equation (4.78) assumes only the values 0 and 1, then any L 2 solution of this equation is dyadic entire. Theorem 4.10 provides an algorithm for the finite verification of the scaling function for belonging to the class of dyadic entire function. For masks whose module takes only two values, according to Corollary 4.4, this algorithm yields a criterion for the solvability of the scaling equation in L 2 (R+ ). In conclusion of the section, applying Theorem 4.10, we obtain Corollary 4.5 If the function φ generates an MRA in L 2 (R+ ), and the mask module of the corresponding scaling equation takes only the values 0 and 1, then this function and the dyadic wavelets generated by it are binary integers.

4.5 Approximation Properties of Dyadic Wavelets In this section, we describe the approximation properties of dyadic wavelets in the Sobolev and Holder ¨ spaces on the half line R+ . We first consider approximations of functions that are smooth in the usual term, i.e., in the Euclidean metric on R+ , and

4.5 Approximation Properties of Dyadic Wavelets

159

then the approximations of W -smooth functions are studied. Since we study bursts with a compact support, it is quite natural to restrict ourselves to approximating the finite functions f . It will be established that (in contrast to classical wavelets on the straight line) all systems of dyadic wavelets on R+ have the same orders of approximation. Namely, in Sobolev spaces W1l , the order of approximation by any system of dyadic wavelets is equal to 1, while in the class of W -smooth functions the order of approximation is +∞. Therefore, to compare the approximation properties, a fine gage—the defect of the approximation is introduced and an explicit formula is obtained for it, which has an interesting analogy with the formula for the order of approximation by classical wavelets. It is known (see, for example, Novikov et al. [1], Theorems 3.2.3 and 3.4.16]) that the order of approximation of a system of wavelets with compact support on a line is equal to the multiplicity of the zero of the mask (trigonometric polynomial) m(ξ ) at point ξ = 21 . For dyadic wavelets, the mask is a Walsh polynomial, i.e., periodic function, piecewise constant on dyadic intervals of the given order n. It is shown that the approximation error is determined by the largest integer k, for which the mask vanishes on the half-interval [ 21 , 21 + 2k−n ). This result allows us to characterize dyadic wavelets with the best approximation properties. Next, we investigate the relationship of the approximation properties of wavelets with their smoothness (or, something is the same, with the smoothness of the scaling function). It is proved that the wavelets with the smallest error of approximation have a maximum smoothness, i.e., binary-integer functions. Recall that for a positive integer l, the Sobolev space W2l (R+ ) of all functions f , for which the derivative f (l−1) is absolutely continuous and f (l) ∈ L 2 (R+ ). For l = 0, we set W20 (R+ ) = L 2 (R+ ). The space of algebraic polynomials of order not above r is denoted by Pr . We will assume that the polynomials are defined on R+ . In particular, P0 is the space of identity constants on R+ . For any function φ ∈ L 2c (R+ ), the family of subspaces {V j } is defined as V j := span{φ j,k : k ∈ Z+ }, j ∈ Z, where φ j,k = φ(2 j x ⊕ k), x ∈ R+ . We note that V j

= V˜ j ∩ L 2c (R+ ), where V˜ j is the set of all possible linear combinations of the form k∈Z+ ak φ(2 j x ⊕ k), ak ∈ C. (The sums are finite since φ has a compact support.) The distance from an arbitrary function f ∈ L 2 (R+ ) to the subspace V j is a quantity given by dist ( f, V j ) := inf f − g . g∈V j

It is clear that the rate of decrease of dist ( f, V j ) depends on the approximated function f , in particular, on its smoothness (by virtue of the inverse theorems of approximation theory). Therefore, it is natural to put the problem for certain classes of functions. For approximations in the class of continuously differentiable functions C l (R), an analogous problem was first investigated by I. Schoenberg [Schoenberq] for the spaces V j , generated by the functions φ(2 j − k), k ∈ Z, where φ is an arbitrary continuous finite function that does not necessarily satisfy the scaling equation.

160

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

I. Schonberg [10] established the connection between the maximum degree of algebraic polynomials generated by integer-valued shifts of the function f , and the order of decreasing distance from an arbitrary function f ∈ C l (R) to the spaces V j . In terms of the Fourier transform of the function, this connection was found explicitly by G. Strang and G. Fix in the monograph [6]. For the Strang–Fix conditions in the theory of wavelets, see, for example, Sect. 3.3. Books (Novikov et al. [1]). In the case, when {ψ j,k } is the system of wavelets defined by the algorithm A0 of Sect. 4.2 and {V j } corresponding to the MRA, for any function f ∈ L 2 (R+ ) the dist ( f, V j ) = f − P j f is true, where P j f is the orthogonal projection of f to V j . The expansion of the projection P j f over the system {ψ j,k } has the form Pj f =



cr,k ψr,k , cr,k =  f, ψr,k .

k∈Z,r < j

Therefore, the approximation properties of the

wavelet system also determine the rate of decrease of the “tail” of the expansion k∈Z,r ≥ j |cr,k |2 or, in fact, the rate of decrease of the coefficients c j,k for j → ∞. The smoothness index of the function f ∈ L 2 (R+ ) is denoted by S f and is defined as S f =:= k + sup{S : f (k) (· + h) − f (k) (·) ≤ Ch S }, where k is the largest integer for which f ∈ W2k (R+ ). For an arbitrary function φ ∈ L 2c (R+ ), the order of approximation νφ is the exact upper bound of the set of numbers γ ≥ 0 such that dist ( f, V j ) ≤ C( f, γ )2− jγ , j ∈ Z,

(4.113)

for any function f ∈ L 2 (R+ ), for which S f > γ . It is seen that the order of approximation νφ will be larger if the distance from any sufficiently smooth finite function f to the space V j is equal to O(2− j γ ) for j → ∞. According to the following proposition, the order of approximation per unit exceeds the largest degree of the space of polynomials, which the function φ generates with its entire shifts. Proposition 4.13 For any function φ ∈ L 2c (R+ ), we have  νφ =

1 + max{r : Pr ⊂ V˜0 }, if P0 ⊂ V˜0 , 0, if P0  V˜0 .

Proof Let supp φ ⊂ [0, 2n−1 ), where n ∈ N. Let us assume that Pr ⊂ V˜0 . Then, at the interval [0, 2n−1 ) any degree polynomial not higher than n is generated by functions φ(· ⊕ k), k = 0, 1, . . . , 2n−1 . From here it follows that V j contains all functions g of the following form: on a given dyadic interval of length 2n−1− j , the function g coincides with a given polynomial of degree not higher than r , and outside of this interval is equal to zero. Hence, the space V j approximates an arbitrary finite

4.5 Approximation Properties of Dyadic Wavelets

161

function f ∈ W2r +1 (R+ ) not worse than a space of splines of order r with nodes on a uniform grid with step h = 2n−1− j . It is well known that the distance from f to this space of splines does not exceed C( f, r )h r +1 . Consequently, dist ( f, V j ) ≤ [2(n−1)(r +1) C( f, r )]2− j (r +1) , j ∈ Z. In this way, if Pr ⊂ V˜0 , then νφ ≥ r + 1. It is assumed now that P0  V˜0 . Let us / V˜0 . By assumption, q ≤ r . The linear span take the smallest q ∈ Z+ for which x q ∈ n−1 − 1 does not contain the restriction x q to of functions φ(· ⊕ k), k = 0, 1, . . . , 2 n−1 q the interval [0, 2 ). Let a = dist (x , L n−1 ) in the space L 2 [0, 2n−1 ). By virtue of finite dimension of space L n−1 we have a > 0. Hence, the distance from the function x q , bounded to any interval till the space V0 is not less than a. We apply the conversion 1 f (·) → 2− jq f (2 j ·), which reduces the L 2 -norm in 2 j (q+ 2 ) , transforms the space V0 into V j , and leaves the function x q in place. We had obtained that the distance from the function x q bounded to any dyadic interval Is(k) of order k = n − 1 − j, till the 1 space V j is not less than 2 j (q+ 2 ) . Hence, for any function g ∈ V j , we have 

2n−1

|x q − g(x)|2 ≥ 2− j (2q+1) a 2 .

0

Adding this inequality over all 2 j intervals of order n − 1 − j, joining the interval [0, 2n−1 ), we obtain 

2n−1

|x q − g(x)|2 ≥ 2 j · 2− j (2q+1) a 2 = 2−2q j a 2 ≥ 2−2r j a 2 .

0

In this way, the distance from the function x q which is bounded on the interval [0, 2n−1 ) till the space V j is not less than 2−r j a. Now taking an arbitrary function f ∈ W2r +1 (R+ ) that coincides with the polynomial x q on the interval [0, 2n−1 ), we find that dist ( f, V j ) ≥ 2−r j a. In this way, if Pr  V˜0 , then νφ ≤ r . The proposition is proved. Corollary 4.6 For any function φ ∈ L 2c (R+ ), the order of approximation νφ is finite and is a nonnegative integer. We show that the order of approximation of an arbitrary dyadic scaling function with compact support is equal to 1. For this, we apply the results to a scaling function φ satisfying the equation φ(x) =

n 2 −1

ck φ(2x ⊕ k), x ∈ R+ .

(4.114)

k=0

We recall that the mask of this scaling equation is the following Walsh polynomial

162

4 Construction of Dyadic Wavelets and Frames Through Walsh Functions 2 −1 1 ck wk (ω). 2 k=0 n

m 0 (ω) =

(4.115)

Theorem 4.15 For any scaling function φ ∈ L 2c (R+ ), satisfying Eq. (4.114), the equality νφ = 1 is true. Proof Since the carrier of the L 2 function φ is determined with an accuracy till the set of measure zero, as per Theorem 4.4, we have suppφ ⊂ [0, 2n−1 ) and the partition of unity property is fulfilled: 

φ(x ⊕ k) = 1, f or a.e. x.

(4.116)

k∈Z+

Consequently, P0 ⊂ V˜0 and with the help of the proposition 1, we obtain νφ ≥ 1. Further, if the function f (x) = x belongs to the space V˜0 , then the coefficients a0 , a1 , . . . , a2n−1 −1 exist for which 2n−1 −1

ak φ(x ⊕ k) = x f or a.e. x.

k=0

Substituting x ⊕ k for x in this equation and adding over all l = 0, 1, . . . , 2n−1 − 1, we obtain n−1 2n−1 2n−1 −1 2 −1 −1 ak φ((x ⊕ l) ⊕ k) = (x ⊕ l). l=0

k=0

l=0

On the other hand, with the help of (4.116), we have n−1 2n−1 −1 2 −1

l=0

ak φ((x ⊕ l) ⊕ k) =

k=0

2n−1 −1 k=0

ak

2n−1 −1

φ((x ⊕ l) ⊕ k) =

l=0

2n−1 −1

ak .

k=0

Consequently, 2n−1 −1

ak =

k=0

2n−1 −1

(x ⊕ l) f or a.e. x.

(4.117)

l=0

Note, that if x ∈ [0, 1) then x ⊕ l = x + l. Therefore, for x ∈ [0, 1) Eq. (4.117) takes the form 2n−1 2n−1 −1 −1 ak = (x + l) = 2n−1 x + 2n−2 (2n−1 − 1), k=0

k=0

4.5 Approximation Properties of Dyadic Wavelets

163

which is impossible in view of the arbitrariness of x ∈ [0, 1). From here it follows that f (x) = x ∈ / V˜0 . In this way, V˜0 does not hold the set P1 . As per Proposition 4.13, we obtain νφ ≤ 1, which completes the proof of the theorem. It is clear from Theorem 4.15 that for approximation function of the Sobolev class W2k (R+ ), all the wavelets determined by the algorithm A0 of Sect. 4.2 are equivalent. Partially, this phenomenon can be explained by the fact that dyadic wavelets are equal as well as the classical Walsh system is intended for approximation of not smooth (in the general sense) but dyadic-smooth functions. The remaining part of the present section is devoted to such approximations. For a scaling function φ ∈ L 2c (R+ ) satisfying Eq. (4.114), the dyadic module of continuity is defined in Sect. 4.3. In this section, for each function f ∈ L 2 (R+ ), we set ω2 ( f ; h) := sup{ f (· ⊕ x) − f (·) : 0 ≤ x < δ}, h > 0, ¨ and we will estimate the smoothness of function f by H0lder indicator α f := sup{α ≥ 0 : ω2 ( f ; h) ≤ Ch α }. We note that, in the given case, the Holder indicator of function, which is different from the constant can be greater than 1. The binary order of approximation by a function φ ∈ L 2c (R+ ) is named as the exact upper bound νφ of numbers γ ≥ 0, such that (4.113) is fulfilled for all f ∈ L 2c (R+ ) for which α f > γ . Proposition 4.14 For any function φ ∈ L 2c (R+ ), there is an alternative, either νφ = 0, or νφ = +∞. In this case, νφ = +∞ if and only if 

φ(x ⊕ k) = c,

(4.118)

k∈Z+

for some constants c = 0. Proof We choose a positive integer n from the condition supp φ ⊂ [0, 2n−1 ). If

n−1 (4.118) is satisfied, then 2k=0 −1 φ(x ⊕ k) = c, for x ∈ [0, 2n−1 ). Consequently, 1[0,2n−1 ) ∈ V0 from where 1[0,2n−1 ) ∈ Vn−1 . Hence, the space V˜n−1 contains all dyadic entire function of order 0. It follows from here that E0 ⊂ V˜n−1 , and therefore Er ⊂ V˜r +n−1 for any r ∈ Z+ . It is known [Golubov et al., Sect. 10.5.2] that for an arbitrary function f ∈ L 1 (R+ ) ∩ L 2 (R+ ) we have the inequalities 1 ω2 ( f ; 2−r ) ≤ dist ( f, Er ) ≤ ω2 ( f ; 2−r ). 2

(4.119)

Consequently, dist ( f ; V˜r +n−1 ) ≤ dist ( f, Er ) ≤ ω2 ( f ; 2−r ). Assuming j=r +n−1 and noting that dist ( f ; V˜r +n−1 ) = dist ( f ; Vr +n−1 ), such that f is finite, we have dist ( f ; V j ) ≤ ω2 ( f ; 2− j+n−1 ).

(4.120)

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4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

If α f ≥ γ , then ω2 ( f ; 2− j+n−1 ) ≤ C( f )2γ (− j+n−1) , from where dist ( f ; V j ) ≤ 2(n−1)γ C( f )2− jγ . This inequality is true for any γ > 0 and f ∈ lc2 (R+ ), under the condition that α f > γ . By virtue of arbitrary γ , we obtain νφ = +∞. Now assume that the function φ does not have the property (4.118). Then, for any coefficients ak , 0 ≤ k ≤ 2n−1 − 1, we have 2n−1 −1 > 0. 1[0,2n−1 ) (·) − φ(· ⊕ k) k=0 In fact, if for some coefficients {ak } the equality is completed 2n−1 −1

φ(x ⊕ k) = 1[0,2n−1 −1) (x),

k=0

ˆ then, by applying the Walsh–Fourier conversion, we obtain a(ω)φ(ω) = 1[0,2n−1 −1) (ω),

2n−1 −1 ˆ ˆ where a(ω) = k=0 wk (ω). Consequently, a(0)φ(0) = 1 and a(k)φ(k) = 0 for ˆ all k ∈ N. It follows from the first equality that φ(0) = 0 and a(0) = 0. Then, for any k ∈ N we have a(k) = a(0) = 0. Now by applying the second

equality, we get ˆ φ(k) = 0 for all k ∈ N. Putting a 1-periodic function Φ(x) := k∈Z+ φ(x ⊕ k), in the Walsh–Fourier series, we obtain  ˆ ˆ ˆ φ(k)w f or x ∈ R+ . Φ(x) = k (x) = φ(0)w 0 (x) = φ(0), k∈Z+

ˆ Consequently, the condition (4.118) is completed with constant c = φ(0), which contradicts the proposition. Therefore, the continuous function {ak }2k=0 −1 n−1

2n−1 −1 → 1[0,2n−1 ) (·) − φ(· ⊕ k) k=0

takes only positive values, from which (by finite dimensionality) it follows 2n−1 −1 =: q > 0. n−1 inf 1 (·) − φ(· ⊕ k) [0,2 ) {ak } k=0 In this way, dist (1[0,2n−1 ) ; V0 ) = q > 0. Then, for any binary interval Is(k) of order k = n − 1 − j, we have j dist (1 Is(k) , V0 ) ≥ 2− 2 q.

4.5 Approximation Properties of Dyadic Wavelets

165

Consequently, for any function g ∈ V j , we have  Is(k)

|g(x) − 1|2 ≥ 2− j q 2 .

Adding these inequalities over all 2 j intervals Is(k) , constituting the interval [0, 2n−1 ), we obtain q − 1[0,2n−1 ) > q. Thus, dist (1[0,2n−1 ) ; V j ) ≥ q. Thus, the characteristic function of the interval [0, 2n−1 ) cannot be approximated with any accuracy by the spaces V j . But since the Hölder indicator α of the function 1[0,2n−1 ) is equal to +∞, we have νφ = 0, which completes the proof. The following theorem is the direct consequence of Theorem 4.4 and Proposition 4.14. Theorem 4.16 For any scaling function φ ∈ L 2c (R+ ), satisfying Eq. (4.114), is true to the equation νφ = +∞. Thus, the order of approximation by any system of dyadic wavelets is equal to +∞. Therefore, the dyadic wavelets come closer to the W -smooth functions better than any classical system of wavelets. On the other hand, Theorem 4.16 shows that the binary order of approximation νφ has a very rough characteristic. A more detailed analysis proposes the constant C( f, γ ) in inequality (4.113) and its growth depending on γ . Taking into account the estimate (4.118), we introduce the following notion. Let for the function νφ = +∞, the equation φ ∈ L 2c (R+ ) is fulfilled. By the defect of approximation bφ , function φ is named the smallest integer number b, such that for all f ∈ L 2c (R+ ) the following inequality takes place dist ( f, V j ) ≥ ω2 ( f ; 2− j+b ).

(4.121)

The correctness of the definition follows from Proposition 4.12, but first we will prove the following lemma. Lemma 4.1 For any l ∈ Z+ , the following properties of the function φ ∈ L 2c (R+ ) are equivalent: (a) 1[0,2l ) ∈ V˜0 ; (b) E−1 ⊂ V˜0 ; (c) The function φˆ at every interval [k, k + 2−l ) is identically equal to zero and it does not vanish at the interval [0, 2−l ). Proof The equivalence of the properties (a) and (b) are evident. We will prove that the property (a) is equal to (c). Let supp φ ⊂ [0, 2n−1 ). If 2n−1 −1 k=0

φ(x ⊕ k) = 1[0,2l ) (x),

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4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

then, applying the conversion of Walsh–Fourier, we obtain ˆ a(ω)φ(ω) = 1[0,2l ) (ω),

(4.122)

n−1 where a(ω) = 2k=0 −1 wk (ω). It follows that the function a(ω) is not changed to zero at the interval [0, 2−l ), and this means that, at all intervals [k, k + 2−l ), k ∈ N. ˆ That is why, the function φ(ω) at these intervals is equal to zero. Conversely, if (c) holds, then, putting  1 , ω ∈ [0, 2−l ), ˆ a(ω) = φ(ω) 0, ω∈ / [0, 2−l ), and continuing with the periodicity on R+ , we obtain (14), which implies (a). The lemma is proved. Proposition 4.15 For any function φ ∈ L 2 (R+ ), satisfying (4.118), the defect of approximation bφ is equal to the minimal number l for which conditions (a)–(c) of Lemma 4.1 are satisfied. Proof If E−1 ⊂ V˜0 for a given l, i.e., satisfying the condition (b), then E j−1 ⊂ V˜ j for any j. Now, applying (4.119) in any function f ∈ L 2c (R+ ), we obtain dist ( f, V j ) ≥ ω2 ( f ; 2− j+1 ). Consequently, for b = l the inequality (4.121) is satisfied. Conversely, / V˜0 , then, as it if property (a) is not satisfied for the number l − 1, that is, 1[0,2l−1 ) ∈ is in the proof of Proposition 4.14, we conclude that dist ( f 0 , Vl−1 ) ≥ q > 0, where f 0 = 1[0,1) . Since ω2 ( f 0 ; 2−l+1+b ) for any b < l − 1, then the inequality (4.121) is not completed when j = l − 1. Therefore b ≥ l − 1. The proposition is proved. Corollary 4.7 For any function φ ∈ L 2c (R+ ), which satisfies the condition (4.118), we have 0 ≤ b ≤ n − 1, where n is the smallest natural number for which the suppφ ⊂ [0, 2n−1 ). It can be seen from Proposition 4.12 that for a scaling function φ, satisfying Eq. ˆ (4.114), the value of bφ is entirely determined by the set of zeros of the function φ, and hence by the zero set of the mask m 0 (ω), given by formula (4.115). A scaling function φ is named as stable, if the system of entire shifts of this function forms a Riesz basis in L 2 (R+ ), as per Theorem 4.6, this is equivalent to linear independence in the shift system {φ(· ⊕ k) : k ∈ Z+ } in L 2 (R+ ). Lemma 4.2 A stable scaling function φ ∈ L 2c (R+ ) has the properties (a)–(c) of 1 ). Lemma 4.1 if and only if, its mask is equal to zero on the interval [ 21 , 21 + 2l+1 Proof If m 0 ( 21 + ω2 ) = 0 for all ω ∈ [0, 2−l ), then from (4.114) for any odd k ∈ N, we obtain     k ω k ω ˆ + ω) = m 0 φ(k + φˆ + 2 2 2 2     1 ω k ω + φˆ + = 0. = m0 2 2 2 2

4.5 Approximation Properties of Dyadic Wavelets

167

If the number k is even, then let us assume that its form k = 2s r , where r is odd. Then, for any ω ∈ [0, 2−l ), we obtain ˆ + ω) = φ(2 ˆ s r + ω) = φ(r ˆ + 2−s ω) φ(k

s 

m 0 (2−i k + 2−i ω) = 0.

i=1

ˆ + 2−s ω) = 0. Thus, the function φˆ is equal to zero on all Since r is odd, then φ(r −l intervals [k, k + 2 ), k ∈ N. Consequently, on the interval [0, 2−l ) it does appear in zero, otherwise would have a periodic zero and, as per Theorem 4.6, would had been unstable, which contradicts the assumption. In this way, property (c) of Lemma 4.1 is satisfied. Conversely, if this property is satisfied, then for any ω ∈ [0, 2−l ) and for any k ∈ Z+ , we have     1 ω 1 ω ˆ φˆ k + + 0 = φ(2k + 1 + ω) = m 0 k + + 2 2 2 2     1 ω 1 ω + . = m0 φˆ k + + 2 2 2 2 ˆ + 1 + ω ) = 0 For If there exists ω ∈ [0, 2−l ), for which m 0 ( 21 + ω2 ) = 0, then φ(k 2 2 any k ∈ Z+ . Consequently, the function φˆ has periodic zero, and this means φ is unstable. Therefore, m 0 ( 21 + ω2 ) = 0 for all ω ∈ [0, 2−l ). The lemma is proved. By Proposition 4.15 and Lemma 4.2, we obtain Theorem 4.17 The defect of approximation bφ by the stable scaling function φ ∈ L 2c (R+ ) (and the corresponding dyadic wavelet system) is equal to the smallest integer r such that the mask m 0 (ω) is zero on the interval [ 21 , 21 + 2−r −1 ). Thus, for dyadic wavelets, the binary order of approximation is one and the same, the defect of approximation is determined by the length of the binary interval at the starting point 21 , at which the mask turns into zero. For example, for the scaling Lang function from Example 4.2, we have bφ = 1. Corollary 4.8 Minimal defect of approximation is equal to zero and is achieved, when the mask is zero identically on the interval [ 21 , 1). For any n, there are dyadic orthogonal wavelets, which are generated by scaling functions of order n and have a minimal approximation defect, i.e., the best approximate properties on the class of W -continuous functions. A necessary and sufficient condition for this is the equality |m 0 (ω)| = 1 for all ω ∈ [0, 21 ). In order to construct such a wavelet ψ in the algorithm A0 of Sect. 4.2, it is sufficient to choose for 0 ≤ l ≤ 2n−1 . Indeed, according to Theorem 4.10 and Corollary 4.8, the following proposition is true: Proposition 4.16 A scaling function φ, satisfying Eq. (4.114), generates a system of dyadic wavelets with a minimal approximation defect, if and only if, when

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4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

 1, ω ∈ [0, 21 ), |m 0 (ω)| = 0, ω ∈ [ 21 , 1). In particular, the Haar splash has a minimal approximation error. It corresponds to the condition n = 1. For every n ≥ 2, there are many systems of wavelets with minimal defect. As a mask m 0 (ω), one can take an arbitrary 1-periodic binary-integer function from En , which take the values ±1 on the interval [0, 21 ) (if the mask is complex valued, then the values of the mask on [0, 21 ) can be arbitrary complex values with Module 1) and equal to zero on [ 21 , 1). Proposition 4.17 A stable scaling function φ has a minimal defect of approximation if and only if, when it belongs to E0 . ˆ Proof We can confine ourselves to the case when φ(0) = 1. If bφ = 0, then, by Corollary 4.8, the mask m 0 (ω) is equal to zero for all ω ∈ [ 21 , 1). Then, by taking any natural number j for any ω > 1, for which ω ∈ [2 j−1 , 2 j ), we obtain m 0 (2− j ω) = 0. But by Theorem 4.4, we have ˆ φ(ω) =

∞ 

m 0 (2−s ω),

s=1

ˆ and also this means φ(ω) = 0 for all ω > 1. Consequently, φ ∈ E0 . Conversely, if ˆ φ ∈ E0 , then φ(ω) = 0 for every ω > 1. Let us suppose that bφ > 0. This means that ˆ ˆ 0 ), from where m 0 (ω0 ) = 0 for some ω0 ∈ [ 21 , 1). Then, 0 = φ(2ω 0 ) = m 0 (ω0 )φ(ω ˆ ˆ φ(ω0 ) = 0. But since φ(ω0 + k) = 0 for all k ≥ 1, we obtain that ω0 -periodic zero ˆ consequently, φ is unstable. The proposition is proved. of the function φ, Thus, dyadic wavelets with minimal defect approximations are binary integers. Conversely, generally speaking, it is incorrect as shown in Proposition 4.18. Proposition 4.18 For any n, there exists a scaling equation of order n, the solution of which generates MRA, is a dyadic entire function, but it has the maximum possible defect of approximation bφ = n − 1. Proof For a given n, we take an arbitrary mask m 0 (ω), which is zero on the intervals [2−n , 21−n ) and [ 21 , 21 + 2−n ), equal to 1 on the intervals [0, 2−n ) and [ 21 + 2−n , 21 + 21−n ), and at the remaining points of the interval [0, 1) does not vanish and satisfies the condition     1 2 2  |m 0 (ω)| + m 0 ω + = 1. 2  By Theorem 4.10, the corresponding scaling function φ generates an MRA. As in the proof of Proposition 4.17, we establish that φ ∈ En−1 . Nevertheless, bφ = n − 1 according to Theorem 4.17. The proposition is proved

4.5 Approximation Properties of Dyadic Wavelets

169

For classic wavelets on R, the smoothness is a sufficient condition for good approximation properties (see, for example, De Boor et al. [7, 8], Jia [9]). In a dyadic case, as from Proposition 4.18, there is no direct connection between the smoothness of the wavelets and the approximation properties. Dyadic entire wavelets (whose smoothness equals ∞) can have the greatest approximation error. At the same time, by Proposition 4.17, all scaling functions belonging to E0 have optimal approximation properties, that is, the smallest approximation error. In particular, Haar approximations are optimal. In connection to this, we note that in some applied problems, when the approximation “signal” f by its projections P j f , the coefficients of Eq. (4.114) (and thus the scaling function φ) at each level j are selected separately with the help of the chosen criterion of optimization for concrete task, and such “adaptive” approach, as a rule, leads to approximations, which are different from Haar approximations (for more details on the adaptive approach, see Sects. 5.3, 5.5, and 7.5).

4.6 Exercise 4.1 If the mask m 0 does not have symmetric zeros, it does not turn into zero at the point ω = 21 − 21n and is nonzero on the interval [0, 41 ), then the system of entire shifts of the solution φ of the scaling equation (4.36) is linearly independent. ˆ = 1, 4.2 Prove that for a given integer L ≥ 0 and a given function φ ∈ L 2c (R+ ), φ(0) the following properties are equivalent: (i) φˆ [r ] (n) = 0 for all r = 0, 1, . . . , L; n ∈ N (modified Strang–Fix conditions);

r (ii) k∈Z+ (x ⊕ k) φ(x ⊕ k) = 1 for each r = 0, 1, . . . , L and x ∈ R+ . For scaling functions on a real line, the Strang–Fix conditions are proved, for example, in Novikov et al. [1, § 3.2]. 4.3 Prove that if a finite L 2 solution φ of scaling equation (4.78) is a dyadic integer, n−1 then supp φˆ ⊂ [0, 22 ].

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4 Construction of Dyadic Wavelets and Frames Through Walsh Functions

References 1. Novikov, I. Ya., Protasov, V. Yu., & Skopina, M. A. (2011). Wavelet theory. Providence: AMS (Moscow, 2006). 2. Golubov, B. I., Efimov, A. V., & Skvortsov, V. A. (2008). Walsh series and transforms. URSS, Moscow; English Transl. of 1st ed. Dordrecht: Kluwer (1991). 3. Lang, W. C. (1998). Fractal multiwavelets related to the Cantor dyadic group. International Journal of Mathematics and Mathematical Sciences, 21, 307–317. 4. Protasov, V. Yu. (2007). Approximation by Dyadic wavelets. Matematicheskii Sbornik, 198(11), 135–152; English Transl. Sbornik: Mathematics, 1665–1681. 5. Schipp, F., Wade, W. R. & Simon, P. (1990). Walsh series. Adam Hilger. 6. Strang, G., & Fix, J. (1973). An analysis of the finite element method. Englewood Clifs, N.J.: Prentice Hall (second edition in 2008). 7. De Boor, C., DeVore, R., & Ron, A. (1991). Approximation from shift-invariant subspaces of L 2 (Rd ), CMS-TSR University of Wisconsin-Madison, 92-2. 8. De Boor, C., DeVore, R., & Ron, A. (1992). The structure of finitely generated shift-invariant spaces in L 2 (Rd ), CMS-TSR University of Wisconsin-Madison, 92-8. 9. Jia, R. Q. (1993). A Berstein type inequality associated with wavelet decomposition. Construction Aroximation, 9, 299–318. 10. Schoenberg, I. J. (1946). Contribution to the problem of approximation of equidistant data by analytic functions, Part A and B. Quarterly of Applied Mathematics, 4(2), pp. 45–99, 112–141.

Chapter 5

Orthogonal and Periodic Wavelets on Vilenkin Groups

5.1 Multiresolution Analysis on Vilenkin Groups As noted in Chap. 1, the Walsh function can be identified with characters of the Cantor dyadic group. This fact was first recognized by Gelfand in 1940s, who offered to Vilenkin study series with respect to characters of a large class of abelian groups which includes the Cantor group as special case see Vilenkin [1], Fine [2], Agaev et al. [3]. For wavelets on Vilenkin groups most of the results relate to the locally compact group G p , which is defined by a fixed integer p ≥ 2. The group G p have a standard interpretation on R+ . Since the case p = 2 corresponds to the Cantor group C , all the results on wavelets on R+ presented in Chap. 4 can be rewritten for wavelets on C . In this section, necessary and sufficient conditions are given for refinable functions to generate an MRA in the space L 2 (G p ). The partition of unity property, the linear independence, the stability, and the orthogonality of “integer shifts” of refinable functions in L 2 (G p ) are also considered. Let us recall that the group G p consists of sequences x = x j , where x j ∈ {0, 1, 2, . . . p − 1} for j ∈ Z and with at most finite number of negative j such that x j = 0. The zero sequence is denoted by θ . For any x = θ , there exists a unique k = k(x) such that xk = 0 and x j = 0 for j < k. The group operation on G p is defined as the coordinatewise addition modulo p: (z j ) = (x j ) ⊕ (y j ) ⇔ z j = x j + y j (modp) f or j ∈ Z and topology on G p is introduced via the complete system of neighborhoods of zero Ul = {(x j ) ∈ G p : x j = 0 for all j ≤ l}, l ∈ Z . The equality z = x  y means that z ⊕ y = x. For p = 2, we have x ⊕ y = x  y. In this case, the group G 2 coincides with the locally compact Cantor group C and the subgroup U0 is isomorphic to the compact Cantor group C; i.e., the topological © Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2_5

171

172

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

Cartesian product of a countable set of cyclic groups with discrete topology. It is well known that C is a perfect nowhere dense totally disconnected metrizable space, and therefore U0 is homeomorphic to the Cantor ternary set. For any p ≥ 2, we let G = G p and U = U0 . One can show that G is self-dual. The duality pairing on G takes x, w ∈ G to ⎛

⎞  2πi χ (x, w) = ex p ⎝ x j w1− j ⎠ . p j∈Z The Lebesgue spaces L q (G), 1 ≤ q ≤ ∞, are defined with the help of the Haar measure μ on Borel subsets of G normalized by μ(U ) = 1. Denote by (·, ·) and ·

the inner product and the norm in L 2 (G), respectively. The Fourier transform of a function f ∈ L 1 (G) ∩ L 2 (G), fˆ(w) =

 f (x)χ (x, w)dμ(x), w ∈ G, G

admits a standard extension to the space L 2 (G). By C0 (G), we denote the set of all continuous complex-valued functions g on G such that, for any ε > 0 there exists a compact set E ⊂ G (depending on g and ε) such that |g(x)| < ε for all x ∈ G \ E. The following properties are well known (see, e.g.,): 1) If f ∈ L (G), then fˆ ∈ C0 (G). 2) The Fourier operator maps L 2 (G) into itself linearly, continuously, and one to one. ˆ (Plancherel’s equality). 3) If f, g ∈ L 2 (G), then ( f, g) = ( fˆ, g) Take a discrete subgroup H = {(x j ) ∈ G : x j = 0 f or j > 0} and define an automorphism A ∈ Aut G by the formula (Ax) = x j+1 . It is easy to see that the quotient H contains p elements. We define a map λ : G → R+ by group A(h) λ(x) =



x j p − j , x = (x j ) ∈ G.

j∈Z

The image of H under λ is the set of nonnegative integers: λ(H ) = Z+ . For every α ∈ Z+ , let h [α] denote the element of H such that λ(h [α] ) = α. Notice that χ (Ax, w) = χ (x, Aw) for all x, w ∈ G. The Walsh function for the group G can be defined by wα (x) = χ (x, w[α] ), α ∈ Z+ , x ∈ G. These functions are continuous on G and satisfy the orthogonality relations  wα (x)wβ (x)dμ(x) = δα,β , α, β ∈ Z+ , U

5.1 Multiresolution Analysis on Vilenkin Groups

173

where δα,β is the Kronecker delta. Also, it is well known that the system {wα } is complete in L 2 (U ). For any positive integer n, let εn (G) denote the collection of all functions defined on G and constant on the sets Un,s := A−n (h[s]) ⊕ A−n (U ), s ∈ Z+ . The elements of the set ε(G) := ∪n εn (G) will be called p-Adic entire functions on G (Golubov et al. [4, § 6.2]). Note that the sets Un,s , 0 ≤ s ≤ p n − 1, are cosets of the subgroup A−n (U ) in the group U . It is easy to check that the functions wα , 0 ≤ α ≤ p n − 1, belong to the class ε(G). As above, the characteristic function of a set E is denoted by 1 E . Any function f ∈ εn (G) can be expressed as the series f (x) =

∞ 

f n,s 1Un,s ,

(5.1)

s=0

where f n,s are the values of f on U n, s (i.e., f n,s = f (A−n (h [ s])))). Besides, note that the proof of the equalities  wβ (x)dμ(x) = 0, 0 ≤ α ≤ pl+1 − 1, pl ≤ β ≤ pl+1 − 1 , l ∈ Z+

(5.2)

Ul,α

is similar to that of relations (1.5.18) from [4]. Using 5.1 and 5.2, we prove the following. Proposition 5.1 The following properties hold: (a) If f ∈ L 1 (G) ∪ ξn (G), then suppose fˆ ⊂ U−n . (b) If f ∈ L 1 (G) and supp fˆ ⊂ U−n , then fˆ ∈ ξn (G) Proof Suppose that f ∈ L 1 (G) ∩ ξn (G). Then, for any w ∈ G, from 5.1, we have fˆ(w) =



f (t)χ (t,¯ w)dμ(t) = G

∞ 



χ (t,¯ w)dμ(x).

f n,α Un

α=0

Setting t = A−n (h [α] ) ⊕ x and noting that A−n (U ) = Un , we obtain  Un,α

χ (t, w)dμ(t) = ξ(A−n (h [α] ), w)

 χ (x, w)dμ(x). Un

For an arbitrary element w = (w j ) from G \ U−n , we set s = min{ j : w j = 0}. We can easily see that s ≤ −n. Let l = −s. Then, for any element x = (x j ) from Un we have

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5 Orthogonal and Periodic Wavelets on Vilenkin Groups

 χ (x, w) = ex p

2πi (xn+1 w−n + xn+2 w−n−1 + · · · + xl+1 w−l ) = wβ (x), p

where β = w−n p n + w−n−1 p n−1 + · · · + w−l pl . Further, splitting Un into the sets Ul,α , and using 5.2 we obtain 

pl−n −1 

χ (x, w)dμ(x) = Un



wβ (x)dμ(x) = 0. Ul,α

α=0

Thus, fˆ(w) = 0 for all w ∈ G \ U−n and assertion (a) is proved. Now suppose that f ∈ L 1 (G) and supp f ⊂ U−n . Then fˆ(w) =

pn −1 

 f (t)χ (t, w)dμ(t) = U−n

 s=0

f (t)χ (t, w)dμ(t). U0,s

Hence, for w = (. . . , 0, 0, w−l , w−l+1 , . . . , w0 , w1 , w2 , . . .). Substituting t = h [s] ⊕ x, we obtain fˆ(w) =

pn −1



cs,β ws∗ (w),

(5.3)

s=0

where cs,β = U f (h [ s]) ⊕ xwβ (x)dμ(x), β=β(w) = w−l pl + w−l+1 pl−1 + · · · + w−l p + w0 . Since ws (w) and β(w) are constant on the sets Un,s , s ∈ Z+ , it follows from 5.3 that fˆ ∈ ξn G. Proposition 5.1 is proved. A function f ∈ L 2 (G) is said to be stable if there exist positive constants A0 and B0 such that   21   21 ∞ ∞ ∞ 

   2 2 |aα | ≤ aα f ·  h [α]  ≤ B0 |aα | , A0   α=0

α=0

α=0

for each sequence {αn } ∈ l 2 . In other words, a function f is stable in L 2 (G) if functions f (·  h), h ∈ H , form a Riesz system in L 2 (G). Note also, that a function f is stable in L 2 (G) with constants A0 and B0 if and only if A0 ≤



| fˆ(w  H )|2 ≤ B0 for a.e ∈ w ∈ G.

(5.4)

h∈H

The proof of this fact is quite similar to that of Theorem 1.1.7 in [5]. We say that a function g : G → C has a periodic zero at a point w ∈ G if g(w ⊕ h) = 0 for all

5.1 Multiresolution Analysis on Vilenkin Groups

175

h ∈ H. The set of all compactly supported functions in L 2 (G) will be denoted by L 2c (G). Proposition 5.2 For any f ∈ L 2c (G), the following properties are equivalent: (a) f is stable in L 2 (G). (b) { f (·  h)|h ∈ H } is a linearly independent system. (c) fˆ does not have periodic zeros. Proof The implication (a) ⇒ (b) follows from the well-known property of the Riesz system (see, e.g., Novikov et al. [5], Theorem 1.1.2). Our next claim is that f ∈ L 1 (G), since f has compact support and f ∈ L 2 (G). Let us choose a positive integer n such that supp f ⊂ U1−n . By Proposition 5.1, then f ∈ ξn−1 (G). Besides, if λ(h) > p n−1 μ{supp f (·  h) ∩ U1−n } = 0. Therefore, the linearly independence of the system and { f (·  h)|h ∈ H } is equivalent to that of the finite system { f (·  h [α] )|α = 0, 1, 2, . . . p n−1 − 1}. Further, if some vector (a0 , . . . , a pn−1 −1 ) satisfies conditions pn−1 −1



aα f (·  α) = 0 and |a0 | + · · · + |a2n−1 −1 | > 0,

(5.5)

α=0

then using the Fourier transform we obtain fˆ(w)

pn−1 −1



aα wα (w) = 0.

α=0

The Walsh polynomial pn−1 −1

W (w) =



aα wα (w)

α=0

is not identically equal to zero; hence among Un−1,s , 0 ≤ s ≤ p n−1 − 1 there exists a set (denote it by X ) for which W (X ⊕ h) = 0, h ∈ H . Since fˆ ∈ ξn−1 (G), it follows that 5.5 holds if and only if there exists a set X = Un−1,s , X ⊂ U , such that fˆ(X ⊕ h) = 0 for all h ∈ H . Thus (b) ⇒ (c). It remains to prove that (c) ⇒ (a). Suppose that fˆ does not have periodic zeros. Then F(w) =



| fˆ(w  h)|2

h∈H

is positive and H -periodic function. Moreover, since fˆ ∈ ξn−1 (G), we see that F is constant on each Un−1,s , 0 ≤ s ≤ p n−1 − 1. Hence, (5.4) is satisfied and Proposition 5.2 is established.

176

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

Proposition 5.3 Let φ ∈ L 2 (G) and the system {φ(·  h) : h ∈ H } be Riesz basis in the closed subspace V of space L 2 (G) to belong to V , it is necessary and sufficient that there exists H -periodic function m f ∈ L 2 (U ) such that ˆ fˆ(w) = m f (w)φ(w) w ∈ G.

(5.6)

Besides, the coefficient of decomposition f for the system {φ(·  h) : h ∈ H } coincide with the corresponding Walsh coefficient of function m f . Proof According to the known properties of Riesz system (see, for example, Theorem 1.1.2 in [5]) the subspace V consists of function of the type f (x) =





ch φ(x  H ) =

ch [α] φ x  h [α] , x ∈ G,

(5.7)

α∈Z+

h∈H

 where h∈H |ch |2 < ∞. Using the Fourier equation, we get f ∈ L 1 (G), since f has compact support and f ∈ L 2 (G). Let us choose a positive integer n such that supp f ⊂ U1−n . By Proposition 5.1, then f ∈ ξn−1 (G). Besides, if λ(h) > p n−1 , fˆ(w) =



ˆ ch χ (h, w)φ(w) =



ˆ ch [α] wα (w)φ(w).

(5.8)

α∈Z+

h∈H

It is evident from here that 5.6 is fulfilled if we assume  wα (w), w ∈ U. m f (w) =

(5.9)

α∈Z+

Conversely if 5.6 is satisfied and m ∈ L 2 (U ), then m f may be written in the form 5.9. From here follows 5.8 and that means even 5.7. The proposition is proved. We say that a function φ ∈ L 2c (U ) is a refinable function, if it satisfies an equation of the type pn −1 

(5.10) aα φ Ax  h [α] . φ(x) = p α=0

The functional equation 5.10 is called the refinement equation. The Walsh polynomials on G, i.e., finite linear combination of the characters, play the same role as the trigonometric polynomials in the real setting. So, we have



ˆ φ(w) = m 0 A−1 w φˆ A−1 w , where the Walsh polynomial

(5.11)

5.1 Multiresolution Analysis on Vilenkin Groups

177

pn −1

m 0 (w) = p



aα wα (w)

(5.12)

α=0

is called the mask of Eq. 5.10 (or the mask of its solution φ). The coefficients of Eq. 5.10 are related to the values bs of m 0 on Un,s by means of the direct and the inverse Vilenkin–Chrestenson transforms: p −1 1  bs wα (A−n h [s] ), 0 ≤ αp n − 1, aα = n p s=0 n

(5.13)

pn −1

bs =



aα wα (A−n h [s] ), 0 ≤ s ≤ p n − 1.

(5.14)

s=0

These transforms can be realized by the fast algorithms (see, for instance, [Schipp et al. [18]]). Thus, any choice of the values of m 0 on Un,s defines also the coefficient of Eq. 5.10. Now, we prove the following analogue of Theorem 4.4 ˆ ) = 1, then Theorem 5.1 Let ϕ ∈ L 2c (G) satisfy Eq. 5.10. If ϕ(θ pn −1



aα = 1 and suppϕ ⊂ U1−n .

α=0

The solution is unique, and is given by the formula ϕ(w) ˆ =



m 0 (A− j w),

(5.15)

and possesses the following properties: 1.  ϕ(h) ˆ = 0 for all h ∈ H \{θ }; 2. h∈H ϕ(x ⊕ h) = 1 for almost every x ∈ G (the partition of unity property). Proof As wα (θ ) = φ(θ ) = 1, after submission in 5.11 and 5.12 the values of w = θ we get the equation pn −1  aα = 1. α=0

Let s be the highest integer such that μ{x ∈ U0,s−1 |φ(x) = 0} > 0, where U0,s−1 = h [s−1] ⊕ U. Let us assume that s ≥ p n−1 + 1. Then the arbitrary element x from U0,s−1 has the form

178

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

x = (. . . , 0, 0, x−k , x−k+1 , . . . , x−1 , x0 , x1 , . . .), x j ∈ {0, 1, . . . , p − 1}, (5.16) where x−k p k + x−k+1 p k+1 + · · · + x−1 p + x0 = s − 1, x−k = 0, k ≥ n − 1.

(5.17)

Let us select this element such that the number λ(x) is not p rational (then the average of the component x j with positive indices is an infinite number other than zero). For any α ∈ {0, 1, . . . , p n − 1}, element y (α) = Ax  h [α] is the form y

(α)

 =

. . . , 0, 0,

(α) (α) (α) y−k−1 , y−k , . . . , y−1 , y0(α) , y1(α) , y2(α) , . . .

, k + 1 ≥ n,

(α) where y−k−1 = x−k = 0 and average of the number y αj , j ≥ 0 is other than zero. Consequently, (5.18) λ(Ax)h [α] > p n , x ∈ U0,s−1 .

If s ≤ p n then from (5.18), we get that φ(Ax)h [α] ) = 0 for x ∈ U0,s−1 . Then, on the strength of (5.10), φ(x) = 0 for any x ∈ U0,s−1 , which contradicts selection s. Hence s ≥ p n + 1. By using this inequality for any α ∈ {0, 1, 2, . . . p n − 1} from 5.16 and 5.17, we get λ(Ax  h [α] ) > p(s − 1) − ( p n − 1) ≥ 2(s − 1) − (s − 2) = s. From here just as above, it follows that φ(x) = 0 for x ∈ U0,s−1 . Hence, s ≤ p n − 1 and φ ⊂ U1−n . Let us prove that φˆ can be represented by the Formula (5.16). As φˆ is finite and belongs to L 2 (G), then it belongs also to L 1 (G). As suppφ ⊂ u 1−n , then as per ˆ ) = 1, Proposition 5.1, we have φˆ ∈ ξn−1 (G). On the strength of the condition φ(θ ˆ we get that φ(w) = 1 for all w ∈ Un−1 . On the other hand, m 0 (w) = 1, w ∈ Un−1 . This means that for any natural number l ˆ ˆ −l−n w) φ(w) = φ(A

l+n 

m 0 (A− j w) =

j=1

∞ 

m 0 (A− j w), w ∈ U−l .

j=1

Hence the equality is true and the solution φ is unique. As φ ∈ L 1 (G) then φˆ ∈ C0 (G). Besides m 0 (As h) = 1 on the strength of equality m 0 (θ ) = 1n 0 . Hence for any h ∈ H \ {θ } j−1  ˆ ˆ ˆ j h) → 0 φ(h) = φ(h) m 0 (As h) = φ(A s=0

when j → +∞. It follows from here that

5.1 Multiresolution Analysis on Vilenkin Groups

179

ˆ φ(h) = 0, h ∈ H \ {θ }.

(5.19)

Using now the Poisson summation formula: 



φ(x ⊕ h) =

h∈H

ˆ φ(h)χ (x, h),

h∈H

ˆ ) = 1, we get and using 5.19 and the condition φ(θ 

φ(x ⊕ h) = 1 ∀x ∈ G.

h∈H

Theorem 5.1 is proved. Proposition 5.4 Let φ ∈ L 2 (G). Then {φ(·  h) : h ∈ H } is an orthonormal system in L 2 (G) if and only if 

ˆ  h)|2 = 1 f ora.e.w ∈ G. |φ(w

h∈H

Proof The function φ(w) =



ˆ  h)|2 |φ(w

h∈H

is H -periodic: φ(w ⊕ h) = φ(w) for all h ∈ H. Furthermore, it has a finite L 1 -norm on U , because    ˆ  h)|2 dμ(w) φ(w)dμ(w) = |φ(w 0≤ U

=

U h∈H

 U ⊕h

h∈H

2 ˆ |φ(w)| dμ(w) =



2 ˆ |φ(w)| dμ(w) < +∞. G

ˆ Denote by {φ(h)} the Fourier coefficients of function φ with respect to the system {χ (h, ·)}. For any h ∈ H by a change of variable η = w  h, we have ˆ φ(h) =



 φ(w)χ (h, w)dμ(w) =

U

=

U

 h∈H

χ (h, w)

2 ˆ |φ(w)| dμ(w)

h∈H



2 ˆ |φ(w)| χ (h, w)dμ(w).

χ (h, η)|η| ˆ dμ(η) = 2

U ⊕h



G

Applying Plancherel’s equality, we get 

ˆ φ(x  h)φ(x)dμ(x) = φ(h), h ∈ H. G

180

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

ˆ To complete the proof, we note that 3.1 is equivalent to φ(h) = δθ,h for h ∈ H. For each l ∈ {0, 1, 2, . . . p − 1} let us denote by δl the sequence w = (w j ) such that w1 = l and w j = 0 for j = 1 (in particular, δ0 = θ .) Note that δl = A−1 wl ∈ U, ) for all l ∈ {0, 1, 2, . . . p − 1}. λ(δl ) = pl and wα (δl ) = ex p( 2παl p Proposition 5.5 Suppose that φ ∈ L 2 (G) satisfies 5.10 and the translates {φ(·  h) : h ∈ H } are orthonormal in L 2 (G). Then p−1 

|m 0 (w ⊕ δl )|2 = 1 f orall ∈ w ∈ G,

(5.20)

l=0

where m 0 is the mask of Eq. 5.10. Proof For l ∈ {0, 1, 2, . . . p − 1}, we put Hl = {h ∈ H : A−1 h  δl ∈ H } such that h 0 = l for h = (h j ) ∈ Hl . Since φˆ is continuous on G, by Proposition 5.4 we obtain  ˆ −1 w ⊕ A−1 h)|2 = 1, l ∈ {0, 1, 2, . . . p − 1}, |φ(A h∈Hl

for all w ∈ G. It follows from this and 5.11 that 1=

2    m 0 (A−1 (w ⊕ h))2  ˆ −1 (w ⊕ h)) φ(A  h∈H

p−1 2   2   −1 −1    ˆ m(A w ⊕ δl ) = |m(A−1 w ⊕ δl )|2 . φ(A h) = h∈H

h∈Hl

l=0

Thus, 5.20 holds. The proof is complete. We recall that a collection of closed subspaces V j ⊂ L 2 (G), j ∈ Z is called a multiresolution analysis (MRA) in L 2 (G) if the following hold: (i) (ii) (iii) (iv) (v)

V j ⊂ V j+1 for all j ∈ Z, ∪V j = L 2 (G) and ∩V j = {0}, f (·) ∈ V j ⇔ f (A·) ∈ V j+1 for all j ∈ Z, f (·) ∈ V0 ⇒ f (·  h) ∈ V0 for all h ∈ H, there is a function φ ∈ L 2 (G) such that the system {φ(·  h) : h ∈ H } is an orthonormal basis of V0 .

The function φ in condition (v) is called a scaling function in L 2 (G). For arbitrary φ ∈ L 2 (G) we set j

φ j,h (x) = p 2 φ(A j x  h), j ∈ Z, h ∈ H,

5.1 Multiresolution Analysis on Vilenkin Groups

181

and V j = span{φ j,h : h ∈ H }, j ∈ Z.

(5.21)

We say that a function φ generates an MRA in L 2 (G) if the system {φ(·  h) : h ∈ H } is orthonormal in L 2 (G) and, in addition, the family of subspaces V j is an MRA in L 2 (G). If a function φ generates an MRA in L 2 (G), then it is a scaling function in L 2 (G). In this case the system {φ j,h \ h ∈ H } is an orthonormal basis of V j for every j ∈ Z and one can define orthonormal wavelets ψ1 , ψ2 , ψ3 , . . . ψ p−1 in such a way that the functions j

ψl, j,h = p 2 ψl (A j x  h), 1 ≤ l ≤ p − 1, j ∈ Z, h ∈ H, form an orthonormal basis of L 2 (G) (see Sect. 5.2) Proposition 5.6 If the system, {φ(·  h) : h ∈ H } is orthonormal basis in V0 , then ∩V j = {0}. Proof From Eq. 5.20, it follows that at each j ∈ Z the system {φ j,h \ h ∈ H } is the orthonormal basis of space V j . Hence, the orthogonal projector P j : L 2 (G) → V j as per the formula  Pj f = ( f, φ j,h )φ j,h , f ∈ L 2 (G). h∈H

Let us assume that f ∈ ∩V j and fix ε > 0. The C0 (G) denotes the set of continuous on G function with compact suppport on L 2 (G). Let us select f 0 ∈ C0 (G) such that

f − f 0 < ε.

f − P j f 0 ≤ P j ( f − f 0 ) ≤ f − f 0 < ε and means

f ≤ P j f 0 + ε

(5.22)

for each j ∈ Z. If supp f 0 ⊂ Ul,n j



(P j f 0 , φ j,h ) = ( f 0 , φ j,h ) = p 2

f o (x)φ(A j x  h)dμ(x), Ul

where the number l depends on f 0 . Using Cauchy–Bunyakovsky inequality, from here we get

P j f 0 = 2

 h∈H

For j < l, we have

|(P j f 0 , φ j,h )| ≤ f 0

2

2

 h∈H

 p

|φ(A j x  h)|2 dμ(x).

j Ul

182

5 Orthogonal and Periodic Wavelets on Vilenkin Groups





 |φ(A j x  h)|2 dμ(x) =

pj

h∈H

Ul

|φ(x)|2 dμ(x), Sl, j

where Sl, j =



{y  h|y ∈ Ul− j }.

h∈H

Consequently,



P j f 0 ≤ f 0

2

1 Sl, j (x)|φ(x)|2 dμ(x).

2

(5.23)

G

It is easy to see that lim 1 Sl, j (x) = 0 x ∈ / H.

j→−∞

According to Lebesgue theorem, we get from 5.23 that lim P j u = 0.

j→−∞

Taking into consideration 5.22, we conclude that f ≤ ε and hence ∩V j = {0}. Proposition 5.7 Let the mask m 0 of the measuring Eq. 5.10 meet the conditions m(θ ) = 1 u

p−1 

|m(w⊕)δl |2 = 1 w ∈ G.

(5.24)

l=0

Then the function φ given by the equation ˆ φ(w) =

∞ 

m(A− j w),

(5.25)

j=1

is the L 2 -solution of Eq. 5.10 when φ ≤ 1. Proof Flow convergence of the product in 5.24 follows from the fact that the mask m 0 is equal to 1 for the set Un,0 (such that for any w ∈ G only a finite number of the factors in 5.24 may be different from 1). Let us represent the right-hand part of Eq. 5.25 by g(w). It follows from 5.24 that |m 0 (w)| ≤ 1 for all w ∈ G. Hence for any s ∈ N, we have s  |m 0 (A− j w)|2 |g(w)|2 ≤ j=1

and consequently,

5.1 Multiresolution Analysis on Vilenkin Groups





s 

|g(w)|2 dμ(w) ≤ U−l

183

|m 0 (A− j w)|2 dμ(w) = 2s

U−l j=1

  s−1

|m 0 (A j w)|2 dμ(w).

U j=0

(5.26) From the equation, pn −1

m 0 (w) =



aα wα (w), wα (w)wβ (w) = wαβ (w),

α=0

is seen that

pn −1

|m 0 (w)|2 =



cα wα (w),

(5.27)

α=0

where the coefficients cα are calculated using acα . Let us substitute 5.27 in the part Equation 5.24 and using the fact that α is smaller than p, then p−1 

wα (δl ) = p,

l=0

and for the remaining α this sum is equal to 0. As a result, we get c0 = for nonzero α, multiple p. It means, the following equation is true

1 p

and cα = 0

p −1 p−1   1 c pα+l w pα+l (w). |m 0 (w)| = + p α=0 l=1 n−1

2

From here, we get s−1 

|m 0 (A j w)|2 = p −s +

σ (s) 

bγ wγ (w), σ (s) ≤ sp n−1 ( p − 1),

γ =1

j=0

where each of the coefficients bγ is equal to the product of certain coefficients c pα+l , l = 1, 2, . . . p − 1. Taking into consideration that  wγ (w)dμ(w) = 0, γ ∈ N, U

we have

  s−1

|m 0 (A j w)|2 dμ(w) = p −s .

U j=0

Substituting in 5.26, for any arbitrary l ∈ N, we derive the equation

184

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

 |g(w)|2 dμ(w) ≤ 1 U−l

and consequently,

 |g(w)|2 dμ(w) ≤ 1.

(5.28)

G

Now let φ ∈ L 2 (G) and φˆ = g. Then, from 5.25, it follows that ˆ −1 w) ˆ φ(w) = m 0 (A−1 w)φ(A and this means φ satisfies Eq. 5.10. Besides, as per Parseval’s identity, we get from 5.28, that φ ≤ 1. Proposition 5.7 is proved. A compact subset E of G is said to be congruent to U modulo H , if μ(E) = 1 and for each w ∈ E, there is an element h ∈ H such that w ⊕ h ∈ U. Let m 0 be the mask of Eq. 5.10. We say that m 0 satisfies the modified Cohen condition, if there exists a compact subset E of G containing a neighborhood of the zero element such that: (1) the set E congruent to U modulo H ; (2) the inequality inf inf |m 0 (A− j w)| > 0 j∈N w∈E

(5.29)

is true. Since E is compact, we note that if m 0 (θ ) = 1 than there exists a number j0 such that m 0 (A− j w) = 1 for all j > j0 , w ∈ E. Therefore, 5.29 holds if the polynomial m 0 (w) does not vanish on the sets A−1 (E), . . . , A− j0 (E). Moreover, we can choose j0 ≤ p n , because m 0 is completely defined by the values bs , 0 ≤ s ≤ p n − 1. Proposition 5.8 Let the mask m 0 of Eq. 5.10 satisfy the conditions m 0 (θ ) = 1 u

p−1 

|m(w⊕)δl |2 = 1 for w ∈ G.

(5.30)

l=0

Suppose that the function φ ∈ L 2 (G) is determined with the help of formula ˆ φ(w) =

∞ 

m 0 (A− j w), w ∈ G.

(5.31)

j=1

The system {φ(·  h) : h ∈ H } is orthonormal in L 2 (G) then and only then when the full m 0 satisfies the modified condition of Cohen. Proof From 5.30 and 5.31, with the help of Proposition 5.5 we derive that the given function φ is the solution of Eq. 5.10 and satisfies the condition of the Theorem 5.1.

5.1 Multiresolution Analysis on Vilenkin Groups

185

Let us assume that the system {φ(·  h) : h ∈ H } is orthonormal in L 2 (G). Let us examine  ˆ  h)|2 . |φ(w (5.32) φ(w) = h∈H

Evidently, the function φ is nonnegative and possess the property H -periodicity. ˆ is constant on From Proposition 5.1, it follows that the function φ (and also φ) the sets Un−1,s . 0 ≤ s ≤ p n−1 − 1, merging of which coincides with U . Now, as in Proposition 5.5 we get that φ(w) = 1 for all w ∈ U . Let us construct the compact set E ⊂ G, which congruent to U modulo H , for which is fulfilled Eq. 5.29. For the arbitrarily fixed w(0) ∈ U , let us select the neighborhood Uk = {(w j ) ∈ G : w j = 0 j ≤ k}, such that,



ˆ (0)  h)|2 ≥ |φ(w

h ∗ ∈Vk

1 , 2

where Vk = Uk ∩ H. The set Vk is finite, but the function φˆ is continuous. Hence, the neighborhood (w(0) ) such that for all ξ ∈ (w(0) ) fulfilled the inequality 

ˆ  h)|2 ≥ |φ(ξ

h∈Vk

1 . 4

Let us find out such neighborhood (w) for all w ∈ U. On the strength of compactness of the subgroup U , from the covering/mask {(w) : w ∈ U }, it is possible to separate the finite covering/mask {(w(α) )}sα=1 . In the structure for each α ∈ {1, 2, . . . s}, there exists a number k(α) such that for all ξ ∈ (w(α) ) there is inequality  1 ˆ  h)|2 ≥ , |φ(ξ 4 h∈V k

where Vk(α) = Uk(α) ∩ H . Suppose V = 

α=1

Vk(α) . Then the inequality

ˆ  h)|2 ≥ |φ(w

h∈V

is true for all w ∈ U. Let c0 =

s

1 4

(5.33)

, where γ is the number of elements of the set ˆ  h)| ≥ c0 is met at V . On the strength of 5.33, for each w ∈ U the inequality |φ(w certain h ∈ V. Let us note that the set 1 √ 2 γ

ˆ ≥ c0 } S0 = {w ∈ U : |φ(w)|

186

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

contains the neighborhood of zero element θ on the strength of the continuous funcˆ ) = 1. From the elements h 1 , . . . h γ forming the set V , tion φˆ and the equation φ(θ let us determine by induction method the sets ˆ  h 1 )| ≥ c0 }, S1 = {w ∈ U \ S0 : |φ(w ˆ  h 2 )| ≥ c0 }, S2 = {w ∈ U \ (S0 ∪ S1 ) : |φ(w   γ −1  ˆ  h γ −1 )| ≥ c0 . Sγ = w ∈ U \ Sl : |φ(w l=1

γ The set E = l=0 is the neighborhood of the zero element of the group G and the congruent to U modulo H . Let us select the number j0 such that m 0 (A− j w) = 1 for all j ≥ j0 and all w ∈ E. According to 5.29, for w ∈ E we get ˆ |φ(w)| =

j0 

ˆ − j0 w)|, |m 0 (A− j w)||φ(A

j=1

ˆ where |φ(w)| ≥ cα . It is seen from here that for the constructed set E Condition 5.29 is true. Conversely, let us assume that there exists a compact set E congruence to U as per modulo H containing the neighborhood of zero element and such that the 5.29 is fulfilled. Let us prove that the system {φ(·  h) : h ∈ H } is orthonormal in L 2 (G). For arbitrary natural k, let us assume [k]

μ

=

k 

m 0 (A− j w)1 E (A−k w), w ∈ G.

j=1

From Condition 5.30, it is seen that m 0 (w) = 1 for all w ∈ Un,0 . Hence, it follows from 5.31 that (5.34) lim μ[k] = m 0 (w), w ∈ G. k→∞

Taking into consideration that A−n w ∈ Un,0 for w ∈ U , we have also m 0 (w) =

n−1 

m(B − j w), w ∈ U.

j=1

On the strength of 5.29, there exists the constant c1 > 0 such that |m 0 (A− j w)| ≥ c1 for j ∈ N, w ∈ E and this means c11−n |m(w)| ≥ 1 E (w) for w ∈ G. From here, we get

5.1 Multiresolution Analysis on Vilenkin Groups

187

k k     [k]    μ (w) = m 0 (A− j w) m 0 (A−k w) m 0 (A− j w)1 E (A−k w) ≤ c1−n 1

j=1

j=1

and taking into consideration 5.31  [k]  μ (w) ≤ c1−n |m 0 (w)|, k ∈ N, w ∈ G. 1 

Now let

(5.35)

 [k] 2 μ (w) ws (w)dμ(w) k ∈ N, s ∈ Z+ .

Ik (s) = G

Assuming ζ = A−k w, we find out 

k 

Ik (s) =

|m 0 (A− j w)|2 ws (w)dμ(w)

Uk,0 j=1

 = p

|m 0 (ζ )|

k

2

U

As U =

 p−1 l=0

k−1 

|m 0 (A j ζ )|2 ws∗ (Ak ζ )dμ(ζ ).

(5.36)

j=1

(A−1 (U ) ⊕ δl ), then 

Ik (s) = p k−1

p−1

|m(A−1 w ⊕ δl )|2

U

k−1 

|m(A j−1 w)|2 ws (Ak−1 w)dμ(w)

j=1

and on the strength of 5.30, it follows Ik (s) = p k−1

  k−2

|m 0 (A j w)|2 ws (Ak−1 w)dμ(w).

U j=0

From here and from 5.36, it follows that Ik (s) = Ik−1 (s). Similarly, when k = 1  I1 (s) = p

 |m(w)|2 ws (Aw)dμ(w) =

U

dμ(w), U

where the last integral is equal to δ0,s on the strength of the system {wα } ∈ L 2 (U ). It means (5.37) Ik (s) = δ0,s k ∈ N, s ∈ Z+ .

188

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

In particular, for all k ∈ N 

 [k] 2 μ (w) dμ(w) = 1.

Ik (0) = G

Using Lebesgue’s theorem on limited convergence, and from 5.34, 5.35, and 5.37 we find out that  2 ˆ |φ(w)| ws (w)dμ(w) = lim Ik (s) = δ0,s . k→∞

G

From here as per Plancheral’s formula, it follows that the system {φ(· ⊕ h) : h ∈ H } is orthonormal in L 2 (G). Proposition 5.7 is proved. Let M ⊂ U and let Tp M =

p−1 

{A−1 w[l] + A−1 (w) : w ∈ M}.

l=0

The set M is said to be blocked (for the mask m 0 ) if it coincides with some union of sets (5.38) Un−1,s = A1−n (w[s] ) ⊕ A1−n (U ), 0 ≤ s ≤ p n−1 − 1, which does not contain the set Un−1,0 and satisfies the condition T p M ⊂ M ∪ Z (m 0 ), where Z (m 0 ) is the set of all zeros of m 0 on U . It is clear that each mask can have only a finite number of blocked sets. ˆ ) = 1. Then Proposition 5.9 Let φ ∈ L 2c (G) be a solution of Eq. 5.10 such that φ(θ φ is not stable if and only if its mask m 0 possess a blocked set. Proof Using Theorem 5.1 and the Proposition 5.1, we have suppφ ⊂ U1−n φˆ ∈ ξn−1 (G). Let us assume that the function φ is unstable. While proving Proposition 5.2, it was established that if the unstable function f ∈ L 2 (G) has a carrier in U1−n , then there exists as set X = Un−1,s such that all the points of the set X are periodic zeros of the Fourier transform fˆ (and any periodic zero w ∈ U for fˆ is in the same set X ). Hence the set   ˆ ⊕ h) = 0 M0 = w ∈ U : φ(w is represented in the form of convergence of some from the sets Un−1,s , 0 ≥ s ≥ ˆ ) = 1, then M0 does not contain Un−1,s . Besides, if w ∈ M0 , then p n−1 − 1. As φ(θ as per Formula 5.11



m 0 A−1 w ⊕ A−1 h φˆ A−1 w ⊕ A−1 h = 0 for all h ∈ H

5.1 Multiresolution Analysis on Vilenkin Groups

189

and consequently, the elements A−1 w ⊕ A−1 w[l] , l = 0, 1, 2, . . . p − 1 belongs either to M0 or Z (m 0 ). Thus, if φ is unstable, then the set M0 is blocking set for m o . Conversely, let the mask m 0 has a blocking set M. Let us show that then each element from M is a periodic zero for φˆ (and, consequently, as per Proposition 5.2 the function φ is ˆ ⊕ h) = 0. unstable). Let us propose that there exists w ∈ M and h ∈ H also that φ(w Let us select the natural number j, for which A− j (w ⊕ h) ∈ Un−1 and then for each r ∈ {0, 1, 2, . . .}, let us find out u r ∈ U, h r ∈ H such that A−r (w ⊕ h) = u r ⊕ h r . Further for each r ∈ {0, 1, 2, . . . j − 1}, let us take l ∈ {0, 1, 2, . . . p − 1} such that A−1 h r = A−1 w[l] ⊕ wr , where wr ∈ H . Then u r +1 ⊕ h r +1 = A−r −1 (w ⊕ h) = A−1 (u r ⊕ h r ) = (A−1 u r ⊕ A−1 w[l] ) ⊕ wr and consequently, u r +1 = w A−1 u r ⊕ A−1 w[l] , l = l(r ) ∈ {0, 1, 2, 3 . . . , p − 1}. It is evident from here that if u r ∈ M, then u r +1 ∈ T p M. Besides, it follows from the equation ˆ ⊕ h) = φ(A ˆ −1 (w ⊕ h)) φ(w

j 

ˆ j) m 0 (A−r (w ⊕ h)) = φ(u

r =1

j 

m 0 (u r )

r =1

that all u r ∈ / Z (m 0 ). Thus, if u r ∈ M then u r +1 ∈ M, u 0 = w ∈ M, it follows from here that all u r ∈ M. this contradicts the fact that u j = A− j (w ⊕ h) ∈ Un−1 and the set M does not contain Un−1 = Un−1,s . Proposition 5.9 is proved. Proposition 5.10 Let φ be the finite L 2 solution of Eq. 5.10 such that θˆ = 1. The system {φ(·  h) : h ∈ H } is orthonormal in L 2 (G) then and only then when the mask m of Eq. 5.10 does not have blocking sets and satisfies the condition p−1 

|m 0 (w ⊕ δl )|2 = 1 w ∈ G.

(5.39)

l=0

Proof If the system {φ(·  h) : h ∈ H } is orthonormal in L 2 (G), then the condition 5.39 is fulfilled (see Proposition 5.5 and the absence of blocking sets follows from Propositions 5.2 and 5.3. Conversely, let mask m 0 does not have blocking sets and Condition 5.39 is fulfilled. As in 5.2, we suppose φ(w) =

 h∈H

ˆ  h)|2 . |φ(w

(5.40)

190

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

As the function φ is not negative and possess the property H -periodicity, on the strength of proposition 5.4 it is sufficient to verify that φ(w) = 1. Let a = in f {φ(w)|w ∈ U }. From Theorem 5.1 and Proposition 5.1, it follows that the function φ just as φˆ is constant on the sets Un−1,s then the function φˆ has a periodic zero and this means φˆ is unstable. According to Proposition 5.2 and 5.3, this contradicts the absence of blocking sets of mask m 0 . This means that the number α is positive. Besides, taking into consideration Theorem 5.1, we have φ(θ ) = 1. Thus 0 ≤ a ≤ 1. We notice that from 5.11 and 5.40 follows the equation p−1    m 0 (A−1 w  δl )2 φ(A−1 w  δl ). φ(w) =

(5.41)

l=0

Now let Ma = {φ(w) = a|w ∈ U }. If 0 < a < 1, then from 5.39 and 5.41, it follows that for any w ∈ Ma the elements belonging to A−1 w  δl , l = 0, 1, 2, . . . p − 1 are the blocking set, which contradicts the condition. Thus φ(w) ≥ 1 for all w ∈ U . From here and from the equation  φ(w)dμ(w) = U

 h∈H

ˆ 2 dμ(w) = |φ(w)|

U ⊕h

    ˆ 2 φ(w) dμ(w) = φ 2 G

as per Proposition 5.7, we get that  φ(w)dμ(w) = 1 U

Repeatedly using the inequality and using the fact that function φ is constant in sets Un−1,s 0 ≤ s ≤ p n−1 − 1, we conclude that φ(w) ≡ 1. Proposition 5.8 is proved. Theorem 5.2 Suppose that the refinement Eq. (5.10) possesses a solution ϕ ∈ L 2c (G) such that ϕ(θ ˆ ) = 1 and the corresponding mask m 0 satisfies the condition (5.30). Then the following are equivalents: (a) The function ϕ generates an MRA in L 2 (G). (b) The mask m 0 satisfies the modified Cohen’s condition. (c) The mask m 0 has no blocked sets. Proof According to Proposition 5.9, we have ∩V j = {0}. Embedding V − j ⊂ V j+1 from the fact that satisfies Eq. 5.10. Let us show that ∪V j = L 2 (G).

(5.42)

Let f ∈ (∪V j )⊥ and ε > 0. Let us select u ∈ L 2 (G) such that and uˆ ∈ C0 (G) with

fˆ − u

ˆ < ε. For the orthogonal projection P j f , the function f in the subspace V j

5.1 Multiresolution Analysis on Vilenkin Groups

191

we have

P j f 2 = (P j f, P j f ) = ( f, P j f ) = 0 and it means, ˆ < ε.

P j u = P j ( f − u) ≤ f − u = fˆ − u

(5.43)

ˆ Let us fix the number j ∈ N such that suppuˆ ⊂ U− j , A− j w ∈ Un−1 for all w ∈ suppu. − 2j −j 2 Using the fact that the system { p wα (A )} is orthonormal and full in L (U− j ) for ˆ − j w), we have the function Γ (w) = n(w) ˆ φ(A   |Γ (w)|2 dμ(w) = |cα (Γ )|2 , (5.44) p− j Uj

α∈Z+

where j

cα (Γ ) = p − 2

 U− j

Γ (w)Wα∗ (B − j w)dμ(w).

Noting that 

ˆ − j w)wα (A− j w) φ(A j x  h [α] )χ (x, w)dμ(x) = p − j φ(A G

we get j

p − 2 (u, , φ j,h ) = p − j

 Γ (w)wα (A− j w)dμ(w). U− j

From here and from (5.42), we find out

P j u 2 =





   ˆ − j 2 |w| ˆ 2 φ(A w) dμ(w).

|(u, φ j,h )|2 =

(5.45)

U− j

h∈H

According to the condition m 0 (w) = Un,0 the number j is selected such that A− j w ∈ ˆ − j w) = 1 for all ˆ From here and from (5.31), it is seen that φ(A Un+1 for w ∈ suppu. w ∈ suppu. ˆ Taking into consideration that suppuˆ ⊂ U− j from 5.43 and 5.45, we get that ˆ = u . ε > P j u = u

Consequently,

f < ε + u < 2ε. Thus,

 ∪ Vj

⊥

= {0},

192

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

and hence 5.42 is true. Thus the implications (b) ⇒ (a) and (c) ⇒ (a) are true. The inverse implications follow Proposition 5.8 and 5.10 and Theorem 5.2 is proved. Taking into consideration the H -periodicity of mask m 0 , we note that Condition (30) is equivalent to the equations b0 = 1, |bl |2 + |bl+ pn−1 |2 + · · · + |bl+( p−1) pn−1 |2 = 1, 0 ≤ l ≤ p n−1 − 1, (5.46) Examples of the function φ satisfying Condition 5.10 and generating MRA in L 2 (G) are given below. Example 5.1 If a0 = . . . a p−1 = 1p and all aα = 0 for α ≥ p then function φ = 1Un−1 is the solution of Eq. 5.10 in particular when n = 1, Haar function satisfies Eq. 5.10; (φ = 1U compare with Example 4.1). Example 5.2 Let p = n = 2 and b0 = 1, b1 = a, b2 = 0, b3 = b, where |a|2 + |b|2 = 1. Let us propose a0 =

(1 + a + b) (1 + a − b) (1 − a − b) (1 − a + b) , a1 = , a2 = , a3 = . 4 4 4 4

When α = 0, the modified Cohen condition is fulfilled on the set E = U and the corresponding solution φ regenerates MRA in L ( G). In particular, when a = 1 and a = −1, we get, respectively, Haar function : φ(x) = 1U (x) and mixed Haar function : φ(x) = 1U (x  h [1] ). If 0 < |a| < 1, then the solution φ is determined as in Example 4.2 by decompositon ⎛ ⎞  ∞  −1 −1 1 j 1U A x ⎝1 + a φ(x) = b w2 j+1 −1 A x ⎠ , x ∈ G. 2 j=0 When a = 0, the set U1,1 is blocked, the function φ is determined by the formula {φ(x) = φ(x  h [1] )} and the system {φ(·  h) : h ∈ H } is linearly dependent (since φ(x) = 21 1U (A−1 x)). Example 5.3 Let p = 2, n = 3 and b0 = 1, b1 = a, b2 = b, b3 = c, b4 = 0, b5 = α, b6 = β, b7 = γ , where |a|2 + |α|2 = |b|2 + |β|2 = |c|2 + |γ |2 = 1. The coefficients a0 , a1 , . . . a7 of Eq. 5.10 are determined simultaneously by the values b0 , b1 , . . . b7 and the expression of these coefficients through the parameters a, b, c, α, β, γ is obtained with the help of 5.13 and has the form

5.1 Multiresolution Analysis on Vilenkin Groups

a0 = a1 = a2 = a3 = a4 = a5 = a6 = a7 =

193

1 (1 + a + b + c + α + β + γ ), 8 1 (1 + a + b + c − α − β − γ ), 8 1 (1 + a − b − c + α − β − γ ), 8 1 (1 + a − b − c − α + β + γ ), 8 1 (1 − a + b − c − α + β − γ ), 8 1 (1 − a + b − c + α − β + γ ), 8 1 (1 − a − b + c − α − β + γ ), 8 1 (1 − a − b + c + α + β − γ ). 8

Blocking sets for mask m(w) =

7 

aα wα∗ (w).

α=0

If a = 0 or c = 0, then the system {φ(·  h)|h ∈ H } is linearly dependent. If, however, a and c are nonzero, then the function regenerates MRA in L 2 (G). Modified Cohen condition is fulfilled for E = U when abc = 0, and in case a = 0, b = 0, c = 0 it takes place for E = B(U3,0 ∪ U3,1 ∪ U3,3 ∪ U3,6 ). In particular, when a = c = 1, we get φ(x) =

1 1U (y)(1 + w1 (y) + bw2 (y) + w3 (y) + βw6 (y)), y = A−2 x, 4

(compare with 4.36). Example 5.4 Let us assume that Eq. 5.46 is fulfilled for certain numbers bs , 0 ≤ s ≤ P n − 1. Using Formula (5.13), let us find out the coefficient of mask pn −1

m 0 (w) =



aα wα (w).

α=0

In the sets Un,s , 0 ≤ s ≤ p n − 1 and the value bs . If it is additionally known that b j = 0 for j ∈ {1, 2, . . . p n−1 − 1}, then Eq. 5.10 with the coefficients found out aα has the solution φ generating KMA in L 2 (G) (modified Cohen condition is fulfilled for E = U ).

194

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

5.2 Compactly Supported Orthogonal p-Wavelets As in the previous section, we denote by G the Vilenkin group G p . Let us show that for each function φ ∈ L 2c (G) satisfying the equation pn −1

φ(x) = p



aα φ(Ax  h [α] )

(5.47)

α=0

and generating KMA {V j } ∈ L 2 (G). We may construct the function ψ (1) , . . . ψ ( p−1) such that the functions ψ (ν) j,h =

j (ν) j ψ (A x  h), 1 ≤ ν ≤ p − 1, j ∈ Z, h ∈ H, 2

form orthogonal basis in L 2 (G). Such function ψ (1) , . . . ψ ( p−1) will be called orthogonal p-wavelets. Proposition 5.11 Let f ∈ L 2 (G) and function φ generate KMA V j ∈ L 2 (G). For the function f to belong to V j , it is necessary and sufficient that there exists h-periodic function m f ∈ L 2 (U ) such that ˆ − j w) w ∈ G. fˆ(w) = m f (A− j w)φ(A This proposition is proved in a way similar to Proposition 5.3. As above, the orthogonal addition V j in V j+1 will be denoted as W j . In the following variant, the principle of unitary prolongation is used in finding the algorithm of the structure of orthogonal p-wavelets along with Theorem 5.2. Theorem 5.3 Let the scaling function φ ∈ L 2c (G) satisfy Eq. 5.47 and generate an MRA {V j }. Let us assume that p−1

M = {m ν (w ⊕ δ)k)}ν,k=0

(5.48)

where m 0 is the mask of Eq. 5.47 and m 1 , m 2 , . . . m p−1 , are certain H -periodic functions from L 2 (U ) is unitary for almost all w ∈ U . Then the functions ψ (ν) determined by the equations ˆ −1 w), ν = 1, 2, . . . p − 1, ψˆ (ν) (w) = m ν (A−1 w)φ(A

(5.49)

belong to W0 and posses orthogonality properties (ψ (ν) )(·  h [α] ), ψ (χ) (·, h [β] ) = δν,χ δα,β ν, χ ∈ {1, 2, . . . p − 1}, α, β ∈ Z+ . Proof As per Proposition 5.11, under Condition of 5.49, we have ψ (ν) ∈ V1 . Further as per Proposition 5.4, the equations

5.2 Compactly Supported Orthogonal p-Wavelets

(ν)

ψ (·), ψ (ν) (·  h) = δθ,h , h ∈ H, ν = 1, 2, . . . p − 1,

195

(5.50)

take place then and only then when ν = 1, . . . p − 1 fulfill the conditions 

|ψˆ (ν) (Aw ⊕ h)|2 = 1, w ∈ G.

(5.51)

h∈H

Using 5.49 and 5.51 in the form 

ˆ ⊕ A−1 h)|2 = 1 |m ν (w ⊕ A−1 h)φ(w

(5.52)

h∈H

for w ∈ G. Assuming h = w[α] , wk,s = δk⊕w[s] where α = ps + k, δk = A−1 w[k] , s ∈ Z+ , 0 ≤ k ≤ p − 1 and using H -periodicity of masks, let us transform the left part of 5.52 as follows: 









m ν w ⊕ A−1 w[α] φˆ w ⊕ A−1 w[α] m ν w ⊕ A−1 w[α] φˆ w ⊕ A−1 w[α]

α∈Z+

=

p−1  k=0

m nu (w ⊕ δk )m ν (w ⊕ δk )



ˆ ⊕ wks )φ(w ˆ ⊕ wks ). φ(w

s∈Z+

From the unitary nature of matrix 5.48 and Proposition 5.4, it follows that the last expression is equal to zero for almost all w ∈ G. Thus, Eq. 5.51 and, it also means, 5.50 are proved. The equations

φ(·  h), ψ (ν) (·) = 0, h ∈ H, ν = 1, . . . , p − 1, and

(ν) ψ (·  h), ψ (χ) (·) = 0, h ∈ H, ν = χ ,

are established similarly. Hence, ψ (ν) ∈ W0 for ν = 1, 2 . . . , p − 1 and the theorem is proved. From Theorems 5.1 and 5.2, we get the following algorithm for making the orthogonal p-wavelets. Algorithm A bs , 0 ≤ s ≤ p n − 1 • Step 1 Select the numbers bs , 0 ≤ s ≤ p n − 1, such that the following equation is satisfied b0 = 1, |bl |2 + |bl+ pn−1 |2 + · · · + |bl+(n+1) pn−1 |2 = 1, 0 ≤ l ≤ p n−1 − 1. (5.53)

196

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

• Step 2 Calculate the coefficients aα , 0 ≤ α ≤ p n − 1 by 5.13 and to verify that the mask pn −1  aα wα (w) m 0 (w) = α=0

satisfies the condition (b) or (c) of Theorem 5.2. • Step 3 Define the function φ ∈ L 2 (G) such that ˆ φ(w) =

∞ 

m 0 (A− j w), w ∈ G.

(5.54)

j=1

• Step 4 Find

pn −1

m ν (w) =



aα(ν) wα (w), 1 ≤ ν ≤ p − 1,

α=0 p−1

such that the matrix (m ν (w ⊕ δk ))ν,k=0 is unitary. • Step 5 Define p-wavelets ψ (1) , . . . , ψ ( p−1) by the formula ˆ −1 w), 1 ≤ ν ≤ p − 1. ψˆ (ν) (x) = m ν (A−1 w)φ(A Using the Fourier’s inverse transform, we have ψ

(ν)

pn −1

(x) = p



aα(ν) φ(Ax  h [α] ), 1 ≤ ν ≤ p − 1.

(5.55)

α=0

We notice that when p = 2, Algorithm A is equivalent to Algorithm A0 examined in Chap. 4. ˆ Let us show how to calculate the values of function φ(w) given by Formula 5.54 by the numbers selected in Step 1. For each 0 ≤ s ≤ p n − 1, let us assume γ (i 1 , . . . i n ) = bs , if s = i 1 p 0 + i 2 p 1 + · · · + i n p n−1 , i j ∈ {0, 1, . . . p − 1}, and determine numerical sequence {dl : l ∈ N} as follows. Let l=

k  j=0

then

μ j p j , μ j ∈ {0, 1, 2, . . . p − 1}, μk = 0, k = k(l) ∈ Z+ ,

(5.56)

5.2 Compactly Supported Orthogonal p-Wavelets

197

dl = γ (μ0 , 0, . . . , 0, 0), k(l) = 0; dl = γ (μ0 , 0, . . . , 0, 0)γ (μ0 , 0, . . . , 0, 0), k(l) = 1; ................................................. dl = γ (μk , 0, . . . , 0, 0)γ (μk−1 , 0, . . . , 0, 0) . . . γ (μ0 , μ1 , μ2 , . . . μn−2 , μn−1 ), k(l) = 1;

if k = k(l) ≥ n − 1. Further, let us denote by M0 the set of all natural l ≥ p n−1 in which the p-decomposition 5.56, among the collections (μ j , μ j+1 , . . . μ j+n−1 ) there is not a single one from the collections (0, 0, . . . , 0, 1), (0, 0, . . . 0, 2), . . . , (0, 0, . . . 0, p − 1) and assume M = {1, 2, . . . p n−1 − 1} ∪ M0 . We notice that when n = 2 the set M has the form ⎧ ⎫ k ⎨ ⎬ M= μ j p j : μ j ∈ {1, 2, . . . p − 1}, k ∈ Z+ . ⎩ ⎭ j=0

There takes place the next generalization of the Proposition 4.8. Proposition 5.12 Let m 0 and dl be determined by bs under Condition 5.53 as shown above. If the function φ is given by Formula 5.54 then ⎧ ⎪ ⎨1, w ∈ Un−1,0 ˆ φ(w) = dl , w ∈ Un−1,l l ∈ M ⎪ ⎩ / M. 0, w ∈ Un−1,l l ∈

(5.57)

Proof If w ∈ Un−1,0 then A− j w ∈ Un,0 for all j ∈ N but m 0 (w) = 1 on Un,0 . From here and from 5.54 follows the first equation in 5.57. Further, according to 5.53 b pn−1 = · · · = b( p−1) pn−1 = 0.

(5.58)

Let us take l ∈ N with Decomposition 5.56 and find out the least number j0 such that p j−1 > l + 1. Then, for any w ∈ Un−1,l when j ≥ j 0 we have A− j w ∈ Un,0 and it means m 0 (A− j w) = 1. From here, according to 5.54 we have ˆ φ(w) =

j0 −1



m 0 (A− j w), w ∈ Un−1,l .

(5.59)

j=1

When l = μ0 (that is, when k(l) = 0), in Formula 5.59 we will have j0 = 2, A−1 w ∈ Un,μ0 and A− j w ∈ Un,0 . Consequently, if l ∈ {1, 2, . . . p − 1} and w ∈ Un−1,l , then

198

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

ˆ φ(w) = b1 = γ (μ0 , 0, . . . 0, 0) = dl . Similarly, when k(l) = l, we have

m 0 (A−1 w) = γ (μ1 , 0, 0 . . . , 0, 0) m 0 A−2 w = γ (μ0 , μ1 , 0 . . . 0, 0), . . . , ˆ and this means that φ(w) = dl . In general case, from Formula 5.56 for any j ∈ N it follows that  μ j−1 μ0 1 l k−n . + μ = + · · · + + μ p + · · · + μ p j−1 j k p n+ j−1 p n p j−1 p Besides, if w ∈ Un−1,l , then A− j w ∈ Un+ j−1,l . If l ∈ / M, then on the strength of 5.58, then among the factors in 5.59 there is zero. If l ∈ M, then Formula 5.59 will have A−1 w ∈ Un,μk A−2 w ∈ Un,μk−1 + pμk ,... , A− j0 w ∈ Uμ,ν0 , where ν0 = μ j0 −1 + μ j0 p + · · · μk p k−n . From here taking into consideration H -periodicity of the polynomial m 0 follows the formula 5.57. Proposition 5.12 is proved. Let us show how to carry out Step 4 of the Algorithm A. When p = 2, it is possible to take just as in the theorem m 1 (w) = −w1 (w)m 0 (w ⊕ δ1 ). For p ≥ 3, let us introduce the notation bs(ν) = m ν (A−n h [s] ), ν = 0, 1, . . . p − 1, s = 0, 1 . . . p n − 1. According to 5.53, we have (0) (0) 2 2 n−1 − 1. b0(0) = |bl(0) |2 + |bl+ pn−1 | + · · · + |bl+( p−1) pn−1 | = 1, l = 1, 2, . . . p

For carrying out Step 4, it is necessary to find out the coefficients b0(ν) , b1(ν) , . . . , b(ν) pn −1 . ν = 1, 2, . . . , p − 1, such that the matrices

(5.60)

5.2 Compactly Supported Orthogonal p-Wavelets

⎛

199

(0) (0) bl(0) bl+ pn−1 . . . . . . bl+( p−1) pn−1

⎞

⎜ (1) (1) ⎟ (1) ⎜ b ⎟ bl+ pn−1 . . . . . . bl+( ⎜ l p−1) pn−1 ⎟ ⎜ ⎟ ⎜ ... ⎟ ... ... ... ... ⎝ ⎠ ( p−1) ( p−1) ( p−1) bl+ pn−1 . . . . . . bl+( p−1) pn−1 bl

(5.61)

will be unitary. For this to happen, it is necessary to use the following proposition based on Householder transformation (see Novikov et al. [5, §2.6]) r 2 Proposition 5.13 Let ck ∈ C, k = 1, 2 . . . r , c = 1, k=0 |ck | = 1, and let ck0 = c c 1−c00 k0 j0 ck , c0 j = c j0 1−c , ck j = δk j − 1−c , j, k = 1, . . . r. Then the matrix is unitary. 00 00 In addition to Proposition 5.13, let us show that the method of unitary prolongation of the matrix proposed in [6]. the unitary nature of the matrix 5.61 is of equal force due to the fact that, for each l, 0 ≤ l ≤ p n−1 − 1, is satisfied the equations: (0) (ν) (0) (ν) bl(0) bl(ν) + bl+ pn−1 bl+ pn−1 + · · · + bl+( p−1) pn−1 bl+( p−1) pn−1 = 0, ν = 1, 2, . . . p − 1 (5.62) (μ) (μ) (μ) (ν) (ν) bl(ν) bl + bl+ b + · · · + b b = 0, ν, μ = 1, 2 . . . p−1 pn−1 l+ pn−1 l+( p−1) pn−1 l+( p−1) pn−1 (5.63) − 1 unknown constant with respect to p( p − These equations form a system p( p+1) 2 1). Let us determine the matrices

⎛

1

c2 c1

⎜ −c2 1 ⎜ c1 ⎜ 0 0 R0 = ! ⎜ |c1 |2 + |c2 |2 ⎝ . . . . . . 0 0 |c1 |

⎛

1 ⎜ 0 ⎜ |c1(1) | ⎜ −c3 R1 = " ⎜ (1) ⎜c (1) 2 2 |c1 | + |c3 | ⎝ . 1. . 0

R p−2

0

c3 c1(1)

0 0 0 ... 0

... ... ... ... ...

...

0 0 ... 0 1 ... ... ... ... 0 0 ...

⎞ ⎛ 0 0 ⎜ ⎟ 0⎟ ⎜0 ⎜ 0⎟ ⎟+⎜ 0 ⎝. . . ⎠ ... 0 0 ⎞

⎛

0 0 ⎜0 ⎜ 0⎟ ⎟ ⎜ ⎜0 0⎟ 0 ⎟+⎜ ⎟ ⎜ ⎜0 ⎠ ⎜ ... ⎝. . . 0 0

0 0 0 ... 0

0 1 0 0 0 ... 0

0 0 1 ... 0

0 0 1 0 0 ... 0

... ... ... ... ...

0 0 0 1 0 ... 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ . . .⎠ 1

0 0 0 0 1 ... 0

... ... ... ... ... ... ...

......................................................................................... ⎛ ⎛ cp ⎞ 0 0 0 ... 1 0 0 . . . ( p−2) c1 0 1 0 ... ⎜ 0 0 0 ... 0 ⎟ ⎜ ( p−2) ⎜ ⎟ ⎜ | |c1 0 0 1 ... ⎜ 0 0 0 ... 0 ⎟ ⎜ ⎜ =" ⎜ ⎟+ ... ... ... ... ⎜ ... ... ... ... ... ⎟ ⎜ ( p−2) 2 ⎜ 2 |c1 | + |c p | ⎝ ⎠ ⎝ 0 0 0 ... −c p 0 0 . . . 1 ( p−2) 0 0 0 ... c1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ . . .⎠ 1

0 0 0 ... 1 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ . . .⎟ ⎟ 0⎠ 0

200

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

(0) where c1 = b0(0) , c2 = b(0) pn−1 ,…c p = b" ( p−1) p(n−1) ! ( p−3) |c(1) | |c | ( p−2) c1(1) = |cc11 | |c1 |2 + |c2 |2 , c1(2) = 1(1) |c1(1) | + |c3 |2 . . . c1 = 1( p−3) c1 c1 " ( p−3) |c1 | + |c p |2 Let us now examine the unitary transformation C P with the matrix R = R p−1 R p−2 . . . R0 . This transformation converts the plane

c1 z 1 + c2 z 2 + · · · + c p z p = 0 into plane 21 = 1. Let us select in C P the orthogonal system {a1 , a2 , . . . a p−1 } consisting of vectors of the type = =

a1 a2 a p−1

(0, a12 , a13 , . . . a1 p )t (0, a22 , a23 , . . . a2 p )t

........................................ = (0, a( p−1)2 , a( p−1)3 , . . . a( p−1) p )t

and find out part of the coefficients 5.60 from the equations

t (ν) (ν) = R −1 aν ν = 1, . . . p − 1 b0 , b pn−1 , . . . , b((0) p−1) pn−1 Using this algorithm successively for l = 1, . . . , p n−1 − 1 and assuming, respectively, (0) (0) c1 = bl(0) , c2 = bl+ pn−1 , . . . c p = bl+( p−1) pn−1 ,

we find out all the remaining coefficients 5.60, which also permit in constructing p-spurs ψ (1) , . . . , ψ ( p−1) Example 5.5 Let p = 3, n = 2. According to Step 1 of the algorithm A, let us select the numbers a, b, c, α, β, γ such that |a|2 + |b|2 + |c|2 = |α|2 + |β|2 + |γ |2 = 1, and then propose b0 = 1, b1 = a, b2 = α, b3 = 0, b4 = b, b5 = β, b6 = 0, b7 = c, b8 = γ . Then for l = 1, we have ⎛

|a|

|a|b a |a|

ac |a|

⎞

⎟ ⎜ |a|b ⎟ ⎜ 0 R = ⎜− a √1−|c|2 √1−|c|2 ⎟ ⎠ ⎝ ! − √ ac 2 − √ bc 2 1 − |c|2 1−|c|

1−|c|

5.2 Compactly Supported Orthogonal p-Wavelets

201

Further, for the numerical values a = 0.900000, b = 0.435889, c = 0.000000, α = 0.900000, β = 0.2064742, γ = 0.383886 

and vectors a1 =

(0, 1, 0)t , l = 0, l = 2 (0, 0, 1)t , l = 1

 a2 =

(0, 0, 1)t , l = 0, l = 2 (0, 1, 0)t , l = 1,

Using the method shown above, we get b0(1) = 0.000000, b1(1) = 0.000000, b2(1) = −0.223607, b3(1) = 1.000000, b4(1) = 0.00000, b5(1) = 0.974679, b6(1) = 0.000000, b7(1) = 1, b8(1) = 0.000000, b0(2) = 0.000000, b1(2) = −0.435890, b2(2) = −0.374166, b3(2) = 0.000000, b4(2) = 0.900000, b5(2) = −0.085840, b6(2) = 1.000000, b7(2) = 0.000000, b8(1) 0.923380. According to 5.55, we find out from here ψ (ν) (x) = 3

∞ 

aαν φ Ax  h [α] , ν = 1, 2

α=0

where aαν is calculated from the values found out with the help of discrete transformation 5.13. From Formula 5.55, it is seen that the calculation of values of ψ (ν) (x) it is sufficient to be able to calculate the values φ(x). For this, the decomposition of function φ(x) in the Walsh–Fourier series may be useful (see Remark of Chap. 4). According to 5.57, we have  ˆ φ(w) = 1Un−1 (w) + dl 1Un−1 (w  A1−n h [l] ). (5.64) l∈M

As 

 (w  A1−n h [l] )χ (x, w)dμ(w) = χ (x, A1−n h [l] ) G

χ (x, w)dμ(w) Un−1

= p 1−n 1U (A1−n x)χ (A1−n x, h [l] ) = p 1−n 1U (A1−n x)wl (A1−n x). The formal application of Fourier inverse transformation to Eq. 5.64 leads to the decomposition

202

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

 φ(x) =

1 p n−1

1U (A1−n x)(1 +



dl wl (A1−n x)), x ∈ G.

(5.65)

l∈M

Example 4.3 shows that the Fourier–Walsh series does not converge absolutely for certain scaling functions φ. Sufficient condition of absolute and uniform of series 5.64 are formulated in analogy with the Lemma 7.3. The series 5.64 gets transformed into the final sum in that and only in that case when the function φ belongs to ε(G). The following criterion takes place. Proposition 5.14 Let φ ∈ L C2 (G) be a solution of Eq. (5.47) such that φ(θ ) = 1. Then φ does not belong to ε(G) if and only if there exists a finite sequence V , N ≤ p n−1 + n − 1, dk ∈ {0, 1, · · · , . p − 1}, such that {dk }k=1 (i) d1 = · · · = dn−1 = 0, dn = 0, (ii) there exists an integer j, n − 1 ≤ j ≤ N − 1, such that d j−s = d N −n for s = 0, · · · , n − 2, (iii) m 0 (ω(0) )m 0 (ω(1) ) · · · m 0 (ω(N −n) ) = 0, where m 0 is the mask of Eq. (5.47) and N such that ω(k) ∈ U are determined by the sequence {dk }k=1 λ(ω(k) ) =

N 

dν+k p −v , k = 0, 1, . . . , N − n.

ν=1

The proof of this proposition is essentially the same as in the case p = 2 (see Theorem 4.13). Example 5.6 Let p = 3, n = 2. Just as in Example 5.5, let us propose b0 = 1, b1 = a, b2 = α, b3 = 0, b4 = b, b6 = 0, b7 = c, b8 = γ , where |a|2 + |b|2 + |c|2 = |α|2 + |β|2 + |γ |2 = 1. Then in conformity with step 2 of Algorithm A the coefficients of the mask m 0 has the form

1 (1 + a + b + c + α + β + Γ ), 9 1 a1 = (1 + a + α + (b + β)ε32 + (c + γ )ε3 ), 9 1 a2 = (1 + a + α + (b + β)ε3 + (c + γ )ε32 ), 9 1 a3 = (1 + (a + b + c)ε32 + (α + β + γ )ε3 ), 9 a0 =

5.2 Compactly Supported Orthogonal p-Wavelets

203

1 (1 + c + β + (a + γ )ε32 + (b + α)ε3 ), 9 1 a5 = (1 + b + γ + (a + β)ε32 + (c + α)ε3 ), 9 1 a6 = (1 + (a + b + c)ε3 + (α + β + γ )ε32 ), 9 1 a7 = (1 + b + γ + (a + β)ε3 + (c + α)ε32 ), 9

a4 =

a8 =

1 (1 + c + β + (a + γ )ε3 + (b + α)ε32 ), 9

where ε3 = ex p(2πi/3). The blocking sets of this mask are as follows: 1. U1,1 when a = c = 0, when α = β = 0, 2. U1,2  3. U1,2 U1,2 when a = α = 0. It is seen from here that the scaling function φ formed from the given parameters regenerate MRA in L 2 (G) in the following cases: 1. a = 0, α = 0, 2. a = 0, α = 0, c = 0, 3. α = 0, a = 0, β = 0. In each of these cases, the summation in 5.65 is carried out as per the set M=

⎧ k ⎨ ⎩

μ j 3 j : μ j ∈ {1, 2}, k ∈ Z+

j=0

⎫ ⎬ ⎭

= {1, 2, 4, 5, 7, 8, 13, 14, 16, 17, 22, 23, 25, · · · }. Let us give the values of certain coefficients d1 = a, d2 = 0, d1 = ab, d5 = aβ, d16 = aβcβ, d17 = aβγ , d22 = abc, d7 = αc, d8 = αγ , d13 = ab2 , d14 = abβ, d41 = ab2 β, d43 = abβc, d23 = αβc, d25 = αcγ , d26 = αγ 2 , d40 = ab3 , 2 d67 = ab2 c, d44 = abβγ , d49 = abβγ , d50 = aβ c, d52 = aβcγ , d53 = aβγ 2 , 2 d68 = αbβc, d70 = αβc , d71 = αβcγ , d76 = abcγ , d77 = αβcγ , . . . With help of Proposition 5.14, we find that the series in (5.65) get transformed into the final sums (and the function φ is step function) in the following two cases: 1. If |α| = 1, |a|2 + |c|2 = 1, b = β = γ = 0, then



φ(x) = (1/3)1v (A−1 x) 1 + aw1 A−1 x + αw2 A−1 x + αcw7 A−1 x . (5.66) 2. If |a| = 1, |α|2 + |β|2 = 1, b = c = γ = 0, then

204

5 Orthogonal and Periodic Wavelets on Vilenkin Groups







φ(x) = (1/3)1v A−1 x 1 + aw1 A−1 x + αw2 A−1 x + aβw5 A−1 x . (5.67) According to Proposition 5.6, if in Algorithm A the selected vectors, all the values of bs with indices (b0 , b1 , . . . , b pn −1 < p n−1 are zero, then the corresponding mask m 0 satisfies the modified condition of Cohen and consequently the function φ determined by formula (54) generates MRA. If there is zero among the components bs , s < p n−1 , then the mask m 0 the conditions (b) or (c) from Theorem 5.2 are verified in Step 2. In particular, it is necessary to clarify if the mask m 0 has a blocked set. In case of small p and n, this task is solved by a simple method as in Example 5.6, but when there is increase in the dimension the computing complexity in the search for blocked sets increases sharply. In recent works by Lukomskii [7, 8], Berdnikov and Lukomskii [9] on class of step scaling functions for the search of similar masks N -valid trees is used. In particular, there is the following proposition (see Lukomskii [7, 8]). Proposition 5.15 Let l ∈ {1, 2, . . . , p − 2} where p > 2 is a simple number. Let us propose that El(0) and El(1) form a division of set {0, 1, . . . , p − 1} such that El(0) = { j1 , j2 , . . . , jl } and El(1) . For fixed j0 ∈ El(1) , j0 = 0 when n = 2 let us select in (5.49) numbers such that bs , s = s1 + s2 p, s1 , s2 ∈ {0, 1, . . . , p − 1}, such that 1. 2. 3. 4.

b0 = 1, |bs | = 1 for all s ∈ El(1) 0, |b j1 + j0 p | = |b j2 + j1 p | = · · · = |b jl + jl−1 p | = 1. bs = 0 in the rest of the cases.

Further let us determine the coefficients of the mask m 0 by Formula (5.13). Then the function φ given by the formula (5.50) belonging to the class εl (G) and generates KMA in L 2 (G). Let us assume that the scaling function φ and mask m 0 are determined as in Proposition 5.15. Let us select the mask p2 −1

m˜ 0 (ω) =



a˜ α wα (ω),

α=0

values bs of which in the sets in Un,s satisfy the condition b˜0 = 1, |b˜k |2 + |b˜k+ p |2 + · · · + |b˜k+( p−1) p |2 = 1, 0 ≤ k ≤ p − 1 while m˜ 0 (ω) = 0 for all values of ω such that |m 0 (ω)| = 1. Then as in step 3 of Algorithm A, let us determine the function φ˜ by the formula ∞

 ˆ˜ φ(ω) = m˜0 (A− j ω). j=1

5.2 Compactly Supported Orthogonal p-Wavelets

205

This function φ˜ generates KMA in L 2 (G) (in fact, if the condition b of Theorem 5.2 is met for m 0 with compact set E, then this is true even for the mask m˜ 0 of the function ˜ We also notice that when p = 3, Proposition 5.15 leads to two conditions φ). 1) b0 = 1, |b1 | = |b5 | = 1, bs = 0 for the remaining s(E 1(0) = {2}, E 1(1) = {0, 1}, j0 = 1);

5.3 Periodic Wavelets on Vilenkin Groups As above, we write G = G p , N = p n , and ε p = exp(2πi/ p). We recall that the automorphism A is defined by (Ax) j = x j+1 where x = (x j ) ∈ G. For 0 ≤ k ≤ N − 1, we let xn, k := A−n h [k] and Uk(n) := xn, k + A−n (U ). It is easily seen that Uk(n) ∩ Ul(n) = ∅ for k = l,

N −1 

Uk(n) = U.

k=0

Further, we shall use the notation wl,(n)k := wl (xn, k ) for 0 ≤ l, k ≤ N − 1. Notice that

N −1 

(n) −sq (n+1) wl,(n)k = wk, l = ε p w pk+s, N q+l , 0 ≤ s, q ≤ p − 1,

wi,(n)l wi,(n)k =

i=0

N −1 

(5.68)

(n) wl,(n)j wk, j = N δl, k , 0 ≤ l, k ≤ N − 1.

(5.69)

j=0

A finite sum D N (x) :=

N −1 

w j (x), x ∈ G,

j=0

is called the Walsh–Dirichlet kernel of order N . It is well known that # N , x ∈ U0(n) , D N (x) = 0, x ∈ U \ U0(n) .

(5.70)

Let us introduce the following spaces (j)

Vn := span{1, w1 (x), . . . , wN−1 (x)}, Wn := span{wjN (x), wjN+1 (x), . . . , w(j+1)N−1 (x)}, ( p−1)

respectively. Note that the orthogonal direct sum of Vn , Wn(1) , . . . , Wn coincides $ $ ( p−1) $ with Vn+1 , that is, for Wn := Wn(1) · · · Wn Wn = Vn+1 . The , we have Vn

206

5 Orthogonal and Periodic Wavelets on Vilenkin Groups ( j)

spaces Vn and Wn will be called the approximation spaces and wavelet spaces , respectively. We can use the discrete Vilenkin–Chrestenson transform to recover v ∈ Vn from the values v(xn, l ), 0 ≤ l ≤ N − 1. Indeed, if v(x) =

N −1 

ck wk (x), x ∈ U,

(5.71)

k=0

then

N −1 1  v(xn, l )wl,(n)k , 0 ≤ k ≤ N − 1. ck = N l=0

(5.72)

Suppose that a = (a0 , a1 , . . . , a N −1 ), where ak = 0, 0 ≤ k ≤ N − 1. Then we set Φ Na (x) :=

N −1 1  ak wk (x), φn, k (x) := Φ Na (x  xn, k ), 0 ≤ k ≤ N − 1, x ∈ G. N k=0

Proposition 5.16 Let v ∈ Vn . Assume that αn, k = αn, k (v) :=

N −1 

(n) al−1 cl wl,k , 0 ≤ k ≤ N − 1,

(5.73)

l=0

where cl are defined as in (5.72). Then v(x) =

N −1 

αn, k φn, k (x).

(5.74)

k=0

Proof According to the orthogonal relations (5.73), we get N −1 

wl,(n)k φn, k (x) = al wl (x), 0 ≤ l ≤ N − 1,

k=0

and, in view of (5.71), (5.72), and (5.73), v(x) =

N −1 N −1  

al−1 cl wl,(n)j φn, j (x)

l=0 j=0

=

N −1 

αn, k φn, k (x).

k=0

Therefore, the expansion in (5.78) is valid for any v ∈ Vn . The proposition is proved. %n,k are defined by Remark 5.1 Suppose that φ

5.3 Periodic Wavelets on Vilenkin Groups

%n,0 (x) = φ

N −1 

207

% % a −1 j w j (x), φn,k (x) = φn,0 (x ⊕ x n, k ), k = 1, . . . , N − 1.

j=0 N −1 N −1 %n,k }k=0 Then {φ is a dual shift basis for {φn,k }k=0 . Indeed, using (5.75) and (5.77), for any v ∈ Vn we have



%n,k ) := (v, φ

%n,k (x) d x = v(x), φ

 &

U

=

U

 & U

cl wl (x)

'& 

l

' %n,0 (x ⊕ xn, k ) d x cl wl (x) φ

l

' (n) a l−1 wl,k wl (x)

d x = αn, k (v),

l

where the last equality follows from the orthogonality of {wk : k ∈ Z+ }. Let b = (b0 , b1 , . . . , b pN −1 ), where bk = 0 for all 0 ≤ k ≤ pN − 1. In particular, we can choose # # ak/ p if k is divisible by p , ak if k ≤ N − 1 , or bk = bk = 1 if k is not divisible by p 1 if 0 ≤ k ≤ pN − 1 . Then we set b Φ pN (x) :=

pN −1 1  b bk wk (x), φn+1, k (x) := Φ pN (x  xn+1, k ), 0 ≤ k ≤ pN − 1, x ∈ G, pN k=0

and define ( j)

ψn, k (x) :=

p−1 

ε−p js φn+1, pk+s (x), 0 ≤ k ≤ N − 1, 1 ≤ j ≤ p − 1.

s=0 ( j)

N −1 is a bases for the corresponding Let us show that, for each j, the system {ψn,k }k=0 ( j) wavelet space Wn . ( j)

Proposition 5.17 Suppose that w ∈ Wn for some j ∈ {1, . . . , p − 1}. Then w(x) =

N −1 

( j)

βn,k ψn,k (x),

(5.75)

k=0

where with the notations as in (5.77), βn,k = βn,k (w) = αn+1, pk (w), 0 ≤ k ≤ N − 1.

(5.76)

208

5 Orthogonal and Periodic Wavelets on Vilenkin Groups ( j)

( j)

Proof Let w ∈ Wn where j ∈ {1, . . . , p − 1}. Then, since Wn Proposition 2.1 we have w(x) =

( j+1)N  −1

cl wl (x) =

l= j N

pN −1 

αn+1, k (w)φn+1, k (x) =

p−1 N −1  

⊂ Vn+1 , as in

αn+1, pk+s (w)φn+1, pk+s (x),

(5.77)

s=0 k=0

k=0

where αn+1, pk+s (w) =

N −1 

(n+1)

b−1 j N +l c j N +l w j N +l, pk+s , c j N +l =

l=0

pN −1 1  (n+1) w(xn+1, l )wl, j N +l . pN l=0

(n+1) Here, in view of (5.72), w(n+1) j N +l, pk+s = ε p w j N +l, pk , and hence js

− js

αn+1, pk+s (w) = ε p

αn+1, pk (w), 0 ≤ k ≤ N − 1, 0 ≤ s ≤ p − 1,

which by (5.77) and (5.78) yields (5.75). The proposition is proved. Let α = 0. For the case where #

#

α if k = 0 or k = pN − 1 , 1 otherwise (5.78) the assertions which are equivalent to Propositions 5.16 and 5.17 can be found in Farkov [10]. Notice that the value α = 1 corresponds to the Haar case (so, we use α = 1 in the sequel). For each l ∈ {0, 1, . . . , N − 1} with p-ary expansion ak =

α if k = 0 or k = N − 1 , 1 otherwise,

l=

n−1 

bk =

ν j p j , ν j ∈ {0, 1, . . . , p − 1},

j=0

we let γ (l) :=

n−1 j=0

ν j . In Case (5.82), we have the following equalities

φn, k (x) =

p−1 

φn+1, pk+s (x) −

s=0

1 φn+1, pk+s (x) = p

&

(1 − α) −γ (k) εp w N −1 (x), N

(5.79)

' p−1 N −1 1 − α  γ (ν)−γ (k) 1  j ( p−s) ( j) εp φn,ν (x) + εp ψn,k (x), φn,k (x) + αN p ν=0

j=1

(5.80) where 1 ≤ k ≤ N − 1, 0 ≤ s ≤ p − 1. Note also, that w N −1 (x) can be expressed as p−1 N −1 N −1   1  γ (s) w N −1 (x) = ε φn, s (x) = γn+1, pk+s φn+1, pk+s (x), α s=0 p k=0 s=0

(5.81)

5.3 Periodic Wavelets on Vilenkin Groups

209

where γn+1, pk+s := w(n+1) N −1, pk+s . For any functions f n ∈ Vn and gn ∈ Wn , we write f n (x) =

N −1 

Cn,k φn,k (x), gn (x) =

k=0

p−1 

gn( j) (x), gn( j) (x) =

j=0

N −1 

( j)

Dn,k ψn,k (x),

k=0

(5.82) where the coefficient sequences ( j)

Cn = {Cn,k }, D(nj) = {Dn,k }, 1 ≤ j ≤ p − 1,

(5.83)

uniquely determine f n and gn , respectively. Let us describe the algorithms, in terms of the coefficient sequences (5.87), for decomposing f n+1 ∈ Vn+1 as the orthogonal ( j) ( j) ( j) sum of f n ∈ Vn and gn ∈ Wn , and for reconstructing f n+1 from f n and gn . As a consequence of (5.80), we observe that φn+1, pk+s (x) =

N −1 

A(n) pk+s,ν φn,ν (x) +

p−1 

ν=0

( j)

B (n) pk+s, j ψn,k (x),

(5.84)

j=1

where ⎧ ⎨ 1/ p + (1 − α)/(αpN ),

A(n) pk+s,ν =

⎩

γ (ν)−γ (k) εp (1

ν = k,

− α)/(αpN ),

ν = k,

−1 j ( p−s) . B (n) pk+s, j = p ε p

Since f n + gn = f n+1 , it follows from (5.82) and (5.84) that N −1 

Cn,ν φn,ν (x) +

ν=0



Cn+1, pk+s

s,k

   ν

( j) ( j) Dn,ν ψn,ν (x) =

j=1 ν=0

=

=

p−1 N −1  

A(n) pk+s,ν φn,ν (x) +

ν=0

p−1 

( j)

B (n) pk+s, j ψn,k (x)

j=1

 Cn+1, pk+s A(n) pk+s,ν

Cn+1, pk+s φn+1, pk+s (x)

s=0 k=0

⎧ N −1 ⎨ ⎩

p−1 N −1  

φn,ν (x) +

s,k

 p−1   j=1

⎫ ⎬ ⎭ 

Cn+1, pk+s B (n) pk+s, j

( j)

ψn,k (x).

s,k

This implies that Cn,ν =

 s,k

A(n) pk+s,ν C n+1, pk+s ,

( j) Dn,ν =

 s,k

B (n) pk+s, j C n+1, pk+s .

(5.85)

210

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

Now, using (5.79) and (5.80), we obtain p−1 N −1  

φn,ν (x) =

Q (n) pk+s,ν φn+1, pk+s (x),

k=0 s=0

where Q (n) pk+s,ν =

⎧ γ (k) ⎨ 1 − ε p (1 − α)γn+1, pk+s /N , ⎩

k = ν,

γ (k)

−ε p (1 − α)γn+1, pk+s /N ,

k = ν.

Therefore, we have 

Cn+1, pk+s φn+1, pk+s (x) =



 Cn,ν

ν

k,s

+

p−1 N −1  

( j) Dn,k

=

k,s

⎩

 Q (n) pk+s,ν φn+1, pk+s (x)

k,s

 p−1 

 ε−p js φn+1, pk+s (x)

s=0

j=1 k=0

⎧  ⎨



Q (n) pk+s,ν C n,ν +

ν



( j)

ε−p js Dn,k

j

and so Cn+1, pk+s =

 ν

Q (n) pk+s,ν C n,ν +



⎫ ⎬ ⎭

φn+1, pk+s (x)

( j)

ε−p js Dn,k .

(5.86)

j

We remark that the decomposition and reconstruction algorithms based on Formulas (5.85) and (5.86) have more simple structure than the similar algorithms constructed in Chui and Mhaskar [11] for the case of trigonometric wavelets. To conclude this section, let us consider the case where p = 2, N = 2n , and # bk =

0 ≤ k ≤ N − 1, ak , a N −k , N ≤ k ≤ 2N − 1;

(5.87)

recall here that all ak = 0. Then, for any k ∈ {0, 1, . . . , N − 1}, φn,k (x) = φn+1,2k (x) + φn+1,2k+1 (x), ψn,k (x) = φn+1,2k (x) − φn+1,2k+1 (x), and thus φn+1,2k (x) =

1 1 [φn,k (x) + ψn,k (x)], φn+1,2k+1 (x) = [φn,k (x) − ψn,k (x)]. 2 2

5.3 Periodic Wavelets on Vilenkin Groups

211

So, under Condition (5.87), instead of (5.85) and (5.86) we obtain the classical Haar discrete transforms.

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform Let us denote by k p the remainder from the division of the integer k by the natural number p, and let [ a ] be the integer part of a number a. For any a ∈ R+ , the digits of the p -adic expansion a=

∞ 

a−ν p

ν−1

ν=1

+

∞ 

aν p −ν

(5.88)

ν=1

are defined by a−ν =  [ p 1−ν a ]  p , aν =  [ p ν a ]  p (so, the finite representation for a p -adic rational a is taken). We can easily see that, for each a ∈ R+ there exists a natural number μ such that a−ν = 0 for all ν > μ as well as that the first sum in (5.88) is equal to [ a ]. The representation (5.92) induces the operation of addition modulo p (or p -adic addition) on R+ as follows: a ⊕ p b :=

∞ ∞   a−ν + b−ν  p p ν−1 + aν + bν  p p −ν , a, b ∈ R+ . ν=1

ν=1

As usual, the equality c = a  p b means that c ⊕ p b = a. For N = p n , we set Z N = {0, 1, . . . , N − 1}. Suppose that the space C N consists of complex sequences x = (. . . , x(−1), x(0), x(1), x(2), . . . ), such that x( j + N ) = x( j) for all j ∈ Z. An arbitrary sequence x from C N is given if the values of x( j) are given for j ∈ Z N ; therefore, the element x is often identified with the vector (x(0), x(1), . . . , x(N − 1)). The space C N is equipped with the following natural inner product: N −1  x( j)y( j). x, y := j=0

For an arbitrary j ∈ Z N , let j ∗ denote the nonnegative integer defined by the condition j ⊕ p j ∗ = 0. For p = 2, we have j ∗ = j, and, for p > 2, the number x the vector from C N j ∗ is p -adic opposite to j. For each x ∈ C N we denote by % ∗ ) for all j ∈ Z . Further, for k, j ∈ Z , we set {k, j} := such that % x ( j) = x( j N N p n k j , where ν−n−1 −ν ν=1 k=

n  ν=1

k−ν p ν−1 ,

j=

n  ν=1

j−ν p ν−1 , k−ν , j−ν ∈ {0, 1, . . . , p − 1}.

212

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

) The Vilenkin–Chrestenson functions w0(N ) , w1(N ) , . . . , w(N N −1 for the space C N are {k, j} defined by the equalities wk(N ) ( j) = ε p p and wk(N ) (l) = wk(N ) (l + N ), where k, j ∈ Z N , l ∈ Z. For n ≥ 2 and p = 2, the Vilenkin–Chrestenson functions coincide with Walsh functions and, in the case n = 1 and p ≥ 2, they are exponential functions: ( p) kj wk ( j) = ε p , k, j ∈ {0, 1, . . . , p − 1} . ) The functions w0(N ) , w1(N ) , . . . , w(N N −1 constitute an orthogonal basis in C N and

wk(N ) 2 = N for all k ∈ Z N . To an arbitrary vector x from C N the Vilenkin– Chrestenson transform by the definition) assigns the sequence ( x of the Fourier ) : coefficients of x in the system w0(N ) , w1(N ) , . . . , w(N N −1

( x (k) :=

N −1 1  x( j)wk(N ) ( j), k ∈ Z N . N j=0

For all x, y ∈ C N , we define the p -convolution x ∗ y by the formula (x ∗ y)(k) :=

N −1 

x(k  p j)y( j), k ∈ Z N .

j=0

By a unit N -periodic impulse we mean the vector δ N from C N defined by the equality # 1, if j is divisible by N , δ N ( j) := 0, if j is not divisible by N . The system of shifts {δ N (·  p k) | k ∈ Z N } is an orthonormal basis in C N and x( j) = (x ∗ δ N )( j) =

N −1 

x(k)δ N ( j  p k),

j ∈ ZN ,

k=0

for all x ∈ C N . For each k ∈ Z N the p -adic shift operator Tk : C N → C N is defined as (Tk x)( j) := x( j  p k), x ∈ C N , j ∈ Z N . It follows from the definitions that, for all x, y ∈ C N , the following relations hold: (N ) )  x (l), (% x )(k) = ( x (k), (T k x)(l) = wk (l) ( (5.89) x (k),  Tk x, Tl y =  x, Tl p k y, k, l ∈ Z N . (5.90)  y, Tk x = y ∗ %

x, y = N  ( x,( y , x ∗y = N( x( y,

For ν = 0, 1, . . . , n, we set Nν = N / p ν and Δν = p ν−1 . The operators D : C N → C N1 and U : C N1 → C N given by the formulas (Dx)( j) := x( pj),

j = 0, 1, . . . , N1 − 1,

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform

#

and (U y)( j) :=

213

y( j/ p) if j is divisible by p , 0 if j is not divisible by p ,

where x ∈ C N and y ∈ C N1 are called the thickening sampling operator and the thinning sampling operator, respectively. Note that D(U y) = y for all y ∈ C N1 . Further, suppose that D 1 = D, U 1 = U and, for ν = 2, . . . , n, we define the operators D ν : C N → C Nν and U ν : C Nν → C N by the formulas ν

ν

(D x)( j) := x( p j),

ν

(U y)( j) :=

#

y( j/ p ν ) if j is divisible by p ν , 0 if j is not divisible by p ν ,

U ν y(l) = where x ∈ C N and y ∈ C Nν For any y ∈ C Nν , the following relation holds:  −ν y(l), l ∈ Z N , where, on the left-hand side, the Vilenkin–Chrestenson transform p ( is taken in C N , while, on the right-hand side, it is taken in C Nν . In addition, U ν (x ∗ y) = U ν (x) ∗ U ν (y) for all x, y ∈ C Nν .

(5.91)

Following the approach from Frazier [12], we introduce the concept of the wavelet basis of the first stage in C N : Definition 5.1 Suppose that u 0 , u 1 , . . . , u p−1 ∈ C N . If the system N1 −1 N1 −1 N1 −1 ∪ {T p k u 1 }k=0 ∪ · · · ∪ {T p k u p−1 }k=0 B(u 0 , u 1 , . . . , u p−1 ) = {T p k u 0 }k=0

is an orthonormal basis in C N , then B(u 0 , u 1 , . . . , u p−1 ) is called the wavelet basis of the first stage in C N generated by the collection of vectors u 0 , u 1 , . . . , u p−1 . The following theorem characterizes all collections of vectors generating wavelet bases of the first stage in C N . Theorem 5.4 The collection of vectors u 0 , u 1 , . . . , u p−1 generates a wavelet basis of the first stage in C N if and only if the matrix ⎛

( u 0 (l) ⎜( u 0 (l + N1 ) N ⎜ ⎜( √ ⎜ u 0 (l + 2N1 ) p⎜ .. ⎝ .

( u 1 (l) ( u 1 (l + N1 ) ( u 1 (l + 2N1 ) .. .

... ( u p−1 (l) ... ( u p−1 (l + N1 ) ... ( u p−1 (l + 2N1 ) .. ... .

⎞ ⎟ ⎟ ⎟ ⎟ (5.92) ⎟ ⎠

u 1 (l + ( p − 1)N1 ) . . . ( u p−1 (l + ( p − 1)N1 ) ( u 0 (l + ( p − 1)N1 ) (

is unitary for l = 0, 1, . . . , N1 − 1. Using Theorem 5.4, for each 1 ≤ m ≤ n we define the following procedure for the construction of a wavelet basis of the first stage in C N . Step 1. Choose complex numbers bl , 0 ≤ l ≤ p m − 1, satisfying the condition

214

5 Orthogonal and Periodic Wavelets on Vilenkin Groups p−1 

|bl+kp m−1 |2 = 1, l = 0, 1, . . . , p m−1 − 1.

(5.93)

k=0

Step 2. Calculate a0 , . . . , a pm −1 by the formulas aj = p

−m+1/2

pm −1



( pm )

bl wl

( j),

j = 0, 1, . . . , p m − 1.

l=0

Step 3. Define a vector u 0 ∈ C N , for which # u 0 ( j) =

a j , 0 ≤ j ≤ p m − 1, 0, p m ≤ j ≤ p n − 1.

(5.94)

Step 4. Find vectors u 1 , . . . , u p−1 ∈ C N such that, for all l = 0, 1, . . . , N1 − 1, Matrix (5.92) is unitary. For p = 2, the last step of this procedure is carried out simply by the formula u 1 ( j) = (−1) j u 0 (1 ⊕ 2 j),

j ∈ ZN .

(5.95)

One of algorithms for the realization of step 4 in the case p > 2 is based on the Haushölder transform [5, § 2.6] and can be described by the formulas u 0 (l + k N1 ) ( u k (l) = (

1 −( u 0 (l) 1 −( u 0 (l)

, ( u k (l + j N1 ) = δk j −

u 0 (l + k N1 ) ( u 0 (l + j N1 )( 1 −( u 0 (l)

,

(5.96) where δk j is the Kronecker delta, k, j = 1, 2, . . . , p − 1 and l = 0, 1, . . . , N1 − 1. Example 5.7 Suppose that N > p . Take m = 1 and b0 = 1, b1 = · · · = b p−1 = 0. Then the system B(u 0 , u 1 , . . . , u p−1 ) is generated by the vectors

( p−1)μ , 0, 0, . . . , 0 , μ = 0, 1, . . . , p − 1. u μ = p −1/2 1, ε pμ , ε2μ p , . . . , εp In particular, for p = 2, we have the Haar basis of the first stage in C N : √ √ √ √ u 0 = (1/ 2, 1/ 2, 0, 0, . . . , 0), u 1 = (1/ 2, −1/ 2, 0, 0, . . . , 0). The following example is obtained by modifying the orthogonal wavelets constructed in Example 5.14; it corresponds to the case m = p = 2, b0 = 1, b1 = a, b2 = 0, b3 = b. Example 5.8 Suppose that a and b are complex numbers such that |a|2 + |b|2 = 1. Suppose that p = 2 and N ≥ 4, , and the vectors u 0 , u 1 ∈ C N are given by the equalities

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform u 0 (0) =

u 1 (0) =

215

1+a+b 1+a−b 1−a−b 1−a+b , u 0 (1) = , u 0 (2) = , u 0 (3) = , √ √ √ √ 2 2 2 2 2 2 2 2

1+a−b 1+a+b 1−a+b 1−a−b , u 1 (1) = − , u 1 (2) = , u 1 (3) = − , √ √ √ √ 2 2 2 2 2 2 2 2

under the condition that u 0 ( j) = u 1 ( j) = 0 for 4 ≤ j ≤ N − 1. Then the vectors u 0 , u 1 generate a wavelet basis of the first stage in C N . Note that, for a = 1, b = 0, the resulting wavelet basis B(u 0 , u 1 ) coincides with the Haar wavelet basis of the first stage described in Example 5.7. The following two examples are similar to Examples 5.3 and 5.4, respectively. Example 5.9 Suppose that p = 2, n > 3, and m = 3. We set ( b0 , b1 , . . . , b7 ) =

1 ( 1, a, b, c, 0, α, β, γ ), 2

where |a|2 + |α|2 = |b|2 + |β|2 = |c|2 + |γ |2 = 1. Then, by relation (5.94), we have 1 1 u 0 (0) = √ (1 + a + b + c + α + β + γ ), u 0 (1) = √ (1 + a + b + c − α − β − γ ), 4 2 4 2 1 1 u 0 (2) = √ (1 + a − b − c + α − β − γ ), u 0 (3) = √ (1 + a − b − c − α + β + γ ), 4 2 4 2 1 1 u 0 (4) = √ (1 − a + b − c − α + β − γ ), u 0 (5) = √ (1 − a + b − c + α − β + γ ), 4 2 4 2 1 1 u 0 (6) = √ (1 − a − b + c − α − β + γ ), u 0 (7) = √ (1 − a − b + c + α + β − γ ). 4 2 4 2

Further, we set u 1 ( j) = u 0 ( j) = 0 for 8 ≤ j ≤ 2n − 1 , and we choose the other components of the vector u 1 so that relation (5.95) are valid, i.e., u 1 (0) = u 0 (1), u 1 (1) = −u 0 (0), u 1 (2) = u 0 (3), u 1 (3) = −u 0 (2), u 1 (4) = u 0 (5), u 1 (5) = −u 0 (4), u 1 (6) = u 0 (7), u 1 (7) = −u 0 (6). The resulting pair u 0 , u 1 generates a wavelet basis of the first stage in C N . Example 5.10 Suppose that p = 3, n > 2, m = 2 and 1 ( b0 , b1 , . . . , b8 ) = √ ( 1, a, α, 0, b, β, 0, c, γ ), 3 where |a|2 + |b|2 + |c|2 = |α|2 + |β|2 + |γ |2 = 1. Then, using (5.93) and (5.94), we obtain

216

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

1 u 0 (0) = √ (1 + a + b + c + α + β + γ ), 3 3

1 u 0 (1) = √ 1 + a + α + (b + β)ε32 + (c + γ )ε3 , 3 3

1 u 0 (2) = √ 1 + a + α + (b + β)ε3 + (c + γ )ε32 , 3 3

1 u 0 (3) = √ 1 + (a + b + c)ε32 + (α + β + γ )ε3 , 3 3

1 u 0 (4) = √ 1 + c + β + (a + γ )ε32 + (b + α)ε3 , 3 3

1 u 0 (5) = √ 1 + b + γ + (a + β)ε32 + (c + α)ε3 , 3 3

1 u 0 (6) = √ 1 + (a + b + c)ε3 + (α + β + γ )ε32 , 3 3

1 u 0 (7) = √ 1 + b + γ + (a + β)ε3 + (c + α)ε32 , 3 3

1 u 0 (8) = √ 1 + c + β + (a + γ )ε3 + (b + α)ε32 , 3 3 where ε3 = exp(2πi/3). We set u 0 ( j) = u 1 ( j) = u 2 ( j) = 0 for 9 ≤ j ≤ 3n − 1 and use formulas (5.96) to define the other components of the vectors u 1 , u 2 ∈ C N so that the matrix ⎛ ⎞ ( u (l) ( u 1 (l) ( u 2 (l) 9 ⎝ 0 u 1 (l + 3) ( u 2 (l + 3) ⎠ ( u 0 (l + 3) ( √ 3 ( u 1 (l + 6) ( u 2 (l + 6) u 0 (l + 6) ( is unitary for l = 0, 1, 2. The resulting collection of the vectors u 0 , u 1 , u 2 generates a wavelet basis of the first stage in C N . Definition 5.2 Suppose that m ∈ N, m ≤ n. By a sequence of orthogonal wavelet filters of the mth stage we mean a sequence of vectors (1) (1) (m) (m) (m) u (1) 0 , u 1 , . . . , u p−1 , . . . , u 0 , u 1 , . . . , u p−1 ,

such that u (ν) μ ∈ C Nν−1 for ν = 1, 2, . . . , m , μ = 0, 1, . . . , p − 1 and the matrices

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform ⎛

(ν)

(ν)

( u 0 (l)

217 ⎞

(ν)

( u 1 (l)

⎜ ⎜ (ν) (ν) ( u 1 (l + Nν ) u (l + Nν ) ⎜( ⎜ 0 N ⎜ (ν) (ν) A(ν) (l) := √ ⎜ ( u (l + 2Nν ) ( u 1 (l + 2Nν ) p⎜ 0 ⎜ ⎜ ... ... ⎝ (ν) (ν) ( u 0 (l + ( p − 1)Nν ) ( u 1 (l + ( p − 1)Nν )

... ( u p−1 (l)

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(ν)

... ( u p−1 (l + Nν ) (ν)

... ( u p−1 (l + 2Nν ) ...

...

...

(ν) ( u p−1 (l

+ ( p − 1)Nν )

are unitary for ν = 1, 2, . . . , m , l = 0, 1, . . . , Nν − 1. Theorem 5.5 Suppose that the collection of vectors u 0 , u 1 , . . . , u p−1 generates a wavelet basis of the first stage in C N . For a given m ∈ N, m ≤ n, set (ν) −1 u (1) μ ( j) = u μ ( j), u μ ( j) = Δν

Δ ν −1 

u (1) μ ( j + k Nν−1 ),

j ∈ Z Nν−1 ,

(5.97)

k=0

where ν = 2, . . . , m, μ = 0, 1, . . . , p − 1. Then the vectors (1) (1) (m) (m) (m) u (1) 0 , u 1 , . . . , u p−1 , . . . , u 0 , u 1 , . . . , u p−1 ,

constitute a sequence of orthogonal wavelet filters of the mth stage. Thus, from a given vector u 0 ∈ C N , defined by (5.93) and (5.94) we can, first, find a wavelet basis of the first stage u 0 , u 1 , . . . , u p−1 , using (5.95) or (5.96), and then, using (5.97) obtain the sequence of orthogonal wavelet filters of the mth stage. Denote by ⊕ the direct sum of the subspaces of the space C N . By the theorem that follows, from any sequence of orthogonal wavelet filters of the mth stage we can construct an orthonormal wavelet basis in C N . Theorem 5.6 Suppose that a sequence of orthogonal wavelet filters of the mth stage is given in the space C N : (1) (1) (m) (m) (m) u (1) 0 , u 1 , . . . , u p−1 , . . . , u 0 , u 1 , . . . , u p−1 . (1) (1) (ν) , ψμ(ν) for ν = Let φ (1) = u (1) 0 , ψμ = u μ , μ = 1, . . . , p − 1, and define φ 2, . . . , m, μ = 1, . . . , p − 1 by the formulas (ν) (ν−1) ∗ U ν−1 u (ν) φ (ν) = φ (ν−1) ∗ U ν−1 u (ν) μ . 0 , ψμ = φ

Further, for ν = 1, . . . , m, μ = 1, . . . , p − 1, we set (μ)

φ−ν,k = T pν k φ (ν) , ψ−ν,k = T pν k ψμ(ν) , k = 0, 1, . . . , Nν − 1, and define the subspaces (μ)

(μ)

Nν −1 Nν −1 , W−ν = span{ψ−ν,k }k=0 , V−ν = span{φ−ν,k }k=0

218

5 Orthogonal and Periodic Wavelets on Vilenkin Groups ( p−1)

(1) W−ν = W−ν ⊕ · · · ⊕ W−ν

.

Then the following expansion holds: C N = W−1 ⊕ W−2 ⊕ · · · ⊕ W−m ⊕ V−m

(5.98)

and, for each ν = 1, 2, . . . , m the following properties are valid : (a) V−ν = V−ν−1 ⊕ W−ν−1 ; Nν −1 is an orthonormal basis in V−ν ; (b) {φ−ν,k }k=0 ( p−1) Nν −1 (1) Nν −1 is an orthonormal basis in W−ν . (c) {ψ−ν,k }k=0 ∪ · · · ∪ {ψ−ν,k }k=0 This theorem justifies the method of constructing subspaces V−1 , . . . , V−n in C N with the following properties: (i) V−ν−1 ⊂ V−ν for all ν ∈ {1, 2, . . . n} ; (ii) for each ν ∈ {1, 2, . . . n}, there exists a vector φ (ν) ∈ V−ν such that the system Nν −1 is an orthonormal basis in V−ν ; {T pν k φ (ν) }k=0 (iii) for each 1 ≤ m ≤ n, relation (5.98) is valid; (ν) ∈ W−ν such that the (iv) for each ν ∈ {1, 2, . . . n} there exist vectors ψ1(ν) , . . . , ψ p−1  p−1 Nν −1 is an orthonormal basis in W−ν . system μ=1 {T pν k ψμ(ν) }k=0 According to the terminology used in the theory of multiresolution analysis and the sequence {φ (ν) }nν=1 in property (ii), it is natural to call a scaling sequence in C N . In particular, for p = 2, n = 3 by Theorem 5.6 we obtain three orthonormal wavelet bases in C8 : (m = 1), {ψ−1,k }3k=0 ∪ {φ−1,k }3k=0 {ψ−1,k }3k=0 ∪ {ψ−2,k }1k=0 ∪ {φ−2,k }1k=0 {ψ−1,k }3k=0 ∪ {ψ−2,k }1k=0 ∪ {ψ−3,0 } ∪ {φ−3,0 }

(m = 2), (m = 3).

In the Haar case (see Example 5.7), these bases consist of the vectors 1 1 φ−1,0 = √ (1, 1, 0, 0, 0, 0, 0, 0), ψ−1,0 = √ (1, −1, 0, 0, 0, 0, 0, 0), 2 2 1 1 φ−1,1 = √ (0, 0, 1, 1, 0, 0, 0, 0), ψ−1,1 = √ (0, 0, 1, −1, 0, 0, 0, 0), 2 2 1 1 φ−1,2 = √ (0, 0, 0, 0, 1, 1, 0, 0), ψ−1,2 = √ (0, 0, 0, 0, 1, −1, 0, 0), 2 2 1 1 φ−1,3 = √ (0, 0, 0, 0, 0, 0, 1, 1), ψ−1,3 = √ (0, 0, 0, 0, 0, 0, 1, −1), 2 2 φ−2,0 =

1 (1, 1, 1, 1, 0, 0, 0, 0), 2

ψ−2,0 =

1 (1, 1, −1, −1, 0, 0, 0, 0), 2

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform

φ−2,1 =

1 (0, 0, 0, 0, 1, 1, 1, 1), 2

1 φ−3,0 = √ (1, 1, 1, 1, 1, 1, 1, 1), 2 2

ψ−2,1 =

219

1 (0, 0, 0, 0, 1, 1, −1, −1), 2

1 ψ−3,0 = √ (1, 1, 1, 1, −1, −1, −1, −1). 2 2

In the general case, the orthogonal projections P−ν : C N → V−ν and Q −ν : C N → W−ν act by the formulas P−ν x =

N ν −1 

x, φ−ν,k φ−ν,k ,

Q −ν x =

p−1 Nν −1   μ=1 k=0

k=0

(μ)

(μ)

x, ψ−ν,k ψ−ν,k , x ∈ C N .

(5.99) Suppose that I is the identity operator on C N . Setting P0 = I , V0 = C N and using Theorem 5.6 for any x ∈ C N , we obtain the equalities x = P−ν x +

ν 

Q −k x,

P−ν+1 x = P−ν x + Q −ν x, ν = 1, 2, . . . , n.

k=1

An arbitrary vector x from C N can be regarded as the input signal a0 = x and, for ν = 1, 2, . . . , m, we can set (μ) u (ν) = D(aν−1 ∗ % u μ(ν) ), μ = 1, . . . , p − 1. aν = D(aν−1 ∗ % 0 ), dν

(5.100)

We can easily see that the components of the vectors aν and dν(μ) are the coefficients of the expansions (5.100) for a chosen x. The application of formulas (5.105) constitutes the phase of the analysis of the signal x and yields the collection of vectors (1) (m) , . . . , d1(m) , . . . , d p−1 , am . d1(1) , . . . , d p−1

(5.101)

The inverse passage from the collection (5.101) to the original vector x constitutes the reconstruction phase and is defined by the formulas aν−1 = u (ν) 0 ∗ U aν +

p−1 

(ν) u (ν) μ ∗ U dμ , ν = m, m − 1, . . . , 1.

(5.102)

μ=1

Therefore, the formulas (5.100) and (5.102) specify the direct and inverse discrete wavelet transforms associated with the sequence of wavelet filters u (1) 0 , (1) (1) (m) (m) (m) u 1 , . . . , u p−1 , . . . , u 0 , u 1 , . . . , u p−1 . It is easily seen that Uk(n) ∩ Ul(n) = ∅ for k = l,

N −1  k=0

Uk(n) = U.

220

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

Further, we shall use the notation wl,(n)k := wl (xn, k ) for 0 ≤ l, k ≤ N − 1. Notice that

N −1 

(n) −sq (n+1) wl,(n)k = wk, l = ε p w pk+s, N q+l , 0 ≤ s, q ≤ p − 1,

wi,(n)l wi,(n)k =

N −1 

i=0

(5.103)

(n) wl,(n)j wk, j = N δl, k , 0 ≤ l, k ≤ N − 1.

(5.104)

j=0

A finite sum D N (x) :=

N −1 

w j (x), x ∈ G,

j=0

is called the Walsh–Dirichlet kernel of order N . It is well known that # N , x ∈ U0(n) , D N (x) = 0, x ∈ U \ U0(n) . ( j)

Vn := span{1, w1 (x), . . . , w N −1 (x)}, Wn

(5.105)

:= span{w j N (x), w j N +1 (x), . . . , w( j+1)N −1 (x)}, ( p−1)

respectively. Note that the orthogonal direct sum of Vn , Wn(1) , . . . , Wn coincides $ $ ( p−1) $ with Vn+1 , that is, for Wn := Wn(1) · · · Wn Wn = Vn+1 . The , we have Vn ( j) spaces Vn and Wn will be called the approximation spaces and wavelet spaces , respectively. We can use the discrete Vilenkin–Chrestenson transform to recover v ∈ Vn from the values v(xn, l ), 0 ≤ l ≤ N − 1. Indeed, if v(x) =

N −1 

ck wk (x), x ∈ U,

(5.106)

k=0

then ck =

N −1 1  v(xn, l )wl,(n)k , 0 ≤ k ≤ N − 1. N l=0

(5.107)

Suppose that a = (a0 , a1 , . . . , a N −1 ), where ak = 0, 0 ≤ k ≤ N − 1. Then we set Φ Na (x) :=

N −1 1  ak wk (x), φn, k (x) := Φ Na (x  xn, k ), 0 ≤ k ≤ N − 1, x ∈ G. N k=0

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform

221

Proposition 5.18 Let u ∈ C N , ν ∈ {0, 1, . . . , n}. The system is orthonormal then Nν −1 CN only then when {T p ν k u}k=0 |( u (l)|2 + |( u (l + Nν )|2 + · · · + |( u (l + ( p ν − 1)Nν )|2 =

pν N2

(5.108)

for all l ∈ Z Nν . (N )  Proof According to (5.89), (T u (l). Using Parseval’s equation, p ν k u)(l) = w p ν k (l) ( we get

 u, T p ν k u

=N

N −1 

(N ) |( u (l)|2 w p ν k (l) = N

l=0

ν −1 p N ν −1

μ=0 l=0

(N )

|( u (l + μNν )|2 wk

( p ν (l + μNν ))

ν

p −1 Nν −1 1  N2  ) = w(N (l) |( u (l + μNν )|2 . ν Nν l=0 p k p ν μ=0

(5.109)

Thus the derivative Nν  u, T p ν k u coincides with the inverse Vilenkin–Chrestenson transform in C Nν of sequence ν

p −1 N2  Φν (l) := ν |( u (l + μNν )|2 , l ∈ Z Nν . p μ=0

It is easy to see that N ν −1 

) w(N p ν k (l) =

l=0

N ν −1 

wl(Nν ) (k) = Nν δ Nν (k), k ∈ Z Nν .

l=0

Consequently, the equation #  u, T

pνk

u =

1 0

k = 0, k = 1, . . . , Nν − 1,

is fulfilled in that and only in that case when p ν −1

 μ=0

|( u (l + μNν )|2 =

pν , l ∈ Z Nν . N2

Besides, using (5.90) for any k, l ∈ Z Nν we have  T p ν k u, T p ν l u =  u, T p ν (l p k) u.

222

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

Nν −1 It means, the orthonormality of the system {T p ν k u}k=0 is equivalent to the condition (5.108). The proposition 5.18 is proved.

Proof of Theorem 5.4. For μ = 0, 1, . . . , pi − 1, λ = 0, 1, 2, . . . , N1 , we have uˆ (2) μ (l)

p−1 N1 −1 N1 −1  1  1  ( pN ) (N1 ) (2) = u ( j)wl ( j) = u (1) ( j + k N1 )w pl 1 ( j) N1 j=0 μ pN1 j=0 k=0 μ

p−1 N1 −1  1  1  (N1 ) (N1 ) u (1) j = 0 N −1 u (1) μ ( j + k N1 )w pl ( j) = μ ( j)w pl ( j). N1 j=0 k=0 N

Using the induction for any ν ∈ {2, . . . , m}, we get ˆ ν−1 ˆ (1) l), μ = 0, 1, . . . , p − 1, l ∈ Z Nν−1 . u (ν) μ (l) = u μ (p Hence for any ν ∈ {1, 2, . . . , m} true is the equation A(ν) (l) = A(1) ( p ν−1l), where, on the strength of Theorem , matrix A(1) ( p ν−1l) is unitary for all l ∈ Z N . Theorem 5.4 is proved. Proof of Theorem 5.5. According to Theorem 5.4, the systems N1 −1 N1 −1 , . . . , {T p k u p−1 }k=0 {T p k u 0 }k=0

are orthonormal in C N then only then when the column vectors ⎞ ( u 0 (l) ⎟ ⎜ ( u 0 (l + N1 ) ⎟ N ⎜ ⎜ u 0 (l + 2N1 ) ⎟ √ ⎜ ( ⎟, ..., p⎝ ⎠ ... ( u 0 (l + ( p − 1)N1 ) ⎛

⎞ ( u p−1 (l) ⎟ ⎜ ( u p−1 (l + N1 ) ⎟ N ⎜ ⎜ u p−1 (l + 2N1 ) ⎟ √ ⎜ ( ⎟ p⎝ ⎠ ... ( u p−1 (l + ( p − 1)N1 ) ⎛

(5.110)

have length equal to 1. Besides , as per Proposition 5.4.2, the condition  T p k u μ , T p j u ν  = 0, μ = ν, k, j ∈ Z N1 , is uniform on the paired orthogonality of vectors. It is to be noted that the number of elements of the system (5.110). B(u 0 , u 1 , . . . , u p−1 ) for the given set of vectors coincides with the dimension of space C N . For μ = 0, 1, . . . , p − 1, l = 0, 1, . . . , N1 − 1,we have ( u μ(2) (l) =

p−1 N1 −1 N1 −1  1  1  ( pN ) (N1 ) u (2) ( j)w ( j) = u (1) ( j + k N1 )w p l 1 ( j) l N1 j=0 μ pN1 j=0 k=0 μ

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform

=

223

p−1 N1 −1  N −1 1  1  (1) (N ) ) u (1) ( j + k N )w ( j) = u ( j)w(N u μ(1) ( p l). 1 pl p l ( j) = ( N j=0 k=0 μ N j=0 μ

Using the induction for any ν ∈ {2, . . . , m} we get u μ(1) ( p ν−1l), μ = 0, 1, . . . , p − 1, l ∈ Z Nν−1 . ( u μ(ν) (l) = ( Hence for any ν ∈ {1, 2, . . . , m} true is the equation where A(ν) (l) = A(1) ( p ν−1l), on the strength of Theorem 5.5 the matrix A(1) ( p ν−1l) is unitary for all l ∈ Z Nν−1 . Theorem 5.5 is proved. Proposition 5.19 Let f ∈ C N u u 0 , u 1 , . . . , u p−1 ∈ C Nν−1 , where 1 ≤ ν ≤ n. AssumNν−1 −1 is orthonormal in C N ,and the system ing that the system {T pν−1 k f }k=0 Nν −1 Nν −1 {T pk u 0 }k=0 ∪ · · · ∪ {T pk u p−1 }k=0 is orthonormal basis of space C Nν−1 . Let us assume g = f ∗ U ν−1 u 0 , h 1 = f ∗ U ν−1 u 1 , . . . , h p−1 = f ∗ U ν−1 u p−1 .

(5.111)

Then the system Nν −1 Nν −1 Nν −1 ∪ {T pν k h 1 }k=0 ∪ · · · ∪ {T pν k h p−1 }k=0 {T pν k g}k=0

(5.112)

is orthonormal in C N . Proof For the arbitrary k ∈ {0, 1, . . . , Nν−1 − 1}, on the strength (5.90), (5.91), and (5.111), we have g ( p ν k) = ( f ∗ U ν−1 u 0 ∗ % f ∗ U ν−1% u 0 )( p ν k)  g, T pν k g = g ∗ % u 0 ))( p ν k) =(f ∗ % f ) ∗ (U ν−1 (u 0 ∗ % or as per the convolution definition,  g, T pν k g =

N −1 



(f ∗ % f )( p ν k  p j) U ν−1 (u 0 ∗ % u 0 ) ( j).

j=0

As U ν−1 (u 0 ∗ % u 0 )( j) = then,

#

u 0 )( j/ p ν−1 ), j if j is divisible by p ν−1 , (u 0 ∗ % 0, j if j is not divsible by p ν−1 ,

Nν−1 −1

 g, T pν k g =



m=0

(f ∗ % f )( p ν k  p p ν−1 m)(u 0 ∗ % u 0 )(m).

224

5 Orthogonal and Periodic Wavelets on Vilenkin Groups N

−1

ν−1 Taking into consideration the orthonormality of the system {T pν−1 k f }k=0 , for the arbitrary m ∈ Z Nν−1 we have

f ∗ % f ( p ν−1 ( pk  p m)) =  f, T pν−1 ( pk p m) f  =

#

1, if m = pk, 0, if m = pk.

Hence # u 0 )( pk) =  u 0 , T pk u 0  =  g, T pν k g = (u 0 ∗ %

1, if k = 0, 0, if k = 1, 2, . . . , Nν − 1.

For each μ ∈ {1, . . . , p − 1} #  h μ , T pν k h μ  =  g, T pν k h μ  =

1, if k = 0, 0, if k = 1, 2, . . . , Nν − 1.

Consequently, the system (5.108) is orthonormal in C N . Proposition 5.19 is proved. Proof of Theorem 5.6. The equation C N = W−1 ⊕ V−1 and the properties (b), (c) for ν = 1 follow from the fact that, as per Theorem 5.4 the system ( p−1)

(1) N1 −1 N1 −1 N1 −1 ∪ {ψ−1,k }k=0 ∪ · · · ∪ {ψ−1,k }k=0 {φ−1,k }k=0

is orthonormal basis in C N . As per the condition, (μ)

φ−2,k = T p 2 k φ (2) , ψ−2,k = T p 2 k ψμ( 2) , k = 0, 1, . . . , N 2 − 1, where φ ( 2) = φ (1) ∗ U (u (02) ), ψμ( 2) = ψ (1) ∗ U (u (μ2) ), μ = 1, . . . , p − 1. Using Proposition 5.19, we notice that the system ( p−1)

(1) N2 −1 N2 −1 N2 −1 ∪ {ψ−2,k }k=0 ∪ · · · ∪ {ψ−2,k }k=0 {φ−2,k }k=0

is orthonormal basis of space V1 . From here we get the property (a) for ν = 1. Now let ν ∈ {2, 3, . . . , m}. Using Proposition 5.18 for f = ψ (ν−1) , u 0 = u (ν) 0 , . . . , u p−1 = u (ν) , we get that the system p−1  Nν −1  ( p−1) (1) Nν −1 Nν −1 ∪ {ψ−ν,k }k=0 ∪ · · · ∪ ψ−ν,k {φ−ν,k }k=0 k=0

is orthonormal in C N . Hence *

+ * + (μ) (μ ) (μ) φ−ν,k , ψ−ν,l = ψ−ν,k , ψ−ν,l = 0

(5.113)

5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform

225

for k, l ∈ Z Nν , μ, μ ∈ {1, . . . , p − 1}, μ = μ .

(5.114)

Further, for arbitrary k ∈ Z Nν , j ∈ Z N we have ν φ−ν,k ( j) = φ (ν) ( j  p p ν k) = (φ (ν−1) ∗ U ν−1 u (ν) 0 )( j  p p k)

=

N −1 

φ (ν−1) ( j  p p ν k  p m) U ν−1 u (ν) 0 (m)

m=0 Nν−1 −1

=



φ (ν−1) ( j  p p ν k  p p ν−1 m) u (ν) 0 (m)

m=0 Nν−1 −1



=

(ν−1) u (ν) ( j). 0 (m)T pν−1 ( pk⊕ p m) φ

m=0

It means, for any k ∈ Z Nν Nν−1 −1

φ−ν,k =



u (ν) 0 (m)φ−ν+1, pk⊕ p m

(5.115)

m=0

Similarly, for all k ∈ Z Nν , μ ∈ {1, . . . , p − 1} we have (μ)

ψ−ν,k =

Nν−1 −1



u (ν) μ (m)φ−ν+1, pk⊕ p m .

(5.116)

m=0

From Formulas (5.114)–(5.116) follow the relations V−ν ⊥ W−ν , V−ν ⊂ V−ν+1 , W−ν ⊂ V−ν+1 and as dim V−ν + dim W−ν = dim V−ν+1 , V−ν+1 is the direct sum of subspaces V−ν and W−ν : (5.117) V−ν+1 = V−ν ⊕ W−ν . From here and from the orthogonality of 5.110 follow the properties (a)–(c) for each ν. For the decomposition of the system 5.98, let us show that the system p−1  p−1  p−1   N1 −1   N2 −1  Nm −1   (μ) (μ) (μ) Nm −1 ψ−1,k ψ−2,k ψ−m,k ∪ ∪ ··· ∪ ∪ {φ−m,k }k=0

μ=1

k=0

μ=1

k=0

μ=1

k=0

(5.118)

226

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

is orthonormal in C N . From the orthonormality of the system 5.110 follow the equal(μ) Nν −1 Nν −1 {ψ−ν, k }k=0 . for ity 5.111 and orthonormality of each of the system {φ−m, k }k=0 (μ) (μ) (μ)

ν < ν , let us examine the elements ψ−ν, k ψ−ν , l . Then ψ−ν , k ∈ W−ν and as per (μ) proof, ψ−ν, k ∈ W−ν , where W−ν ⊂ V−ν+1 ⊂ · · · ⊂ V−ν . Taking into consideration Nν −1 (μ) Nν −1 (μ) (5.117),we get from here that the sets {ψ−ν, k }k=0 and {ψ−ν , l }l=0 are orthonor(μ) Nν −1 Nm −1 , mal. Similarly, the set {φ−m, k }k=0 is orthonormal to any from the sets {ψ−ν, k }k=0 1 ≤ ν ≤ m. It has to be noted that the number of elements of the sets (5.118) coincides with the dimension of the space C N . Theorem 5.6 is proved.

5.5 Application to the Coding of Fractal Functions It is well known that a quick discrete wavelet transform breaks down the signals into low-frequency(approximating) and high-frequency(detailing) components with their subsequent incomplete sample; the inverse transformation regenerates the signal(see for example, [13, § 7.3]).The results of the calculation experiments given below show that the discrete wavelet transformation associated with wavelets from Sects. 5.2, 5.3, and 5.4 may be used for coding of fractal functions. In this section, the p -adic operations ⊕ p and  p on R+ will be denoted by ⊕ and , respectively.The notations N , Nν , and ε p are understood as before. In Sect. 5.4. for any complex vector b = (b0 , b1 , . . . , b N −1 ), satisfying the condition |bl |2 + |bl+N1 |2 + · · · + |bl+( p−1)N1 |2 = 1, 0  l  N1 − 1,

(5.119)

is constructed an orthonormal basis in space of N -periodic complex sequences C N . Let us denote by G( p, n) the set of all vectors b, satisfying Condition 5.116. Using any method of unitary expansion of matrix, for any arbitrary fixed vector b(0) = (s) (b0(0) , b1(0) , . . . , b(0) N −1 ) from G( p, n) , let us find out the numbers bk , 0  k  N − 1, 1  s  p − 1, such that the matrices ⎛

bl(0) ⎜ (1) ⎜ b Ml = ⎜ l ⎝ ... ( p−1) bl

(0) bl+N 1 (1) bl+N 1 ... ( p−1) bl+N1

⎞ (0) . . . bl+( p−1)N1 ⎟ (1) . . . bl+( p−1)N1 ⎟ ⎟ , 0  l  N1 − 1, ... ... ⎠ ( p−1) . . . bl+( p−1)N1

are unitary. Further, with the help of Vilenkin–Christenson transformation, let us calculate the coefficients ck(s)

N −1 1  (s) (N ) = b w ( j), 0  k  N − 1, 0  s  p − 1, N1 j=0 j k

(5.120)

5.5 Application to the Coding of Fractal Functions

227

and propose bk(s) = ck(s) = 0 k  N . The orthogonal discrete wavelet transform O( p, n), is associated with vector b(0) , and is defined by the formulae a j−1,k =



(0) (1) cl pk a j,l , d j−1,k =

l∈Z+

 l∈Z+

( p−1)

(1) cl pk a j,l , . . . , d j−1,k =



( p−1)

cl pk a j,l ,

l∈Z+

(5.121) where the coefficients ck(0) of the low-frequency filter and the coefficients ck(1) , . . . , ( p−1) of high-frequency filter are determined by Formula 5.118 (compare with [7, ck Sect. 7.3.2]). The dimension of vector a j with the components a j,l is proposed by the multiple number of the inlet canals p. The transformation 5.519 translates vector a j ( p−1) into approximating vectors a j−1 and detailing vectors d (1) j−1 , . . . , d j−1 , the dimensions of which are p times less than the dimensions of the vector a j . The inversion formula of this transformation is  (0) ( p−1) ( p−1) (1) (1) cl pk a j−1,k + cl (5.122) a j,l = pk d j−1,k + · · · + cl pk d j−1,k . l∈Z+

√ √ The discrete transformation O(2, 1), associated with the vector b(0) = (1/ 2, 1/ 2), coincides with classic Haar transform. Here the formulae (5.118) is written in the form a j,2k + a j,2k+1 a j,2k − a j,2k+1 , d j−1,k = . a j−1,k = √ √ 2 2 For arbitrary p and n the discrete Haar transform corresponds to the values b0(0) = √ √ b1(0) = · · · = b(0) bk(0) = 0 when k  p; then ck(s) = (1/ p) p−1 = 1/ p, exp(2πiks/ p) when 0  k, s  p − 1. In particular p = 3 we have (0)

(0)

(0)

(1)

(2)

c0 = c1 = c2 = c0 = c0 =

√

√ √ 3 3 3 2 (1) (2) (1) (2) , c1 = c2 = ε3 , c2 = c1 = ε , 3 3 3 3

(5.123) where ε3 = exp(2πi/3). The discrete transform O(2, 2), associated with vector b(0) = (1, a, 0, b), 0 < a  1, a 2 + b2 = 1. was suited by Lang [14–16]. For this transformation the nonzero coefficients in (5.118) and (5.119) are determined by the formulae 1+a+b 1+a−b , c1(0) = −c0(1) = , √ √ 2 2 2 2 1−a−b 1−a+b = c3(1) = , c3(0) = −c2(1) = . √ √ 2 2 2 2

c0(0) = c1(1) = c2(0)

While processing the discrete signal, the selected discrete wavelet transform is ( p−1) used iteratively, after the first step the vectors d (1) j−1 , . . . , d j−1 remain in memory, but the vectors a j−1 changes to approximating vector a j−2 and detailing vectors

228

5 Orthogonal and Periodic Wavelets on Vilenkin Groups ( p−1)

d (1) j−2 , . . . , d j−2 etc. The dimensions of the vectors decrease by p times in each step. As a result, after j0 steps, we get the vectors ( p−1)

( p−1)

(1) a j− j0 , d (1) j−1 , . . . , d j−1 ; . . . ; d j− j0 , . . . , d j− j0 .

The vector a j is regenerated by inverse transformation of these vectors. The freedom in selecting the vector b(0) helps in adapting the transformation O( p, n) to the signal being processed by root mean square, entropy or other criteria. In a recent work, it is shown that for coding certain fractural signals the discrete wavelet transform O( p, n) is preferred in comparison with discrete Haar transform and zone encoding method. This result is given as an example of coding values generalized by Weierstrass function. Va,b (x) =

∞ 

a k eb

k

πi x

, 0 < a  1, b  1/a.

k=1

As the source of four sources of signals we selected the values of function Va,b (X ) in 24 nodes of uniform division of interval (0, 1) for the pair of indices a and b given in table . (N ) (N ) ) be the matrix composed of numbers wl,k = wk (l/N ) , 0  Let TN = (wl,k l, k  N − 1 . . As is usual, we designate the matrix complexly joined with the matrix TN∗ TN . The direct and inverse discrete, multiplicative transforms in the space C N , are written in the form x, ( x = N −1 TN∗ x, x = TN (

x ∈ CN .

Method of zone coding Z ( p) is used in [4, § 11.3], consists in using direct discrete multiplicative transform to vector x with subsequent zeroing out vector ( x x ( p n−1 ), ( x ( p n−1 + 1), . . ., ( x ( p n − 1). Then of arbitrary p n − p n−1 components: ( the vector % x is obtained after application of inverse direct multiplicative transform. √ The error in regeneration is evaluated by the values δ = ||x − % x ||2 / N . In calculation experiments, the orthogonal discrete spike transformation O(3, 1)O(3, 2) were compared in Rodionov [17] not only with the method of zone coding Z (3), but also with discrete Haar transform H (3), determined by the coefficients 5.120. The results of the calculations are given in Table 5.1. Let us now assume that the spaces Vn and Wn are determined as in § 5.3 when p = 2N = 2n : Vn = span{1, w1 (x), . . . , w N −1 (x)}, Wn = span{w N (x), w N +1 (x), . . . , w2N −1 (x)}, N −1 N −1 , {ψn,k }k=0 are given by the parameter α under the and basis systems {φn,k }k=0 condition (5.78). We notice that for the decomposition coefficients

5.5 Application to the Coding of Fractal Functions

229

Table 5.1 Root mean square compression errors of functions Va,b with the help of sound coding method, discrete Haar transform and orthogonal spike transformations ł Z (3) H (3) O(3, 1) O(3, 2) V0.6,9 V0.8,6 V0.8,9 V0.9,4

0.3774 0.6660 0.6572 1.3786

0.2835 0.5393 0.4599 0.9706

w N −1 (x) =

2N −1 

0.2068 0.4826 0.3448 0.8763

0.1310 0.4629 0.2934 0.8269

γn+1,k φn+1,k (x).

k=0

The recurrent formulae γ2,0 = γ2,1 = 1, γ2,2 = γ2,3 = −1, γn+1,k = γn,k , γn+1,N +k = −γn,k , n ≥ 2, 0 ≤ k ≤ N − 1. According to (5.82) and (5.83), for any function f n ∈ Vn gn ∈ Wn we have the equalities N −1 N −1   Cn,k φn,k (x), gn (x) = Dn,k ψn,k (x), (5.124) f n (x) = k=0

k=0

and sequence of coefficients Cn = {Cn,k }, Dn = {Dn,k }

(5.125)

uniquely defined by f n and gn , respectively. When p = 2 from (5.85), (5.86) we get Cn,k =



(n) Al,k Cn+1,l ,

Dn,k =

l

Cn+1,l =



(n) Bl,k Cn+1,l .



(n) (n) Pl,k Cn,k + Q l,k Dn,k .

k

In these formulae (n) Al,k =

⎧ ⎨ 1/2 + (1 − α)/(2α N ), ⎩

(5.126)

l

k = [l/2],

(1 − α)(−1)γ (k)−γ ([l/2]) /(2α N ), k = [l/2],

(5.127)

230

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

(n) Bl,k =

and (n) Pl,k =

⎧ ⎨ (−1)l /2, ⎩

0,

k = [l/2],

k = [l/2],

⎧ ⎨ 1 + γn+1,k (1 − α)(−1)γ (l)+1 /N , ⎩

γn+1,k (1 − α)(−1)γ (l)+1 /N , (n) = Q l,k

⎧ ⎨ (−1)l , ⎩

0,

k = [l/2], k = [l/2].

k = [l/2], k = [l/2].

For any function f n ∈ Vn and any n 0 ∈ {1, 2, . . . , n − 1} with the help of Formuals (5.121), (5.123) after substitution of n by n − 1, n − 2, . . . , n − n 0 we get the decomposition n0  f n = f n−n 0 + gn− j j=1

and, correspondingly the vector Cn gets transformed into a set of vectors Cn−n 0 , Dn−1 , . . . , Dn−n 0 . Thus is direct discrete diadic periodic spur transformation with parameter α (let us call it Wα ). The corresponding inverse transformation Wα−1 is based on Formulas (5.122), (5.124) and helps in regenerating the function f n and vectors Cn as per the set of vectors Cn−n 0 , Dn−1 , . . . , Dn−n 0 . The method of coding consisting of the following steps will be called the Method A. 1. To use direct discrete wavelet transformation as input mass. 2. Select from the wavelet coefficients a certain number (for example, 10 or 1%) having high modulus, the remaining coefficients equal to zero. 3. Use inverse wavelet transform for the obtained coefficients file. 4. Calculate from the regenerated file % Cn , the average of root mean square deviation , m -1  % %n,k ), δ(Cn , Cn ) := . (Cn,k − C m k=1 Cn . where m is the dimension of vectors Cn % Method B is different from the Method A by the fact that here the step 2 is replaced by uniform quantization of wavelet coefficients Let us remember that uniform quantization of arbitrary vector x = (x1 , . . . , xm ) takes place by the formula

5.5 Application to the Coding of Fractal Functions

# xj =

231

0, |x j | < Δ, Δ([x j /Δ] + 0.5sign(x j )), |x j | ≥ Δ,

where Δ -step in quantization, in Method C given below, we assume

 Δ = Δt =

max x j − min x j /t, t ∈ N.

1≤ j≤m

1≤ j≤m

If the vector x represents the alphabet of the spatial source without storage/memory, then the entropy of this source identified by the formula H (x) =

m 

− p j log2 ( p j ),

j=1

where p j − probability with which the source yields the value x j . In practice, the probability is substituted by relative frequency. For arbitrary vectors x = (x1 , . . . , xm ), y = (y1 , . . . , yk ), . . . , z = (z 1 , . . . , z s ) the vector (x1 , . . . , xm , y1 , . . . , yk , . . . , z 1 , . . . , z s ). is denoted by (x, y, . . . , z) Method C is the coding method consisting of the following five steps. 1. To select parameters n 0 ∈ N, α ∈ {0.01, 0.02, . . . , 0.99, 1}, t ∈ N p ∈ {1, 2, . . . , 100}. 2. To transform the given vector Cn with the help of mapping Wα into a set of vectors Cn−n 0 , Dn−1 , . . . , Dn−n 0 . 3. To quantize uniformly the vector (Cn−n 0 , Dn−1 , . . . , Dn−n 0 ) with the step Δt . 4. To apply the transform Wα−1 to the vector obtained in step 3 and as a result getting of vector % Cn . 5. To verify the condition p · max |Cn,k |. Cn ) < δ(Cn , % 100 0≤k≤N −1

(5.128)

If Condition (5.162) is not satisfied, then change t by t + 1 and repeat steps 3–5. 6. To calculate the value of parameter α, in which the entropy of the vector obtained after quantization in step 3 is the minimum.

232

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

Table 5.2 Root mean square error when coding Weierstrass function by Method A p (a, b) α Wα Haar Db4 85 85 85 95 95 95

(0.7,5) (0.9,3) (0.9,7) (0.7,5) (0.9,3) (0.9,7)

0.555 0.765 0.95 0.68 0.36 0.525

0.6142 0.8983 1.5876 0.6448 1.0690 1.9078

0.6223 0.9008 1.5882 0.6733 1.0927 1.9184

0.7495 6.9543 3.0923 3.6479 3.9443 6.7063

Table 5.3 Root mean square error when coding Weierstrass function by Method B t (a, b) α Wα Haar Db4 10 10 10 100 100 100

(0.7,5) (0.9,3) (0.9,7) (0.7,5) (0.9,3) (0.9,7)

0.09 0.06 0.17 0.77 0.39 0.73

0.2389 0.4593 0.5243 0.0754 0.1308 0.1476

0.2662 0.4754 0.5419 0.0799 0.1384 0.1537

0.2773 0.4714 0.5376 0.0819 0.1437 0.1709

A similar method is employed even when using discrete Haar and Daubechies transformations (Haar transformation is obtained when α = 1). We notice that in examples given below the value of p shows also the percentage of zero spike coefficients when the methods A and B are used. Weierstrass function with parameters a,b is determined by the formula Wa,b (x) =

∞ 

a n cos(bn π x), 0 < a < 1, b ≥ 1/a.

n=1

The results obtained by the methods A, B, and C in steps of digitization 0.005 on section [0,1.28] and pairs of parameters (a, b) = (0.7, 5), (0.9, 3), (0.9, 7). are given in Tables 5.2, 5.3, and 5.4. It is clear that the periodic dynamic spur with optimum values of parameter α give result in many cases better than the classic spurs. Let us now use Method A for the fractal functions of Van-der-waerden V (x). Riemann R(x),Takagi T (x), and Hankel G (x), determined as follows: 1. Let the function f 0 (x) have a period 1 and on section [0,1] defined by the formula # x, x ∈ [0, 1/2), f 0 (x) = 1 − x, x ∈ [1/2, 1] (the value of f 0 (x) is equal to distance between x and the nearest integer point). Then

5.5 Application to the Coding of Fractal Functions

233

Table 5.4 Entropy when coding Weieratrass function by Method C p (a, b) α Wα Haar 95 95 95 90 90 90

(0.7,5) (0.9,3) (0.9,7) (0.7,5) (0.9,3) (0.9,7)

0.48 0.45 0.67 0.52 0.41 0.51

4.0248 3.9607 4.3553 2.7626 2.6586 3.3664

Table 5.5 Root mean square error in 95% zeroing Function α Wα Van-der-Waerden Riemann Takagi Hankel

0.7850 0.4400 0.8950 0.3950

0.0605 0.1637 0.0992 0.1722

V (x) = 2. Riemann functoin: R(x) =

T (x) =

Db4

0.2486 0.4509 0.4039 0.3006

0.0594 0.1607 0.0991 0.1600

(5.129)

∞  sin(n 2 π x)

.

(5.130)

,

(5.131)

0 1 . sin sin(π nx)

(5.132)

n2

∞  t0 (2n−1 x) n=1

4.2059 4.0335 4.3612 3.5927 2.8044 3.2403

Haar

∞  f 0 (4n x) 4n n=1

n=1

3. Takagi function:

4.0678 4.0120 4.5534 2.8200 2.6803 3.4892

Db4

2n

where t0 (x) = 2|x − [x + (1/2)]|. 4. Hankel function: G (x) =

∞  sin(π nx) n=1

n2

/

Table 5.5 contains the results obtained by Method A with initial vectors Cn of length 256 obtained from the calculation of the values of the functions (5.126)– (5.128) on the section [0,1.28] with step 0.005 and function (5.129) on section [0,01, 1.29]. It is seen that periodic dyadic spurs with optimum values of parameters α surpass Haar spur and give results approximately the same as Daubechies Db4.

234

5 Orthogonal and Periodic Wavelets on Vilenkin Groups

References 1. Vilenkin, N. Y. (1947). On a class of complete orthonormal systems. Izv. Akad. Nauk Sssr, Ser. Mat. (No. 11, pp. 363–400). English Translation, American Mathematical Society Translations, 28 (Series 2), 1–35 (1963). 2. Fine, N. J. (1949). On the Walsh functions. Transactions of the American Mathematical Society, 65(3), 372–414. 3. Agaev, G. H., Vilemkin, N. Y., Dzhafarli, G. M., & Rubinstein, A. I. (1981). Multiplicative systems of functions and analysis on 0 dimensional groups. Baku: ELM [In Russian]. 4. Golubov, B. I., Efimov, A. V., & Skvortsov, V. A. (2008). Walsh series and transforms (English Transl. Of 1st ed.). Moscow: Urss; Dordrecht: Kluwer (1991). 5. Novikov, I. Y., Protasov, V. Y., & Skopina, M. A. (2011). Wavelet theory (Moscow, 2006). Providence: AMS. 6. Farkov, Y. A., & Rodionov, E. A. (2011). Algorithms for wavelet construction on Vilenkin groups. P-Adic Numbers, Ultrametric Analysis and Applications, 3(3), 181–195. 7. Lukomskii, S. F. (2014). Step refinable functions and orthogonal MRA on P-Adic Vilenkin groups. Journal of Fourier Analysis and Applications, 20(1), 42–65. 8. Lukomskii, S. F. (2015). Riesz multiresolution analysis on zero-dimensional groups. Izvestiya: Mathematics, 79(1), 145–176. 9. Berdnikov, G. S., & Lukomskii, S. F. N-valid trees in wavelet theory on Vilenkin groups. http:// arxiv.Org/abs/1412.309v1. 10. Farkov, Y. A. (2011). Periodic wavelets on the p-Adic Vilenkin group. P-Adic Numbers, Ultrametric Analysis, and Applications, 3(4), 281–287. 11. Chui, C. K., & Mhaskar, H. N. (1993). On trigonometric wavelets. Construction Approximately, 9, 167–190. 12. Frazier, M. W. (1999). An Introduction to wavelets through linear algebra. New York: Springer. 13. Mallat, S. (1999). A wavelet tour of signal processing. New York, London: Academic Press. 14. Lang, W. C. (1996). Orthogonal wavelets on the cantor dyadic group. SIAM Journal on Mathematical Analysis, 27(1), 305–312. 15. Lang, W. C. (1998). Fractal multiwavelets related to the Cantor dyadic group. International Journal of Mathematics and Mathematical, 21, 307–317. 16. Lang, W. C. (1998). Wavelet analysis on the Cantor dyadic group. Houston Journal of Mathematics, 24, 533–544. 17. Rodionov, E. A., & Farkov, Y. A. (2009). Estimates of the smoothness of Dyadic orthogonal wavelets of Daubechies type. Mathematical Notes, 86(3), 407–421. 18. Schipp, F., Wade, W. R., & Simon, P. (1990). Walsh series. New York: Adam Hilger.

Chapter 6

Haar–Vilenkin Wavelet

6.1 Introduction Haar System The Haar system H = (Hn , n ∈ N) is defined as follows: H0 = 1. For n, r ∈ N and 0 ≤ r < 2n the function Hn is defined on [0, 1) by ⎧ n ⎨ 2 2 x ∈ I (2r, n + 1) n H2n +r (x) = −2 2 x ∈ I (2r + 1, n + 1) ⎩ 0 other wise where I (2r, n + 1) = [2r 2−(n+1) , (2r + 1)2−(n+1) )   2r 2r + 1 = n+1 , n+1 . 2 2 It can be extended to R by the periodicity of period 1. Each Haar function is continuous from the right and the Haar system H is orthonormal on [0, 1). Haar–Vilenkin System The following system which is a generalization of Haar system is connected with the name of Vilenkin. Very often it is termed as a generalized Haar system or a Haar-type Vilenkin system (Schipp et al. [1]). Let m = (m k , k ∈ N) be a sequence of natural numbers such that m k ≥ 2, N denotes the set of nonnegative integers. Let M0 = 1 and Mk = m k−1 Mk−1 , k ∈ P. Let P denote the set of positive integers and let k ∈ P can be written as k = Mn + r (m n − 1) + s − 1. © Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2_6

(6.1) 235

236

6 Haar–Vilenkin Wavelet

where n ∈ N, r = 0, 1, . . . , Mn − 1 and s = 1, 2, . . . , m n − 1. This expression is unique for each k ∈ P. Let us write an arbitrary element t ∈ [0, 1) in the form t=

∞  k=0

tk , (0 ≤ tk < m k ). Mk+1

(6.2)

It may be noted that there exist two such expressions (6.2), for so-called m-adic rational numbers. In such cases we use the expression which contains only a finite number of terms different from zero. Define the function system (h n , n ∈ N) by h 0 = 1 and √ h k (t) =

≤ t < rM+1n . other wise

n Mn exp 2πist mn

r Mn

0

(6.3)

This system can be extended to R (the set of real numbers) by periodicity of period 1: h k (t + 1) = h k (t), t ∈ [0, 1). It can be checked that {h k (t)} is a complete orthonormal system in L 2 (R). It is clear that h k (t) = χ[ Mr

, r +1 ] (t) n Mn



Mn exp

2π istn . mn

Certain properties of this system have been recently studied. In the recent years, various extensions and concepts related to Haar wavelet has been studied. In this chapter, we study basic properties of Haar–Vilenkin wavelets and Haar–Vilenkin scaling function [2, 3]. We prove that the system {ψa,b }, a, b ∈ Z 2 is an orthonormal basis

in L (R), while the convergence properties of expansion of f namely the series a∈Z b∈Z  f, Dm an Tb h k (t)Dm an Tb h k (t) for arbitrary coefficients (k fixed) are studied.

6.2 Haar–Vilenkin Wavelets 6.2.1 Haar–Vilenkin Mother Wavelet The function h k (t) as defined in (6.3) can also be written as ⎧√ Mn ⎪ ⎪ ⎪√ 2πis ⎪ ⎪ ⎨ √ Mn exp m n Mn exp 4πis h k (t) = mn ⎪ ⎪ ⎪ ··· ⎪ ⎪ √ ⎩ M exp 2πis(m n −1) n

mn

r Mn r Mn r Mn

≤ t < Mr n + M1n+1 + M1n+1 ≤ t < Mr n + + M2n+1 ≤ t < Mr n +

r Mn

+

m n −1 Mn+1

≤t
=

R

ψa,b (t)ψa,b (t) dt = 0.

If b = b , then



< ψa,b , ψa,b > =

  ψa,b (t)2 dt

ψa,b (t)ψa,b (t) dt = I

Ia,b

a,b = Ia,b

m an Mn

dt = 1.

Next we will show the orthonormality between the scales. Suppose a, a ∈ Z with a = a , say a > a and let b, b ∈ Z. Then we have the possibilities 1. Ia ,b ∩ Ia,b = φ. In this case ψa,b (t)ψa ,b (t) = 0∀t and < ψa,b , ψa ,b >= ψa,b (t)ψa ,b (t) dt = 0. Ia,b

2. If a > a then either the intervals Ia,b and Ia ,b are disjoint or Ia,b is contained in the one of the m n subintervals     1 1 2





A , A + a

, A + a

,... , A + a

m n Mn+1 m n Mn+1 m n Mn+1   mn − 1 1 . . . A + a

, A + a

m n Mn+1 m n Mn where A =

r m an Mn

+

b m an

.

In each case, we will get < ψa,b , ψa ,b >=

ψa,b (t)ψa ,b (t) dt = 0. Ia,b

242

6 Haar–Vilenkin Wavelet

Thus ψa,b , a, b ∈ Z is an orthonormal system in L 2 (R).



In order to show that {ψa,b }a,b∈Z is an orthonormal basis in L 2 (R), let us consider the two families of subspaces of L 2 (R). S p = span{ψa,b }a< p,b∈Z

(6.11)

L p = {Set of all functions which are constant on intervals I p,b for b ∈ Z} (6.12) Both of these families have the following properties: · · · ⊂ S−2 ⊂ S−1 ⊂ S0 ⊂ S1 ⊂ S2 ⊂ · · ·

(6.13)

f (t) ∈ S p ⇔ f (2t) ∈ S p+1

(6.14)

f (t) ∈ S0 ⇔ f (t + k) ∈ S0 for k ∈ Z

(6.15)

In order to prove that {ψa,b } is an orthonormal basis in L 2 (R)it remains to prove that L p = Sp∀ p ∈ Z Lemma 6.2 For all p ∈ Z, we have L p = S p . Proof From (6.14) above it suffices to show that L 0 = S0 . Since each ψa,b for a < 0 is constant on any interval [u + Mr n , u + rM+1n ) we see that

S0 ⊂ L 0 . Also each function in L 0 can be written as u∈Z au χ[u+ Mr ,u+ rM+1 ) , Hence n n by (6.15) it suffices to show that χ[ Mr , rM+1 ) ∈ S0 . n

n

To show this, let us consider the series   m a/2 m an h k (m an t). n ψa,0 = a 0, M ∈ Z, we write M  

|f , ψj,λ |2 ≤ Bf 2 .

j=−M λ∈Λ

Using Lemma 8.2, we get M 2N −1 + ! 1   ˆ )−j ξ + p/2) fˆ (ξ + (2N )j p/2)ψ((2N 4N j=−M p=0 R  ˆ )−j ξ + l/2)d ξ fˆ (ξ + (2N )j l/2)ψ((2N × l∈Z

 +

R

×

ˆ )−j ξ + p/2)e−πirp/N fˆ (ξ + (2N )j p/2)ψ((2N 

! 

fˆ (ξ + (2N )

j

ˆ l/2)ψ((2N )−j ξ

+ l/2)e

πirl/N

dξ

≤ Bfˆ 2 .

l∈Z

(8.48) Let ξ0 ∈ R, given ε > 0, we consider fˆ (ξ ) =



√1 , 2ε

ξ ∈ (ξ0 − ε, ξ0 + ε) 0, otherwise

(8.49)

For sufficiently small ε, j ≥ −M , the intervals (ξ0 − ε − (2N )j l/2, ξ0 + ε − (2N )j l/2), l ∈ Z are mutually disjoint, we have fˆ (ξ + (2N )j l/2)fˆ (ξ + (2N )j m/2) = 0, l = m; l, m ∈ Z.

(8.50)

8.3 Nonuniform Wavelet Frames in L2 (R)

309

The inequality (8.48) becomes j M 2N −1  1   ξ0 +ε−(2N ) p/2 ˆ ˆ |f (ξ + (2N )j p/2)|2 |ψ((2N )−j ξ + p/2)|2 d ξ ≤ Bfˆ 2 (8.51) 2N ξ0 −ε−(2N )j p/2

j=−M p=0

or

 ξ0 +ε−(2N )j p/2 M 2N −1 1   1 ˆ |ψ((2N )−j ξ + p/2)|2 d ξ ≤ B. 2N j=−M p=0 2ε ξ0 −ε−(2N )j p/2

(8.52)

By taking ε → 0 and M → ∞ consecutively, we have 2N −1 1  ˆ |ψ((2N )−j ξ )|2 d ξ ≤ B 2N j∈Z p=0

⇒



ˆ |ψ((2N )−j ξ )|2 d ξ ≤ B

(8.53)

j∈Z

which is the right inequality of (8.46). To prove left inequality of (8.46) (2.19), consider  |f , ψj,λ |2 = I1 + I2 , (8.54) j∈Z λ∈Λ

    where I1 = j>−M λ∈Λ |f , ψj,λ |2 and I2 = j≤−M λ∈Λ |f , ψj,λ |2 . By the frame condition, I1 ≥ A − I2 . As we have shown that I1 → Sψ (ξ ), it remains to prove that I2 → 0 as M → ∞. Using (8.34), (8.35), and (8.38), we obtain I2 =



|f , ψj,λ |2

j≤M λ∈Λ

 (2N ) 2 1  = 2 j≤−M 0

j

 (2N ) 2 1  + 2 j≤−M 0

j

 (2N ) 2 1  ≤ 2 j≤−M 0

j

 (2N ) 2 1  + 2 j≤−M 0

j

 2     j −j ˆ ) ξ + l/2) d ξ fˆ (ξ + (2N ) l/2)ψ((2N    l∈Z  2   r   ˆ )−j ξ + l/2)eπi N l  d ξ fˆ (ξ + (2N )j l/2)ψ((2N    l∈Z  2     j −j ˆ ) ξ + l/2)| d ξ |fˆ (ξ + (2N ) l/2)ψ((2N    l∈Z  2   rk   ˆ |fˆ (ξ + (2N )j l/2)ψ((2N )−j ξ + l/2)eπi N l | d ξ    l∈Z

310

8 Wavelets Associated with Nonuniform Multiresolution …

=

  0

j≤−M

=

(2N )j 2

 

(2N )j 2

 2     ˆ )−j ξ + l/2)| d ξ |fˆ (ξ + (2N )j l/2)ψ((2N    l∈Z # "  j −j ˆ ˆ |f (ξ + (2N ) l/2)ψ((2N ) ξ + l/2)|

0

j≤−M

l∈Z

" ×



# −j j ˆ ˆ |f (ξ + (2N ) m/2)ψ((2N ) ξ + m/2)| d ξ.

m∈Z

Since the function can continue with I2 ≤

  j≤−M l∈Z

=

 l∈Z

ˆ )−j ξ + l/2)| is 21 (2N )j -periodic, we |fˆ (ξ + (2N )j l/2)ψ((2N

(2N )j 2 (l+1) (2N )j 2 l

 

j≤−M m∈Z R

ˆ |fˆ (ξ )ψ((2N )−j ξ )|



ˆ |fˆ (ξ + (2N )j m/2)ψ((2N )−j ξ + m/2)|d ξ

m∈Z

ˆ ˆ |fˆ (ξ )ψ((2N )−j ξ + m/2)|d ξ. )−j ξ )||fˆ (ξ + (2N )j m/2)ψ((2N

By Cauchy–Schwarz inequality, we get I2 ≤

   j≤−M m∈Z

R

ˆ |fˆ (ξ )ψ((2N )−j ξ )|2 d ξ

1/2  R

ˆ |fˆ (ξ + (2N )j m/2)ψ((2N )−j ξ + m/2)|2 d ξ

1/2

Let ξ0 ∈ R and fˆ be defined as above. If ξ + (2N )j m/2 ∈ (ξ0 − ε, ξ0 + ε), then |(2N )j m/2| < ε for fixed j ≤ −M . By the hypothesis of fˆ , the number of summation 2ε . Thus index m is bounded by (2N )j    (2N )−j (ξ0 +ε) 2ε  2 −j ˆ ˆ )|2 d ξ. ˆ I2 ≤ |f (ξ )ψ((2N ) ξ )| d ξ = |ψ(ξ −j (ξ −ε) (2N )j j≤−M R (2N ) 0 j≤−M (8.55) Also for any ξo = 0, given η > 0, a positive integer M may be chosen so that 

∞ 2(2N )M ξ0 1+2N

ˆ )|2 d ξ < η. |ψ(ξ

(8.56)

−1 It is clear that for 0 < ε < 2N ξ , the intervals ((2N )−j (ξ0 − ε), (2N )−j (ξ0 + 2N +1 0 ε)), j ∈ Z are mutually disjoint. It follows that

8.3 Nonuniform Wavelet Frames in L2 (R)

 

I2 ≤ ≤

j≤−M  ∞

311 (2N )−j (ξ0 +ε)

(2N )−j (ξ0 −ε)

2(2N )M ξ0 1+2N

ˆ )|2 d ξ |ψ(ξ

ˆ )|2 d ξ |ψ(ξ

< η.



This completes the Theorem.

We establish a sufficient condition for the system {ψj,λ (t)}j∈Z,λ∈Λ to be wavelet frame for L2 (R). For this, we prove the following lemma. Lemma 8.3 Let f be in L2 (R) such that fˆ ∈ Cc (R). If ψ ∈ L2 (R) and ess supξ ∈R  −j 2 ˜ j∈Z |ψ((2N ) ξ )| < ∞ then 

 |f , ψj,λ | = 2

j∈R λ∈Λ

R

|fˆ (ξ )|2



ˆ |ψ((2N )−j ξ )|2 d ξ + Rψ (f )

(8.57)

j∈Z

where Rψ (f ) = R0 + R1 + · · · + R2N −1 and for 0 ≤ p ≤ 2N − 1, Rp is given by  ! 1  ˆ )−j ξ + p/2) Rp = fˆ (ξ + (2N )j p/2)ψ((2N 4N j∈Z R l=p

! r ˆ )−j ξ + l/2)(1 + eπi N (l−p) ) d ξ. fˆ (ξ + (2N )j l/2)ψ((2N (8.58)

Proof By (8.33), (8.34), and (8.34), we have 

|f , ψj,λ |2

j∈Z λ∈Λ

 1  ˆ 2 |fˆ (ξ )|2 |ψ((2N )−j ξ )|2 d ξ 4N R j∈Z  ! r 1  ˆ ˆ fˆ (ξ )ψ((2N )−j ξ ) fˆ (ξ + (2N )j l/2)ψ((2N )−j ξ + l/2)(1 + eπ i N l )d ξ + 4N j∈Z l=0 R  1  ˆ 2 |fˆ (ξ + (2N )j /2)|2 |ψ((2N )−j ξ + 1/2)|2 d ξ + 4N R j∈Z  ! 1  ˆ fˆ (ξ + (2N )j /2)ψ((2N + )−j ξ + 1/2) 4N j∈Z l=1 R ! r ˆ × fˆ (ξ + (2N )j l/2)ψ((2N )−j ξ + l/2)(1 + eπ i N (l−1) ) d ξ +

=

.. . +

 1  ˆ 2 |fˆ (ξ + (2N − 1)(2N )j /2)|2 |ψ((2N )−j ξ + (2N − 1)/2)|2 d ξ 4N R j∈Z

312

8 Wavelets Associated with Nonuniform Multiresolution … +

 ! 1   ˆ )−j ξ + (2N − 1)/2) fˆ (ξ + (2N − 1)(2N )j /2)ψ((2N 4N R j∈Z l=2N −1

! r ˆ )−j ξ + l/2)(1 + eπ i N (l−2N +1) ) d ξ. × fˆ (ξ + (2N )j l/2)ψ((2N

Since for 0 ≤ p ≤ 2N − 1,  j∈Z R

|fˆ (ξ + (2N )j p)|2 |ψ((2N )−j ξ + p)|2 d ξ =

 j∈Z R

|fˆ (ξ )|2 |ψ((2N )−j ξ )|2 d ξ.

(8.59) Therefore, we obtain 

 |f , ψj,λ |2 =

j∈R λ∈Λ

R

|fˆ (ξ )|2



|ψ((2N )−j ξ )|2 d ξ + R0

j∈Z

R1 + · · · + R2N −1 

which is the desired result. We establish a sufficient condition for nonuniform wavelet frame for L2 (R). Theorem 8.8 Let ψ ∈ L2 (R) be such that A=

⎧ ⎨

inf

|ξ |∈[1,2N ] ⎩

ˆ |ψ((2N )−j ξ )|2 −



ˆ ˆ )−j ξ + l/2)| |ψ((2N )−j ξ )ψ((2N

j∈Z l=0

j∈Z

⎫ ⎬ ⎭

> 0,

and B=

sup

⎧ ⎨ 

|ξ |∈[1,2N ] ⎩ j∈Z

ˆ ˆ |ψ((2N )−j ξ )ψ((2N )−j ξ + l/2|

l∈Z

⎫ ⎬ ⎭

< ∞.

(8.60)

Then {ψj,λ }j∈Z,λ∈Λ is a wavelet frame with bounds A and B. Proof Let f ∈ L2 (R) such that fˆ is continuous and compactly supported. We estimate Rψ (f ). By (8.58), we have 1 |Rp | = 4N

    !   ˆ )−j ξ + p/2) fˆ (ξ + (2N )j p/2)ψ((2N   j∈Z l=p R

!  r  ˆ )−j ξ + l/2)(1 + eπ i N (l−p) ) d ξ  × fˆ (ξ + (2N )j l/2)ψ((2N   ! 1   ˆ ˆ ≤ )−j ξ + p/2)  f (ξ + (2N )j p/2)ψ((2N 2N j∈Z l=p R !  ˆ )−j ξ + l/2)  d ξ. × fˆ (ξ + (2N )j l/2)ψ((2N

8.3 Nonuniform Wavelet Frames in L2 (R)

313

Applying Cauchy–Schwarz’ inequality on the integral, we get |Rp | ≤

1/2  1  ˆ ˆ |fˆ (ξ + (2N )j p/2)|2 |ψ((2N )−j ξ + p/2)ψ((2N )−j ξ + l/2)|d ξ 2N R j∈Z l =p

 ×

R

ˆ ˆ )−j ξ + l/2)|d ξ |fˆ (ξ + (2N )j l/2)|2 |ψ((2N )−j ξ + p/2)ψ((2N

1/2 .

Again using Cauchy–Schwarz’ inequality on the summation over l, we obtain ⎫1/2 ⎧  ⎬ 1  ⎨ j 2 ˆ −j −j ˆ ˆ |Rp | ≤ |f (ξ + (2N ) p/2)| |ψ((2N ) ξ + p/2)ψ((2N ) ξ + l/2)|d ξ ⎭ ⎩ 2N R j∈Z

l=p

×

⎧ ⎨  ⎩

l=p

R

ˆ ˆ |fˆ (ξ + (2N )j l/2)|2 |ψ((2N )−j ξ + p/2)ψ((2N )−j ξ + l/2)|d ξ

⎫1/2 ⎬ ⎭

1  = IJ . 2N j∈Z

where I=

J =

⎧ ⎨  ⎩

l=p R

⎧ ⎨  ⎩

l=p

R

ˆ ˆ )−j ξ + l/2)|d ξ |fˆ (ξ + (2N )j p/2)|2 |ψ((2N )−j ξ + p/2)ψ((2N

⎫1/2 ⎬ ⎭

ˆ ˆ )−j ξ + l/2)|d ξ |fˆ (ξ + (2N )j l/2)|2 |ψ((2N )−j ξ + p/2)ψ((2N

,

⎫1/2 ⎬ ⎭

On changing the variable ξ → ξ + 21 (2N )j (p − l) in J , J =

=

⎧ ⎨  ⎩

l=p

R

⎧ ⎨  ⎩

l  =p R

ˆ ˆ )−j ξ + p/2)|d ξ |fˆ (ξ + (2N )j p/2)|2 |ψ((2N )−j ξ + (2p − l)/2)ψ((2N

ˆ ˆ )−j ξ + l  /2)ψ((2N |fˆ (ξ + (2N )j p/2)|2 |ψ((2N )−j ξ + p/2)|d ξ

⎫1/2 ⎬ ⎭

⎫1/2 ⎬ ⎭

= I.

We get  1  ˆ ˆ )−j ξ + l/2)ψ((2N |fˆ (ξ + (2N )j p/2)|2 |ψ((2N )−j ξ + p/2)|d ξ 2N j∈Z l=p R  1  ˆ ˆ )−j ξ + (l − p)/2)|d ξ |fˆ (ξ )|2 |ψ((2N )−j ξ )ψ((2N = 2N R

|Rp | ≤

j∈Z l=p

.

314

8 Wavelets Associated with Nonuniform Multiresolution … =

1 2N



|fˆ (ξ )|2

R



ˆ ˆ )−j ξ + (l − p)/2)|d ξ. |ψ((2N )−j ξ )ψ((2N

j∈Z l=p

Therefore, we can write 2N −1   1  ˆ ˆ |Rψ (f )| ≤ )−j ξ + (l − p)/2)|d ξ. |fˆ (ξ )|2 |ψ((2N )−j ξ )ψ((2N 2N p=0 R j∈Z l=p

(8.61) From (8.57) and (8.61), we have 

 |f , ψj,λ |2 ≤

j∈Z λ∈Λ

R

|fˆ (ξ )|2



ˆ |ψ((2N )−j ξ )|2 d ξ

j∈Z

2N −1   1  ˆ ˆ |fˆ (ξ )|2 |ψ((2N )−j ξ )ψ((2N )−j ξ + (l − p)/2)|d ξ 2N p=0 R j∈Z l =p ⎛ ⎞   2 −j −j ˆ ⎝ ˆ ˆ = |f (ξ )| |ψ((2N ) ξ )ψ((2N ) ξ + l/2|⎠ d ξ (8.62)

+

R

j∈Z l∈Z

and  j∈Z λ∈Λ

 ≥

R

|f , ψj,λ |2 ⎛



|fˆ (ξ )|2 ⎝

ˆ |ψ((2N )−j ξ )|2 −



⎞ ˆ ˆ )−j ξ + l/2)|⎠ d ξ |ψ((2N )−j ξ )ψ((2N

j∈Z l=0

j∈Z

(8.63) Taking supremum in (8.62) and infimum in (8.63), we get Af  ≤



|f , ψj,λ |2 ≤ Bf 2 .

j∈Z λ∈Λ

This completes the theorem.



8.4 Exercises 8.1 Prove Theorem 8.1. 8.2 State and prove Analogue of Cohen’s conditions for NUMRA. 8.3 Show that Collection {N m/2 ψk (N m x λ)}λ ∈ Λ+ , m ∈ Z, k = 1, 2, . . . , N − 1 as defined in Lemma 8.1 forms a complete orthonormal system for L2 (R+ ). 8.4 For parameters N and r a s in Definition 8.3 and for m10 (ξ ) = m11 (ξ ) = 1/2, ξ ∈ [0, 2). Show that the system of functions generates a NUMRA in L2 (R+ ).

8.4 Exercises

315

8.5 Give definition of nonuniform wavelet frames. 8.6 Prove that the system {ψj,λ }j∈Z,λ∈Λ forms a Bessel sequence for B given by (8.60).

References 1. Sharma, V., & Manchanda, P. (2013). Wavelet packets associated with nonuniform multiresolution analysis on positive half-line. Asian-European Journal of Mathematics, 6(1), 1350007 (1–16). 2. Meenakshi, M., Manchanda, P., & Siddiqi, A. H. (2012). Wavelets associated with non uniform multiresolution analysis on positive half line. International Journal of Wavelets, Multiresolution and Information Processing 10(2), 1250018(1–27). 3. Sharma, V., & Manchanda, P. (2015). Nonuniform wavelet frames in L2 (R). Asian European Journal of Mathematics, 8(2), 1550034(1–15).

Chapter 9

Orthogonal Vector-Valued Wavelets on R+

9.1 Introduction We have considered the notion of vector-valued multiresolution analysis (VMRA) on positive half line R+ and studied associated vector-valued wavelets and wavelet packets. Xia and Suter in 1996 generalized the concept of multiresolution analysis (MRA) on R to vector-valued multiresolution analysis (VMRA) on R and studied associated vector-valued wavelets. Farkov [1] introduced MRA on R+ . In this chapter, we have introduced vector-valued multiresolution analysis (VMRA) on R+ , where the associated subspace V0 of L 2 (R+ , C N ) has an orthonormal basis, a family of translates of a vector-valued function Φ, i.e, {Φ(x  l)}l∈Z+ , where Z+ is the set of nonnegative integers. The necessary and sufficient condition for the existence of associated vector-valued wavelets has been obtained and the construction of vectorvalued multiresolution analysis (VMRA) on R+ has been presented. Lang [2–4] introduced the concept of compactly supported orthogonal wavelets on the locally compact Cantor dyadic group C . These wavelets turn out to be certain lacunary Walsh series on the R+ . Later on, Farkov [5] extended these results on the wavelet analysis to the locally compact Abelian group G which is defined for an integer p ≥ 2 and coincides with C when p = 2. Farkov [1] has presented the general construction of all compactly supported orthogonal p-wavelets in L 2 (R+ ). He proved necessary and sufficient conditions for scaling filters with p n many terms ( p, n ≥ 2) to generate a p-MRA in L 2 (R+ ). Farkov’s approach is connected with Walsh–Fourier theory. Thereafter, Protasov and Farkov [6] constructed dyadic compactly supported wavelets in L 2 (R+ ). Farkov et al. [7] provided an algorithm for computing compactly supported biorthogonal wavelets on positive half line. Xia and Suter (1996) initiated the study of vector-valued multiresolution analysis and orthogonal vector-valued wavelets on R. They unveiled that vector-valued wavelets can be viewed as a class of generalized multiwavelets and multiwavelets can be yielded from component functions in vector-valued wavelets. Associated with VMRA, they have described vector-valued scaling functions, which leads to © Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2_9

317

9 Orthogonal Vector-Valued Wavelets on R+

318

vector-valued wavelet functions. They carried out vector-valued wavelet transform for vector-valued signals where correlation exists between their components. In the present chapter, we discuss vector-valued MRA p-analysis on positive half line and associated vector-valued wavelets. These results are analog of earlier results cited above. Some results on vector-valued nonuniform MRA [8–10] are also presented.

9.2 Vector-Valued Multiresolution Analysis Let s be a constant and 2 ≤ s ∈ Z. By L 2 (R, Cs ) we denote the set of all vector-valued functions f(t), i.e.,   L 2 (R, Cs ) = f(t) = ( f 1 (t), f 2 (t), . . . , f s (t))T : t ∈ R, f k (t) ∈ L 2 (R), k = 1, 2, . . . , s ,

where T means the transpose and Cs denote the s-dimensional complex Eucledian space. The space L 2 (R, Cs ) is called vector-valued function space. For f ∈ L 2 (R, Cs ), f denotes the norm of the vector-valued function f and is defined as f =

 s  

1/2

R

k=1

| f k (t)|2 dt

.

(9.1)

For a vector-valued function f ∈ L 2 (R, Cs ) the integration of f(t) is defined as 

 R

f(t) dt =

 R

f 1 (t),

T

 R

f 2 (t), . . . ,

R

f s (t)

.

The Fourier transform of f(t) is defined by ˆ )= f(ξ



f(t)e−2πitξ dt.

R

For any two vector-valued functions f, g ∈ L 2 (R, Cs ) the inner product < f, g > is defined as  f(t)g∗ (t) dt, (9.2) < f, g >= R

where * denotes the transpose and the complex conjugate. The inner product in (9.2) is vector-valued, but still satisfies the property for an inner product: the linearity < f, a1 g1 + a2 g2 >= a1 < f, g1 > +a2 < f, g2 >,

9.2 Vector-Valued Multiresolution Analysis

and the commutativity

319

< f, g >=< g, f >∗ .

A sequence {fk (t)} ∈ L 2 (R, Cs ) is said to be orthonormal if it satisfies < fk (.), fn (.) >= δk,n Is , k, n ∈ Z,

(9.3)

where δk,n denotes the Kronecker symbol such that δk,n = 1 when k = n and δk,n = 0 when k = n, Is denotes the identity matrix of order s × s. Definition 9.1 A sequence {fk (t)} ∈ L 2 (R, Cs ), k ∈ Z is called an orthonormal basis for L 2 (R, Cs ) if it satisfies (9.3) and moreover for any f ∈ L 2 (R, Cs ) there exists a sequence of s × s constant matrices {Fk }k∈Z such that f(t) =



Fk fk (t), t ∈ R,

(9.4)

k∈Z

where the multiplication Fk fk (t) for each fixed t is the s × 1 matrix multiplication, and the convergence for infinite summation is as same as of the norm . defined by (9.1) for the vector-valued function space. Let {fk (t)}k∈Z is an orthonormal basis for L 2 (R, Cs ). Then the expansion (9.4) for any f ∈ L 2 (R, Cs ) is unique and Fk =< f, fk >, k ∈ Z.

(9.5)

We also called the expansion (9.4), the Fourier expansion of f. The corresponding Parseval equality is < f, f >=



Fk Fk∗ .

(9.6)

k∈Z

From Eq. (9.6), it is clear that < f, f >= 0 if and only if f = 0 where 0 is the zero vector. Let φ(t) = (φ1 (t), φ2 (t), . . . , φs (t))T ∈ L 2 (R, Cs ) satisfy the following refinement equation:  φ(t) = Pk φ(2t − k), (9.7) k∈Z

where {Pk }k∈Z is a s × s constant matrix sequence. Define a closed subspace V j ⊂ L 2 (R, Cs ) by V j = clos L 2 (R,Cs ) (span{φ(2 j t − k) : k ∈ Z}), j ∈ Z.

(9.8)

Vector-valued multiresolution analysis defined by Xia and Suter (1996) is as follows:

9 Orthogonal Vector-Valued Wavelets on R+

320

Definition 9.2 φ(t) defined by (9.7) generates a vector-valued multiresolution analysis {V j } j∈Z of L 2 (R, Cs ), if the sequence {V j } j∈Z defined in (9.8) satisfies the following: 1. · · · ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ · · · , 2. ∩ j∈Z V j = {0}, ∪ j∈Z V j is dense in L 2 (R, Cs ), where 0 is the zero vector of L 2 (R, Cs ), 3. h(t) ∈ V0 if and only if h(2 j t) ∈ V j ∀ j ∈ Z, 4. there exists φ(t) ∈ V0 such that the sequence {φ(t − k), k ∈ Z} is an orthonormal basis of V0 . The vector-valued function φ(t) is called a scaling function of the vector-valued multiresolution analysis. ˆ On taking the Fourier transform on both sides of (9.7), and assuming that φ(η) is continuous at zero, we have ˆ ˆ φ(η) = α(η/2)φ(η/2), η ∈ R, where α(η) =

1 Pk exp{−ikη}. 2 k∈Z

(9.9)

(9.10)

Let W j , j ∈ Z denote the orthogonal complement of V j in V j+1 and there exists a vector-valued function Ψ (t) ∈ L 2 (R, Cs ) such that the translations and dilations of Ψ (t) form a Riesz basis of W j , i.e., W j = clos L 2 (R,Cs ) (span{Ψ (2 j t − k) : k ∈ Z}), j ∈ Z.

(9.11)

Since Ψ (t) ∈ W0 ⊂ V1 , there exists a unique finitely supported sequence {Bk }k∈Z of s × s constant matrices such that  Bk φ(2t − k). (9.12) Ψ (t) = k∈Z

Let β(η) =

1 Bk .exp {−ikη}. 2 k∈Z

(9.13)

Then the Eq. (9.12) becomes ˆ Ψˆ (η) = β(η/2)φ(η/2), η ∈ R.

(9.14)

The following theorem proves the existence of orthogonal vector-valued wavelets: Theorem 9.1 Let φ(t) defined in (9.7) is an orthogonal vector-valued scaling function. Assume that Ψ (t) ∈ L 2 (R, Cs ) and α(η) and β(η) are defined, respectively, by (9.10) and (9.13). Then Ψ (t) is an orthogonal vector-valued wavelet function

9.2 Vector-Valued Multiresolution Analysis

321

associated with φ(t) if and only if α(η)β(η)∗ + α(η + π )β(η + π )∗ = O, η ∈ R. β(η)β(η)∗ + β(η + π )β(η + π )∗ = Is , η ∈ R. The following theorem proved by Chen and Cheng (2007) present an algorithm for the construction of compactly supported orthogonal vector-valued wavelets: Theorem 9.2 Let φ(t) ∈ L 2 (R, Cs ) be a 3-coefficient compactly supported orthogonal vector-valued scaling functions satisfying the following refinement equation: φ(t) = P0 φ(2t) + P1 φ(2t − 1) + P2 φ(2t − 2). Assume that there exists an integer n, 0 ≤ n ≤ 2, such that the matrix A defined in the following equation, is not only an invertible matrix but also a Hermitian matrix: A2 = (2Is − Pn Pn∗ )−1 Pn Pn∗ . Define



j = n, B j = A Pj , B j = −A−1 P j , j = n, j, n ∈ {0, 1, 2}

and Ψ (t) = B0 φ(2t) + B1 φ(2t − 1) + B2 φ(2t − 2). Then Ψ (t) is an orthogonal vector-valued function associated with φ(t).

9.3 Vector-Valued Multiresolution p-Analysis on R+ Let N ≥ 2 be an integer and C N denotes N -dimensional complex Euclidean space, IN and O represent N × N identity matrix and the zero matrix, respectively. L 2 (R+ , C N ) represents the set of square integrable vector-valued functions f(t) on positive half line, R+ i.e,  T   , L 2 (R+ , C N ) = f(t) = f 1 (t), f 2 (t), , . . . , f N (t) where t ∈ R+ , f v (t) ∈ L 2 (R+ ), v = 1, 2, . . . N and T denotes transpose. The space L 2 (R+ , C N ) is called vector-valued function space. For f ∈ L 2 (R+ , C N ), f L 2 (R+ ,CN ) is the norm of the function f, i.e,

9 Orthogonal Vector-Valued Wavelets on R+

322

f L 2 (R+ ,C N )

 N   = v=1

R+

| f v (t)|2 dt

and integration of f is given by 

 R+

f(t)dt =

 R+

f 1 (t)dt,

T

 R+

f 2 (t)dt, . . . ,

R+

f N (t)dt

.

The Walsh–Fourier transform of f(t) is defined by ˜ f(w) =

 R+

f(t)χ (k, w)dt =

where f˜i (w) =



f˜1 (w), f˜2 (w), . . . , f˜N (w).

T

,

(9.15)

 R+

f i (t)χ (t, w)dt,

1 ≤ i ≤ N.

For any f ∈ L 2 (R+ , C N ), we have 

 R+

f(t)f(t  k)dt =

R+

˜ f(w) ˜ ∗ χ (k, w)dw f(w)

(9.16)

For two vector-valued functions f, h ∈ L 2 (R+ , C N ), their symbol inner product is defined by 

f, h L 2 (R+ ,CN ) = f(t)h(t)∗ dt, R+

where ‘*’ means complex conjugate and transpose. The product defined above is matrix-valued (usually it is a scalar-valued) and it satisfies properties of an inner product such as linearity and commutativity. Definition 9.3 A sequence {fk (t)}k∈Z+ ⊂ U ⊆ L 2 (R+ , C N ) is called an orthonormal set of U, if it satisfies

fk (t), fn (t) = δk,n I N , (9.17) where δk,n is the Kronecker delta such that δk,n = 1 when k = n and δk,n = 0 when k = n. A vector-valued function f(t) ∈ U ⊆ L 2 (R+ , C N ) is said to be orthogonal in U if its translations, i.e., {f(t  k)}k∈Z+ satisfy

f(t  k), f(t  n) = δk,n I N , k, n ∈ Z+ .

(9.18)

9.3 Vector-Valued Multiresolution p-Analysis on R+

323

Definition 9.4 A sequence of vector-valued functions {fk (t)}k∈Z+ ⊂ U ⊆ L 2 (R+ , C N ), is called an orthonormal basis of U if it satisfies (9.17) and for any h(t) ∈ U, there exists a unique sequence {Ak }k∈Z+ whose each element is an N × N constant matrix such that  Ak fk (t). h(t) = k∈Z+

The multiresolution analysis approach is one of the main approaches in the construction of wavelets. We introduce vector-valued multiresolution p-analysis on positive half line and give the definition for associated orthogonal vector-valued wavelets [1, 11]. Definition 9.5 A vector-valued multiresolution p-analysis in L 2 (R+ , C ) is a sequence of closed subspaces V j , j ∈ Z of L 2 (R+ , CN ) such that the following hold: N

(a)  V j ⊂ V j+1 , j ∈ Z,  N 2 (b) j V j is dense in L (R+ , C ) and j V j = {0}, where 0 is the zero vector of 2 + N L (R , C ), (c) f(t) ∈ V j if and only if f( pt) ∈ V j+1 , (d) f(t) ∈ V0 ⇒ f(t ⊕ k) ∈ V0 for all k ∈ Z+ , (e) there exists a function Φ ∈ V0 called scaling function such that its translations {Φk (t) = Φ(t  k), k ∈ Z+ }, form an orthonormal basis for V0 . Now Φ(t) ∈ V0 ⇒ Φ( pt) ∈ V1 , by (e),  δ0,k I N =

R+

 =

Φ(t)Φ(t  k)dt √

R+

√ pΦ( pt) pΦ( pt  k)dt.

√ So, Φ1,k (t) = { pΦ( pt  k)}k∈Z+ form an orthonormal basis for V1 . Therefore, the space V j is defined by j

V j = clos L 2 (R+ ,CN ) (span{ p 2 Φ( p j t  k)}, k ∈ Z+ ), j ∈ Z.

(9.19)

Now Φ = (φ1 (t), φ2 (t), . . . , φ N (t))T ∈ V1 , we write Φ(t) =



Rk Φ( pt  k) , t ∈ R+ ,

(9.20)

k∈Z+

where {Rk }k∈Z+ is a sequence of N × N constant matrices. If the sequence {Rk }k∈Z+ is finite, we say that Φ(t) is a compactly supported vector-valued function.

9 Orthogonal Vector-Valued Wavelets on R+

324

On taking Walsh–Fourier transform, we have ˜ ˜ Φ(w) = R(w/ p)Φ(w/ p), w ∈ R+ , where R(w) =

1  Rk χ (k, w). p k∈Z

(9.21)

(9.22)

+

Noting that χ (k, w + l) = χ (k, w), k, l ∈ Z+ , so R(w) is 1-periodic function of w. By (9.21), we have    w w w Φ R 2 p p p   w w ˜ . . . Φ(0) R =R p p2  ∞  w ˜ = Φ(0). R l p l=1

˜ Φ(w) =R

We get R(0) = I N

or



Rk = I N .

k∈Z+

For j ∈ Z, let W j denote the orthocomplement subspace of V j in V j+1 and there exist p − 1 vector-valued functions Ψm (t) ∈ L 2 (R+ , C N ), m ∈ Λ, where Λ = {1, 2, . . . , p − 1}, such that their translations and dilations form a Riesz basis of W j , i.e, j

W j = clos L 2 (R+ ,C N ) (span{ p 2 Ψm ( p j t  k)}, m ∈ Λ, k ∈ Z+ ), j ∈ Z.

(9.23)

For each m ∈ Λ, Ψm (t) ∈ W0 ⊂ V1 , there exist p − 1 finitely supported sequences of N × N constant matrices {Sk(m) }k∈Z+ such that Ψm (t) =



Sk(m) Φ( pt  k), m ∈ Λ, t ∈ R+ .

(9.24)

k∈Z+

By taking Walsh–Fourier transform, the refinement equation (9.24) becomes ˜ Ψ˜ m (w) = S (m) (w/ p)Φ(w/ p), w ∈ R+ , m ∈ Λ, where S (m) (w) =

1  (m) Sk χ (k, w). p k∈Z+

(9.25)

(9.26)

9.3 Vector-Valued Multiresolution p-Analysis on R+

325

If Φ(t) ∈ L 2 (R+ , C N ) is an orthogonal vector-valued scaling function, then it follows from (9.18) that

Φ(t), Φ(t  k) = δ0,k I N , k ∈ Z+ . (9.27) Definition 9.6 The collection of functions Ψm (t) ∈ L 2 (R+ , C N ), m ∈ Λ are orthogonal vector-valued wavelet functions associated with the orthogonal vector-valued scaling function Φ(t), if they satisfy

Ψm (t), Φ(t  k) = O, m ∈ Λ, k ∈ Z+

(9.28)

and the family {Ψm (t  k), m ∈ Λ}k∈Z+ is an orthonormal basis of the subspace W0 . Therefore,

Ψm (t  l), Ψn (t  k) = δl,k δm,n I N , m, n ∈ Λ, l, k ∈ Z+ .

(9.29)

The following lemma gives a characterization in the frequency domain of an orthogonal vector-valued function f(t). Lemma 9.1 Let f(t) ∈ L 2 (R+ , C N ). Then f(t) is an orthogonal vector-valued function if and only if  ˜ + l)f(w ˜ + l)∗ = I N , w ∈ R+ . f(w (9.30) l∈Z+

Proof Let f(t) = tion, then



f 1 (t), f 2 (t), . . . , f N (t)  δ0,k I N =

R+

T

∈ L 2 (R+ , C N ) is an orthogonal func-

f(t)f(t  k)∗ dt,

By identity (9.16),  δ0,k I N = =

R+

 l∈Z+



=

˜ f(w) ˜ ∗ χ (k, w)dw f(w)

1

l+1

˜ f(w) ˜ ∗ χ (k, w)dw f(w)

l



˜ + l)f(w ˜ + l)∗ χ (k, w)dw f(w

0 l∈Z +

So, f(t) ∈ L 2 (R+ , C N ) is orthogonal ⇔

 l∈Z+

˜ + l)f(w ˜ + l)∗ = I N . f(w

Lemma 9.2 If Φ(t) ∈ L 2 (R+ , C N ), defined by (9.20), is an orthogonal vectorvalued scaling function, then for every k ∈ Z+ , we have

9 Orthogonal Vector-Valued Wavelets on R+

326



Ru (Ru⊕ pk )∗ = pδ0,k I N .

(9.31)

u∈Z+

Proof On using (9.20) in the relation (9.27), we have δ0,k I N = Φ(t  k), Φ(t)   = Ru Φ( pt  pk  u) Φ( pt  v)∗ Rv∗ dt u∈Z+

=

R+

u∈Z+ v∈Z+

=

v∈Z+

  R+

Ru Φ( pt  pk  u)Φ( pt  v)Rv∗ dt

1  Ru Φ(t  pk  u), Φ(t  v) Rv∗ p u,v∈Z+

1  = Ru (Ru⊕ pk )∗ . p u∈Z +

9.4 The Existence of Orthogonal Vector-Valued Wavelets on R+ In this section, we start with considering the existence of compactly supported orthogonal vector-valued wavelets on R+ . The necessary and sufficient condition for the vector-valued wavelets associated with VMRA on R+ is derived as follows: Theorem 9.3 Let Φ(t) ∈ L 2 (R+ , C N ) defined in (9.20) be an orthogonal vectorvalued scaling function. Suppose that Ψm (t) ∈ L 2 (R+ , C N ), m ∈ Λ and R(w) and S (m) (w) are defined by (9.22) and (9.26), respectively. Then Ψm (t), m ∈ Λ are orthogonal vector-valued wavelet functions associated with Φ(t) if and only if p−1 

 R

l=0

w+l p



S (m)



w+l p

∗

= O, m ∈ Λ, w ∈ R+ ,

(9.32)

and p−1  l=0

S

(m)



w+l p

S

(n)



w+l p

∗

= δm,n I N , m, n ∈ Λ, w ∈ R+ . (9.33)

Proof First, we prove the necessary part of the theorem. By Lemma 9.1 and (9.28), we have

9.4 The Existence of Orthogonal Vector-Valued Wavelets on R+

327

O = Φ(t), Ψm (t)  ˜ + l)Ψ˜ m (w + l)∗ Φ(w = l∈Z+

  w+l w + l ∗ (m) w + l ∗ Φ˜ S p p p l∈Z+   ∗   ∗  w w w w + n Φ˜ + n Φ˜ + n S (m) +n = R p p p p l= pn    ∗  w 1 w 1 w 1 + R + + n Φ˜ + + n Φ˜ + +n p p p p p p l= pn+1 ∗  1 w + +n ×S (m) p p . =





R

w+l p



Φ˜



. . +

  ∗ p−1 w p−1 w p−1 w + + n Φ˜ + + n Φ˜ + +n p p p p p p l= pn+( p−1)  ∗ w p−1 ×S (m) + +n p p 



R

On simplification, we get ⎞ ⎛  ∗  ∗    w w w w ⎝ + n Φ˜ + n ⎠ S (m) Φ˜ O=R p p p p l= pn ⎞ ⎛     ∗ w w w 1 ⎝  1 w 1 1 ∗ Φ˜ +R + + + n Φ˜ + + n ⎠ S (m) + p p p p p p p p l= pn+1 . . . .

⎞ ⎛  ∗  p−1 ⎝  p−1 w p−1 w w + + + n Φ˜ + +n ⎠ Φ˜ +R p p p p p p l= pn+ p−1  w p−1 ∗ ×S (m) + p p   p−1  w+l ∗ w+l (m) S = R . p p 

l=0

Again from (9.29) and Lemma 9.1, for m, n ∈ Λ, we get

9 Orthogonal Vector-Valued Wavelets on R+

328

δm,n I N =



Ψ˜ m (w + l)Ψ˜ n (w + l)∗

l∈Z+

   w+l w+l w + l ∗ (m) w + l ∗ S Φ˜ Φ˜ p p p p l∈Z+ ⎛ ⎞  ∗     ∗ w w w ⎝ w + n Φ˜ + n ⎠ S (n) = S (m) Φ˜ p p p p l= pn ⎞ ⎛  ∗    w w w 1 ⎝  1 1 1 ∗ (m) w + + + n Φ˜ + + n ⎠ S (n) + +S Φ˜ p p p p p p p p l= pn+1 . =



S (m)



. . .

⎞ ⎛  ∗   p − 1 p − 1 w p − 1 w w ⎝ + + + n Φ˜ + +n ⎠ Φ˜ + S (m) p p p p p p l= pn+ p−1  p−1 ∗ w (n) + ×S p p   p−1  w+l w+l ∗ S (n) = S (m) . p p 

l=0

Conversely, suppose identities (9.32) and (9.33) hold. By similar steps performed above, we have   p−1   w+l ∗ w+l ˜ + l)Ψ˜ m (w + l)∗ = Φ(w S (m) R = O, p p l=0 l∈Z +

and 

Ψ˜ m (w + l)Ψ˜ n (w + l)∗ =

p−1 

S

(m)

l=0

l∈Z+



w+l p

S

(n)



w+l p

∗

= δm,n I N .

Therefore,

Φ(t), Ψm (t  k) =

 l∈Z+



=

1

l+1

˜ Φ(w) Ψ˜ m (w)∗ χ (k, w)dw

l



Ψ˜ (w + l)Ψ˜ m (w + l)∗ χ (k, w)dw

0 l∈Z +

= O, m ∈ Λ, k ∈ Z+ .

9.4 The Existence of Orthogonal Vector-Valued Wavelets on R+

329

and 

1

Ψm (t), Ψn (t  k) =



Ψ˜ m (w + l)Ψ˜ n (w + l)∗ χ (k, w)dw

0 l∈Z +

= δ0,k δm,n I N , m, n ∈ Λ, k ∈ Z+ . Thus, Φ(t) and Ψm (t), m ∈ Λ are mutually orthogonal and {Ψm (t), m ∈ Λ} is a finite family of orthogonal vector-valued functions. This proves the orthogonality of {Ψm (t  k), m ∈ Λ}k∈Z+ . We now prove the completeness of {Ψm (t  k), m ∈ Λ}k∈Z+ in W0 . For any f ∈ W0 ⊂ V1 , there exists finitely supported sequence of N × N constant matrices {Ak }k∈Z+ such that  f= Ak Φ( pt  k). (9.34) k∈Z+

Thus

˜ ˜ f(w) = A (w/ p)Φ(w/ p),

where A (w) =

(9.35)

1  Ak χ (k, w). p k∈Z+

/ V0 means Also for f ∈ W0 and f ∈  f(t)Φ(t  k)∗ dt = O, k ∈ Z+ .

(9.36)

R+

This is equivalent to



˜ + l)Φ(w ˜ + l)∗ = O. f(w

(9.37)

l∈Z+

On using (9.21), (9.35) and Lemma 9.1, we have p−1  l=0

 A

w+l p



 R

w+l p

∗

= O, w ∈ R+ .

Let     ∗    , . . . , A w+pp−1 , A1 (w) = A wp , A w+1 p and     ∗    w+ p−1 , . . . , R . R1 (w) = R wp , R w+1 p p

(9.38)

9 Orthogonal Vector-Valued Wavelets on R+

330

For i = 1, 2, . . . , p − 1, we set       ∗  (i) w+ p−1 Si (w) = S (i) wp , S (i) w+1 , . . . , S . p p It is clear that for any w ∈ R+ , the identities (9.32) and (9.33) imply that the column vectors in the pN × N matrix R1 (w) and column vectors in the pN × N Si (w) are orthogonal and all these vectors form an orthonormal basis for the pN -dimensional complex Euclidean space C pN . The identity (9.38) implies that the column vectors in the pN × N matrix A1 (w) and column vectors in the pN × N R1 (w) are orthogonal. Then, for each m ∈ Λ, there exist p − 1 matrices L (m) (w), m ∈ Λ whose all entries are 1-periodic functions of w such that A (w) =



L (m) (w)S (m) (w), w ∈ R+ .

(9.39)

m∈Λ

Therefore, by (9.35) ˜ f(w) =



L (m) (w/ p)S (m) (w/ p)Φ(w/ p)

m∈Λ

=



L (m) (w/ p)Ψ˜ m (w).

(9.40)

m∈Λ

By the orthonormality of {Ψm (t  k), m ∈ Λ}k∈Z+ , we get  R+

˜ pw)f( ˜ pw)∗ dw = f(

 l∈Z+



=

1

l+1

l





L (m) (w)Ψ˜ m ( pw)Ψ˜ m ( pw)∗ L (m) (w)∗ dw

m∈Λ

L (m) (w)L (m) (w)∗ dw.

0 m∈Λ

This proves that L (m) (w) has Walsh–Fourier series expansion. Let constant N × N matrices {Q (m) k }k∈Z+ , m ∈ Λ be its Walsh–Fourier coefficients. Then f=



Q (m) k Ψm (t  k).

k∈Z+ m∈Λ

This proves the completeness of {Ψm (t  k), m ∈ Λ}k∈Z+ in W0 . By Walsh–Fourier analysis and (9.22), (9.26), identities (9.32) and (9.33) are equivalent to, respectively,

9.4 The Existence of Orthogonal Vector-Valued Wavelets on R+



331

(m) ∗ Rv (Sv⊕ pk ) = O, m ∈ Λ, k ∈ Z+

(9.41)

v∈Z+



(n) ∗ Sv(m) (Sv⊕ pk ) = pδm,n δ0,k I N , m, n ∈ Λ, k ∈ Z+ .

(9.42)

v∈Z+

Theorem 9.3 implies that a vector-valued multiresolution p-analysis in L 2 (R+ , C N ) provides vector-valued scaling function Φ, which leads to associated p-1 orthogonal vector-valued wavelet functions Ψm (t), m ∈ Λ such that its dilations and translations Ψ j,k,m (t) = p j/2 Ψm ( p j t  k), j ∈ Z, k ∈ Z+ , m ∈ Λ form an orthonormal basis for L 2 (R+ , C N ). The following theorem deals with the construction of vector-valued scaling function Φ. Theorem 9.4 Let R(w) =

1  Rk χ (k, w), p k∈Z +

be a matrix-valued scaling filter satisfying following conditions: (a) p−1 

 R

l=0

w+l p



 R

w+l p

∗

= IN .

(9.43)

(b) There exists a constant C > 0 and integer M > 0, such that for w ∈ (0, p M ) ∞   M   w    w       0, let μ M (w) =

M 

 R

l=1

w pl



˜ Φ(0)1 (0, p M ) (w),

(9.47)

where 1(0, p M ) (w) is the characteristic function of a subset (0, p M ) of R+ . Now  μ M (w)μ M (w)∗ χ (k, w)dw R+

    w w w ˜ R . . . R Φ(0) R p p2 pM 0 ∗  ∗    ∗ w w w ˜ ∗R × Φ(0) R . . . R χ (k, w)dw pM p M−1 p  ∗  M−1  1  M−1   M M−l ∗ M−l =p R( p w) R(w)R(w) R( p w) χ (k, p M w)dw 

pM

=

0

 =p

l=1

1 p

M 0

 +p

R( p

M−l

l=1

2 p

M 1 p

 M−1 

l=1



 M−1 

w) R(w)R(w)

∗

R( p

R( p

M−l

w) R(w)R(w)

∗

l=1

 M−1 

χ (k, p M w)dw

w)

l=1

 M−l

∗

 M−1 

∗ R( p

M−l

χ (k, p M w)dw

w)

l=1

. .  +p

. 1

M p−1 p

 = pM

1 p

0

×

 M−1  l=1

.

 M−1 

 R( p

M−l

w) R(w)R(w)

l=1

 M−1 

R( p M−l w)

l=1

R( p M−l w)

∗

  p−1  l=0

∗

 M−1  l=1

∗ R( p

M−l

w)

χ (k, p M w)dw

   l ∗ l R w+ R w+ p p

χ (k, p M w)dw

9.4 The Existence of Orthogonal Vector-Valued Wavelets on R+

333

˜ Φ(0) ˜ ∗ = I N , we have Using (9.43) and Φ(0)  R+

μ M (w)μ M (w)∗ χ (k, w)dw 

 M−1 

p M−1

= 0

 R

l=1

 =

R+

 =

R+ 1

w pl

  M−1 

 R

l=1

w pl

∗ χ (k, w)dw

μ M−1 (w)μ M−1 (w)∗ χ (k, w)dw . . . μ1 (w)μ1 (w)∗ χ (k, w)dw

 =

I N χ (k, w)dw

0

= δ0,k I N . From (9.47), we get that μk (w) converges to Φ(w) pointwise. In view of (9.44), we have ∗ ∗ ˜ ˜ ˜ ˜ Φ(w)  ≤ (C + 1)Φ(w) Φ(w) , w ∈ R+ . μ M (w)μ M (w)∗ − Φ(w)

Since all matrix norms are equivalent, there exists a constant C1 > 0 such that μ M μ∗M − Φ˜ Φ˜ ∗  ≤ C1

 R+

∗ 2 ˜ ˜ μ M (w)μ M (w)∗ − Φ(w) Φ(w)  dw.

By the Dominated convergence theorem, we get μ M μ∗M − Φ˜ Φ˜ ∗  → 0 as → ∞. Therefore,  R+

∗ ˜ ˜ Φ(w) Φ(w) χ (k, w)dw = lim



M−→∞ R +

μ M (w)μ M (w)∗ χ (k, w)dw

= δ0,k I N .

This proves the orthonormality of Φ(t  k), k ∈ Z+ .

9 Orthogonal Vector-Valued Wavelets on R+

334

9.5 Vector-Valued Nonuniform Multiresolution Analysis Definition 9.7 Given integers N ≥ 1 and r odd with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime, we say that ϕ ∈ L 2 (R, Cs ) generates a VNUMRA {V j } j∈Z of L 2 (R, Cs ), if the sequence {V j } j∈Z satisfies: (a) (b) (c) (d) (e)

· · · ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ · · · , ∪ j∈Z V j is dense in L 2 (R, Cs ), ∩ j∈Z V j = {0}, where 0 is the zero vector of L 2 (R, Cs ), ϕ(t) ∈ V j if and only if ϕ(2N t) ∈ V j+1 ∀ j ∈ Z. there exists ϕ(t) ∈ V0 such that the sequence {ϕ(t − λ), λ ∈ Λ} is an orthonormal basis of V0 where Λ = {0, r/N } + 2Z. The vector-valued function ϕ(t) is called a scaling function of the VNUMRA.

Note that when N = 1, one recovers from the above definition the definition of vector-valued multiresolution analysis with dilation factor equal to 2. Let ϕ(t) = (ϕ1 (t), ϕ2 (t), . . . , ϕs (t))T ∈ L 2 (R, Cs ) satisfy the following refinement equation:  ϕ(t) = Aλ ϕ(2N t − λ) (9.48) λ∈Λ

where {Aλ }λ∈Λ is s × s constant matrix sequence that has only finite number of terms. Define a closed subspace V j ∈ L 2 (R, Cs ) by   V j = clos L 2 (R,Cs ) span{ϕ (2N ) j t − λ , λ ∈ Λ} , j ∈ Z

(9.49)

Given a VNUMRA let Wm denotes the orthogonal complement of Vm in Vm+1 , for any integer m. It is clear from the conditions (a), (b) and (c) of the Definition 9.7 that L 2 (R, Cs ) = ⊕m∈Z Wm . As is the case in the standard situation, the main purpose of VNUMRA is to construct orthonormal basis of L 2 (R, Cs ) given by appropriate translates and dilates of a finite collection of functions, called the associated wavelets. Definition 9.8 A collection {Ψ k }k=1,2,...,2N −1 of functions in V1 will be called a set of wavelets associated with a given VNUMRA if the family of functions {Ψ k (x − λ)}k=1,2,...,2N −1, λ∈Λ is an orthonormal system of W0 . On taking the Fourier transform on both sides of Eq. (9.48), we have ˆ ˆ ϕ(2N η) = A(η)ϕ(η), η∈R

(9.50)

9.5 Vector-Valued Nonuniform Multiresolution Analysis

where A(η) =

335

1  Aλ e−2πiλη . 2N λ∈Λ

(9.51)

Since Λ = {0, r/N } + 2Z, we can write that A(η) = A1λ + A2λ e−2πir/N η

(9.52)

where {A1λ } and {A2λ } are s × s constant symmetric matrix sequences. Then  η   η  ˆ ϕˆ ϕ(η) =A 2N 2N  η   η  η ˆ =A A . . . ϕ(0) A 2N (2N )2 (2N )3  ∞  η ˆ = ϕ(0). A (2N )k k=1

(9.53)

Equation (9.53) implies that A(0) = Is or



A λ = Is ,

(9.54)

λ∈Λ

where Is denotes the identity matrix of order s × s.    W j = clos L 2 (R,Cs ) span{Ψ k (2N ) j t − λ , λ ∈ Λ, k = 1, 2, . . . , 2N − 1} , j ∈ Z.

Since Ψ k (t) ∈ W0 ⊂ V1 there exists a uniquely {Bλ,k }λ∈Λ,k=1,2,...,2N −1 of s × s constant matrices such that Ψ k (t) =



supported

Bλ,k ϕ(2N t − λ)

(9.55) sequence

(9.56)

λ∈Λ

On taking the Fourier transform on both sides of Eq. (9.56), we have ˆ Ψˆ k (2N η) = Bk (η)ϕ(η), where Bk (η) =

1  Bλ,k e−2πiλη . 2N λ∈Λ

(9.57)

(9.58)

Since Λ = {0, r/N } + 2Z, we can write that 1 2 Bk (η) = Bλ,k + Bλ,k e−2πir/N η ,

(9.59)

9 Orthogonal Vector-Valued Wavelets on R+

336

1 2 where {Bλ,k } and {Bλ,k } are s × s constant symmetric matrix sequences.

Lemma 9.3 If ϕ(t) ∈ L 2 (R, Cs ) defined by Eq. (9.48) is an orthogonal vectorvalued scaling function then we have 

Am A∗2N (λ−σ )+m = 2N δλ,σ Is ∀λ, σ ∈ Λ,

(9.60)

m∈2Z

where δλ,σ denotes the Kronecker’s delta. Proof Since ϕ ∈ L 2 (R, Cs ) defined by Eq. (9.48) are orthogonal vector-valued scaling functions, therefore for λ, σ ∈ Λ, we have 

ϕ(x − λ)ϕ ∗ (x − σ ) d x   = Au ϕ(2N x − 2N λ − u) ϕ ∗ (2N x − 2N σ − v)∗ Av d x

δλ,σ Is =

R

u∈Λ

=

R



 Au

u∈Λ v∈Λ

=

R

v∈Λ

ϕ(2N x − 2N λ − u)ϕ ∗ (2N x − 2N σ − v)∗ d x

Av

 1  Au ϕ(x − 2N λ − u)ϕ ∗ (x − 2N σ − v)∗ d x Av . 2N u∈Λ v∈Λ R

Taking u = 2m, v = 2n where m, n ∈ Z, we have  δλ,σ Is =

R

ϕ(x − λ)ϕ ∗ (x − σ ) d x

1  Au < ϕ(x − 2N λ − u), ϕ(x − 2N σ − v) > A∗v 2N u∈Λ v∈Λ 1  = A2m < ϕ(x − 2N λ − 2m), ϕ(x − 2N σ − 2n) > A∗2n 2N m∈Z n∈Z 1   = Am < ϕ(x − 2N λ − m), ϕ(x − 2N σ − n) > A∗n 2N m∈2Z n∈2Z 1  = Am A∗2N (λ−σ )+m . 2N =

m∈2Z



Therefore,

Am A∗2N (λ−σ )+m = 2N δλ,σ Is ∀λ, σ ∈ Λ.

m∈2Z

Taking u =

r N

+ 2m, v = 2n where m, n ∈ Z, we have

9.5 Vector-Valued Nonuniform Multiresolution Analysis

337

1  Au < ϕ(x − 2N λ − u), ϕ(x − 2N σ − v) > A∗v 2N u∈Λ v∈Λ 1  r = A Nr +2m < ϕ(x − 2N λ − − 2m), ϕ(x − 2N σ − 2n) > A∗2n 2N m∈Z n∈Z N 1   r − m), ϕ(x − 2N σ − n) > A∗n = A r +m < ϕ(x − 2N λ − 2N m∈2Z n∈2Z N N 1  = Am A∗2N (λ−σ )+m . 2N m∈2Z

δλ,σ =

Thus, in both the cases, we get 

Am A∗2N (λ−σ )+m = 2N δλ,σ Is ∀λ, σ ∈ Λ, ∀λ, σ ∈ Λ.

m∈2Z

 Lemma 9.4 Consider a VNUMRA as in Definition 9.7. Let Ψ 0 = ϕ, B0 (.) = A(.) and suppose that there exists 2N-1 functions Ψ k , k = 1, 2, . . . , 2N − 1 in V1 . Then the family of functions {Ψ k (x − λ)}λ∈Λ,k=0,1,2,...,2N −1 will form an orthonormal system for V1 iff for k, l ∈ {0, 1, 2, . . . , 2N − 1} 2N −1  p=0

 Bk

ξ p + 2N 4N



Bl∗



ξ p + 2N 4N

= δk,l Is .

(9.61)

Proof First, we will prove the necessary condition. By the orthonormality of Ψ k (t) ∈ L 2 (R, Cs ), k = 0, 1, 2, . . . , 2N − 1 (or the orthonormality of VNUMRA V j ), we have in the time domain < Ψ k (. − λ), Ψ l∗ (. − σ ) >=

 R

Ψ k (t − λ)Ψ l∗ (t − σ ) dt = δk,l δλ,σ Is where

where λ, σ ∈ Λ and k, l ∈ {0, 1, 2, . . . , 2N − 1}. Equivalently, in the frequency domain, we have  R

∗ e−2πi(λ−σ )w Ψˆ k (w)Ψˆ l (w) dw = δk,l δλ,σ Is where λ, σ ∈ Λ, k, l ∈ {0, 1, 2, . . . , 2N − 1}.

Taking λ = 2m, σ = 2n where m, n ∈ Z, we have

9 Orthogonal Vector-Valued Wavelets on R+

338

 δk,l δm,n Is = =

R R

= Let wk,l (ξ ) = Therefore,

 j∈Z

∗

e−2πi(2m−2n)ξ Ψˆ k (ξ )Ψˆ l (ξ ) dξ ∗ e−4πi(m−n)ξ Ψˆ k (ξ )Ψˆ l (ξ ) dξ

e−4πi(m−n)ξ

[0,N )



∗ Ψˆ k (ξ + N j)Ψˆ l (ξ + N j) dξ,

j∈Z

∗ Ψˆ k (ξ + N j)Ψˆ l (ξ + N j).



e−4πi(m−n)ξ wk,l (ξ ) dξ ⎤ ⎡  2N −1    p ⎦ dξ, e−4πi(m−n)ξ ⎣ wk,l ξ + = 2 [0,1/2) p=0

δk,l δm,n Is =

Therefore, we obtain

[0,N )

2N −1  p=0

Also on taking λ =  0=



(9.62)

+ 2m and σ = 2n where m, n ∈ Z, we have ∗

R

=

r N

 p = 2δk,l Is . wk,l ξ + 2

e−2πiξ.2(m−n) e−2πiξr/N Ψˆ k (ξ )Ψˆ l (ξ ) dξ e−4πiξ(m−n) e−2πiξ. N r

[0,N )



∗ Ψˆ k (ξ + N j)Ψˆ l (ξ + N j) dξ

j∈Z

e−4πiξ(m−n) e−2πiξ. wk,l (ξ ) dξ ⎤ ⎡  2N −1  p r r e−4πiξ(m−n) e−2πi N ξ ⎣ e−πi p N wk,l (ξ + )⎦ dξ. = 2 [0, 21 ) p=0

=

r N

[0,N )

We conclude that 2N −1  p=0

α p wk,l (ξ +

p ) = 0 where α = e−πir/N . 2

Also we have 2N −1  j=0

    j j ˆ∗ j ˆ = . wk,l ξ + Ψk ξ + Ψl ξ + 2 2 2 j∈Z

(9.63)

9.5 Vector-Valued Nonuniform Multiresolution Analysis

339

Therefore, Eq. (9.62) reduces to  p ˆ ∗  p Ψl ξ + = 2δk,l Is . Ψˆ k ξ + 2 2 p∈Z



(9.64)

Also     j j ∗ Ψˆ k 2N ξ + Ψˆ l 2N ξ + 2 2 j∈Z      j j j j = Bk ξ + ϕˆ ξ + ϕˆ ∗ ξ + Bl∗ ξ + 2 2 2 2 j∈Z  ˆ + n N )ϕˆ ∗ (ξ + n N )Bl∗ (ξ + n N ) Bk (ξ + n N )ϕ(ξ =

wk,l (2N ξ ) =



j=n.2N

    1 1 1 1 Bk ξ + n N + ϕˆ ξ + n N + ϕˆ ∗ ξ + n N + Bl∗ ξ + n N + 2 2 2 2 j=n.2N +1      2 2 2 2 Bk ξ + n N + + ϕˆ ξ + n N + ϕˆ ∗ ξ + n N + Bl∗ ξ + n N + 2 2 2 2 j=n.2N +2      3 3 3 3 Bk ξ + n N + ϕˆ ξ + n N + ϕˆ ∗ ξ + n N + Bl∗ ξ + n N + + 2 2 2 2 

+

j=n.2N +3

+···

   2N − 1 2N − 1 2N − 1 Bk ξ + n N + ϕˆ ξ + n N + ϕˆ ∗ ξ + n N + 2 2 2 j=n.2N +(2N −1)  2N − 1 Bl∗ ξ + n N + 2 ⎡ ⎤  ˆ + n N )ϕˆ ∗ (ξ + n N )⎦ Bl∗ (ξ ) = Bk (ξ ) ⎣ ϕ(ξ +



j=n.2N

 +Bk

1 ξ+ 2

 +Bk ξ +  +Bk ξ +

2 2 3 2







⎡ ⎣ ⎡ ⎣ ⎡ ⎣



 j=n.2N +1

 ϕˆ ξ

 j=n.2N +2

 ϕˆ ξ

 j=n.2N +3

+··· ⎡  2N − 1 ⎣ +Bk ξ + 2

⎤  1 ⎦ ∗ 1 ϕˆ ξ + n N + Bl ξ + 2 2 ⎤   2 ⎦ ∗ 2 2 Bl ξ + + nN + ϕˆ ∗ ξ + n N + 2 2 2 ⎤   3 ⎦ ∗ 3 3 Bl ξ + + nN + ϕˆ ∗ ξ + n N + 2 2 2

1 ϕˆ ξ + n N + 2

 j=n.2N +(2N −1)

Ψˆ





∗



2N − 1 ξ + nN + 2



⎤ 2N − 1 ⎦ ξ + nN + Ψˆ 2  2N − 1 Bl∗ ξ + 2 ∗



9 Orthogonal Vector-Valued Wavelets on R+

340

     1 1 2 2 = 2 Bk (ξ )Bl∗ (ξ ) + Bk ξ + Bl∗ ξ + + Bk ξ + Bl∗ ξ + + ··· 2 2 2 2   2N − 1 2N − 1 Bl∗ ξ + · · · + Bk ξ + 2 2   2N −1  j j =2 Bk ξ + Bl∗ ξ + . 2 2 j=0

Therefore, we have  p∈Z

  2N −1   ξ ξ p p p ˆ ∗  p Ψl ξ + =2 + Bl∗ + . Bk Ψˆ k ξ + 2 2 2N 4N 2N 4N p=0

By using (9.64), we conclude that 2N −1 

 Bk

p=0

 Bk

ξ 2N



Bl∗



ξ 2N



 + Bk

· · · + Bk



ξ p + 2N 4N

1 ξ + 2N 4N





Bl∗



2N − 1 ξ + Bl∗ 2N 4N

Bl∗







ξ p + 2N 4N

1 ξ + 2N 4N



= δk,l Is , i.e.,

 + Bk

2N − 1 ξ + 2N 4N



2 ξ + 2N 4N



Bl∗



2 ξ + 2N 4N

+ ···

= δk,l Is .

Now we will prove the sufficiency. By Eq. (9.57), for ξ ∈ R   j j ∗ Ψˆk ξ + Ψˆ l ξ + 2 2 j∈Z    ξ j ξ + + = Bk ϕˆ 2N 4N 2N j∈Z ⎡   ξ n ⎣  ξ + = Bk ϕˆ 2N 2 2N j=n.2N ⎡  ξ n 1 ⎣  + + +Bk 2N 2 4N



j 4N



n + 2

ϕˆ ∗





ϕˆ ∗

ξ j + 2N 4N





Bl∗



ξ j + 2N 4N



⎤  ξ n ⎦ ∗ ξ n + + Bl 2N 2 2N 2

⎤ ξ n 1 ⎦ + + ϕˆ 2N 2 4N j=n.2N  ξ n 1 + + + ··· Bl∗ 2N 2 4N ⎡ ⎤    ξ ξ n 2N − 1 ⎣  n 2N − 1 n 2N − 1 ⎦ ξ ∗ + + + + + + +Bk ϕˆ ϕˆ 2N 2 4N 2N 2 4N 2N 2 4N j=n.2N  n 2N − 1 ξ + + Bl∗ 2N 2 4N      1 1 ξ ξ ξ ξ ∗ ∗ Bl + Bk + Bl + + = 2 Bk 2N 2N 2N 4N 2N 4N 

ξ n 1 + + 2N 2 4N



ϕˆ ∗



9.5 Vector-Valued Nonuniform Multiresolution Analysis 

2 ξ + 2N 4N = 2δk,l Is . Bk



Bl∗



2 ξ + 2N 4N



+ · · · + Bk

341

 2N − 1 2N − 1 ξ ξ + Bl∗ + 2N 4N 2N 4N

It proves the orthonormality of {Ψ k (t − λ), λ ∈ Λ, k = 0, 1, 2, . . . , 2N − 1}.



Now, we have the following result on the existence of a vector-valued wavelet function: Theorem 9.5 Suppose {Ψ k (t − λ)}λ∈Λ, k=0,1,...,2N −1 is the system as defined in Lemma 9.4 and orthonormal in V1 . Then this system is complete in W0 ≡ V1  V0 . Proof Since the system {Ψ k (t − λ)}λ∈Λ, k=0,1,...,2N −1 is orthonormal in V1 . By Lemma 9.4 we have  Bk

ξ 2N



Bl∗



ξ 2N



 + Bk 

· · · + Bk

1 ξ + 2N 4N



Bl∗



2N − 1 ξ + Bl∗ 2N 4N





1 ξ + 2N 4N



 + Bk

2N − 1 ξ + 2N 4N



2 ξ + 2N 4N



Bl∗



2 ξ + 2N 4N

+ ···

= δk,l Is .

We will now prove its completeness. For fk ∈ W0 ∃ the constant matrices {Fλ,k } such that fk (t) =



Fλ,k ϕ(2N t − λ), k = 0, 1, . . . , 2N − 1.

λ∈Λ

Thus fˆk (ξ ) = Fk where Fk (ξ ) =



ξ 2N

 ξ ϕˆ , 2N

(9.65)

1  Fλ,k e−2πiλξ . 2N λ∈Λ

On the other hand, f ∈ / V0 and f ∈ W0 implies  R

fk (t)ϕ ∗ (t − λ) dt = 0, λ ∈ Λ.

This condition is equivalent to   j j ϕˆ ∗ ξ + = 0 ξ ∈ R. fˆk ξ + 2 2 j∈Z



Therefore, from Eqs. (9.50) and (9.65) we have  j∈Z

 Fk

j ξ + 2N 4N

   j j j ξ ξ ξ + + A∗ + = 0, ξ ∈ R. ϕˆ ϕˆ ∗ 2N 4N 2N 4N 2N 4N

9 Orthogonal Vector-Valued Wavelets on R+

342

As similar to the identity (9.61) in Lemma 9.4, we have  Fk

  ξ ξ ξ 1 1 ∗ A + Fk + A + + ··· 2N 2N 4N 2N 4N   ξ ξ 2N − 1 2N − 1 A∗ = 0. · · · + Fk + + 2N 4N 2N 4N

ξ 2N



∗



(9.66)

 ξ   ξ   ξ ∗ Let Fk  2N = F , F ξ + 1 , . . . , Fk 2N + 2N4N−1 ,  ξ   ξ k 2N  ξ k 2N1 4N  ξ ∗ A1 2N = A 2N , A 2N + 4N , . . . , A 2N + 2N4N−1 ,  ξ   ξ  ξ  ∗ ξ 1 , . . . , Bk 2N + 2N4N−1 . Then the Bk  2N = Bk 2N , Bk 2N + 4N Eq. (9.61) implies that for any ξ ∈ R, the column vectors in 2N s × s matrix A1 (ξ ) and the column vectors in 2N s × s matrix Bk  (ξ ) are orthogonal for k = 0, 1, . . . , 2N − 1 and these vectors form an orthogonal basis of 2N s dimensional complex Eucledian space C2N s . The identity (9.66) implies that the column vectors in 2N s × s matrix Fk  (ξ ) and the column vectors in A1 (ξ ) are orthogonal. Thus there exists an s × s matrix Pk (ξ ) such that Fk (ξ ) = Pk (ξ )Bk (ξ ), ξ ∈ R, k = 0, 1, . . . , 2N − 1. Therefore, from Eqs. (9.57) and (9.65), we have 

 ξ ξ ϕˆ 2N 2N    ξ ξ ξ Bk ϕˆ = Fk 2N 2N 2N  ξ = Pk Ψˆ k (ξ ). 2N

fˆk (ξ ) = Fk

By using the orthonormality of {Ψ k (t − λ) : λ ∈ Λ}, we have   ∗ fˆk (2N ξ )fˆk∗ (2N ξ ) dξ = Pk (ξ )Ψˆ k (2N ξ )Ψˆ k (2N ξ )Pk∗ (ξ ) dξ. R

R

Therefore, we have  R

fˆk (2N ξ )fˆk∗ (2N ξ ) dξ = 2

 0

1/2

Pk (ξ )Pk∗ (ξ ) dξ.

This shows that Fk (ξ ) has the Fourier series expansion and let the constant s × s matrices {Q λ,k }λ∈Λ,k=0,1,...,2N −1 be its Fourier coefficients. Therefore, fk (t) =

 λ∈Λ

Q λ,k Ψ k (t − λ).

9.5 Vector-Valued Nonuniform Multiresolution Analysis

343

It proves the completeness of {Ψ k (t − λ)}λ∈Λ, k=0,1,...,2N −1 in W0 .



If Ψ 0 , Ψ 1 , . . . , Ψ 2N −1 ∈ V1 are as in Theorem 9.5, one can obtain from them an orthonormal basis for L 2 (R, Cs ) by following the standard procedure for construction of wavelets from a given MRA. It can be easily checked that for every m ∈ Z, the collection {(2N )m/2 Ψ k ((2N )m x − λ)}λ∈Λ,k=0,1,...,2N −1 is a complete orthonormal system for Vm+1 . Given a VNUMRA, since Wm the orthogonal complement of Vm in Vm+1 , m ∈ Z and L 2 (R, Cs ) = ⊕m∈Z Wm where ⊕ denotes the orthogonal direct sum with the inner product of L 2 (R, Cs ). From this, it follows immediately that the collection {(2N )m/2 Ψ k ((2N )m x − λ)}λ∈Λ,m∈Z,k=0,1,2,...,2N −1 forms a complete orthonormal system for L 2 (R, Cs ). When N = 1, we recover the usual construction of vector-valued wavelets from vectorvalued multiresolution analysis.

9.5.1 Construction of Vector-Valued Nonuniform Multiresolution Analysis We are going to construct VNUMRA starting from a matrix-valued function A(ξ ) which is of the form (9.67) A(ξ ) = A1λ + e−2πir/N ξ A2λ , where the integers r and N satisfy N ≥ 1, 1 ≤ r ≤ 2N − 1, r is odd, r and N are relatively prime and A1λ and A2λ are s × s constant symmetric matrix sequences. The scaling function ϕ associated with the given VNUMRA should satisfy the scaling relation ˆ ˆ ), ξ ∈ R. ϕ(2N ξ ) = A(ξ )ϕ(ξ (9.68) Also assume that the following condition holds: 

  ξ ξ ξ 1 1 ∗ A +A + A + + ··· 2N 2N 4N 2N 4N   ξ 2N − 1 2N − 1 ξ + A∗ + = Is . ··· + A 2N 4N 2N 4N

ξ A 2N

,i.e.,



∗



2N −1  j=0

 A

j ξ + 2N 4N



A∗



ξ j + 2N 4N

= Is .

(9.69)

9 Orthogonal Vector-Valued Wavelets on R+

344

  ! ξ Also assume that A(0) = Is in order for the infinite product ∞ k=1 A (2N )k to converge pointwise. The remaining part of this section will be devoted to the construction of a VNUMRA starting from a matrix function A(ξ ) of the form (9.67) that satisfies (9.69) as well as A(0) = Is . We obtain a matrix-valued function ϕ which satisfies the scaling relation ˆ ˆ ). ϕ(2N ξ ) = A(ξ )ϕ(ξ

(9.70)

At this point, it is necessary to determine whether the collection of functions {ϕ(x − λ)}λ∈Λ where Λ = {0, r/N } + 2Z is an orthonormal system in L 2 (R, Cs ). Theorem 9.6 If A(ξ ) is of the form (9.68) which satisfies A(0) = Is together with the conditions (9.69), Aλ = A∗λ for λ ∈ Λ (or A(ξ ) = A∗ (ξ ) ∀ ξ ∈ R). There exists a constant c > 0 and a compact set K ⊂ R that contains a neighborhood of the origin and      ξ  ≥ C ∀ξ ∈ K ∀ k ≥ 1 A (9.71)  k (2N )  then the solution ϕ(t) in the matrix dilation equation (9.48) is a vector-valued scaling function for VNUMRA. Thus the corresponding ϕ j,k (t), j ∈ Z, λ ∈ Λ form an orthonormal basis of L 2 (R, Cs ). Proof In order to prove this proposition, we need only to prove the orthonormality of {ϕ(t − λ) : λ ∈ Λ}. Since Aλ = A∗λ for λ ∈ Λ, therefore, we have A∗ (ξ ) = A(−ξ ) and ϕˆ ∗ (ξ ) = ˆ ϕ(−ξ ) ∀ξ . For any integer k ≥ 1, define ⎡

⎤  ξ ξ ⎣ ⎦ . μk (ξ ) = A χK (2N ) j (2N )k j=1 k 



Since 0 belongs to interior of K by assumption, therefore μk → ϕˆ pointwise as k → ∞. There is also a constant B > 0 such that |A(ξ ) − A(0)| ≤ B|ξ | ∀ ξ ∈ R and thus |A(ξ )| ≥ 1 − B|ξ | ∀ ξ . Since K is bounded, we can find an integer k0 ≥ 1 such that and k > k0 . There exists a constant C > 0 such that ˆ )| ∀ ξ ∈ R. χ K (ξ ) ≤ C|ϕ(ξ

B |ξ | (2N )k


= δλ,σ Is , λ, σ ∈ Λ. Proof We will prove the result by induction on n. If n = 0 then the Eq. (9.80) follows directly from hypothesis. # n $ < Suppose 0 ≤ n < (2N )l for some integer l. Then, for some (2N )l−1 ≤ 2N # n $ l 2 where [x] denotes the greatest integer of x and order n = 2N 2N + k, k = 0, 1, 2, . . . , 2N − 1. Therefore, by induction, we have & n n γ [ 2N ] (. − λ), γ [ 2N ] (. − σ ) = δλ,σ Is .

%

(9.81)

9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties

349

We obtain %

&

γ (. − λ), γ (. − σ ) = n

n

 R



e−2πi(λ−σ )ξ γˆ n (ξ )γˆ n (ξ )∗ dξ e−2πi(λ−σ )ξ

[0,N )

Let wn (ξ ) =





γˆ n (ξ + N j)γˆ n (ξ + N j)∗ dξ,

j∈Z

γˆ n (ξ + N j)γˆ n (ξ + N j)∗ .

j∈Z

Therefore, by using (9.74) and (9.81), we obtain wn (2N ξ )  n = γˆ (2N (ξ + j/2))γˆ n (2N (ξ + j/2))∗ j∈Z

=



Q (k) (ξ + j/2)γˆ

#

n 2N

j∈Z

=



Q (k) (ξ + n N )γˆ

$

#

((ξ + j/2))γˆ

n 2N

$

#

(ξ + n N )γˆ

n 2N

#

$

n 2N

((ξ + j/2))∗ Q (k) (ξ + j/2)∗ $

(ξ + n N )∗ Q (k) (ξ + n N )∗

j=n.2N

 # $ # $ n n 1 1 Q (k) ξ + n N + γˆ 2N ξ + n N + γˆ 2N ξ + n N + 2 2 j=n.2N +1  # $ # $  n n 2 2 + Q (k) ξ + n N + γˆ 2N ξ + n N + γˆ 2N ξ + n N + 2 2 j=n.2N +2 ...  # $  n 2N − 1 2N − 1 + Q (k) ξ + n N + γˆ 2N ξ + n N + 2 2 j=n.2N +(2N −1)   # n $ 2N − 1 ∗ (k) 2N − 1 ∗ Q ξ + nN + γˆ 2N ξ + n N + 2 2 

+

⎡ = Q

(k)

(ξ ) ⎣



γˆ

#

n 2N

$

(ξ + n N )γˆ

#

n 2N

$

 1 ∗ Q (k) ξ + n N + 2  ∗ 2 2 ∗ Q (k) ξ + n N + 2 2

1 2

∗

⎤ ∗⎦

(ξ + n N )

Q (k) (ξ )∗

j=n.2N

⎤ ⎡   # $ ∗ # n $  n 1 1 1 1 ∗ (k) (k) ⎦ ⎣ 2N 2N ξ+ ξ + nN + ξ + nN + ξ+ Q γˆ γˆ +Q 2 2 2 2 j=n.2N +1 ⎤ ⎡   # $ # n $ n 2 ⎣  2 2 ∗ ⎦ (k) 2 ∗ ξ+ Q γˆ 2N ξ + n N + γˆ 2N ξ + n N + +Q (k) ξ + 2 2 2 2 j=n.2N +2

+··· +Q

(k)

⎡  2N − 1 ⎣ ξ+ 2

 j=n.2N +(2N −1)

γˆ

#

n 2N

$

ξ + nN +

2N − 1 2



⎤  # n $ 2N − 1 ∗ ⎦ (k) 2N − 1 ∗ ξ+ Q γˆ 2N ξ + n N + 2 2

9 Orthogonal Vector-Valued Wavelets on R+

350

     1 1 ∗ 2 2 ∗ Q (k) ξ + Q (k) ξ + = 2 Q (k) (ξ )Q (k) (ξ )∗ + Q (k) ξ + + Q (k) ξ + + ··· 2 2 2 2 ∗   (2N − 1) (2N − 1) Q (k) ξ + · · · + Q (k) ξ + 2 2 =2

2N −1 

Q (k) (ξ + j/2)Q (k) (ξ + j/2)∗ .

j=0

Also 2N −1 

wn (ξ + p/2) =

p=0



γˆ n (ξ + j/2)γˆ n (ξ + j/2)∗

j∈Z

=2

2N −1 

Q (k)



j=0

ξ j + 2N 4N



Q (k)



ξ j + 2N 4N

∗

.

If λ = 2m 1 , σ = 2m 2 , where m 1 , m 2 ∈ Z, by using (9.78) we have & γ n (. − λ), γ n (. − σ )  e−2πi(2m 1 −2m 2 )ξ γˆ n (ξ )γˆ n (ξ )∗ dξ = R  = e−4πi(m 1 −m 2 )ξ γˆ n (ξ )γˆ n (ξ )∗ dξ R   n = γˆ (ξ + N j)γˆ n (ξ + N j)∗ dξ e−4πi(m 1 −m 2 )ξ %

[0,N )

 =

[0,N )

 =

e−4πi(m 1 −m 2 )ξ wn (ξ ) dξ ⎤ ⎡ 2N −1  e−4πi(m 1 −m 2 )ξ ⎣ wn (ξ + p/2)⎦ dξ

[0,1/2)

 =2

j∈Z

p=0

e−4πi(m 1 −m 2 )ξ

[0,1/2)

2N −1 

Q (k)

j=0



ξ j + 2N 4N



Q (k)



ξ j + 2N 4N

∗ dξ

= δm 1 ,m 2 Is = δλ,σ Is . When λ = 2m 1 , σ = 2m 2 + r/N where m 1 , m 2 ∈ Z, we obtain by using (9.79) < γ n (. − λ), γ n (. − σ ) >

9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties

 =

[0,N )

 =

351

e−4πi(m 1 −m 2 )ξ e−2πiξr/N wn (ξ ) dξ ⎞ ⎛ 2N −1  e−4πi(m 1 −m 2 )ξ e−2πiξr/N ⎝ e−πi pr/N wn (ξ + p/2)⎠ dξ

[0,1/2)

p=0

= 0.  Theorem 9.8 For any n 1 , n 2 ∈ Z+ and λ, σ ∈ Λ, we have < γ n 1 (. − λ), γ n 2 (. − σ ) >= δn 1 ,n 2 δλ,σ Is , where {γ n (t) : n ∈ Z+ } is VNUWP with respect to orthogonal vector-valued scaling function φ(t). Proof If n 1 = n 2 then the result follows by Theorem 9.5. If n 1 = n 2 , without loss of generality we can assume that n 1 > n 2 . Write 'n ( 'n ( 1 2 + k, n 2 = 2N + l, n 1 = 2N 2N 2N where k, l ∈ # n{0, $ 1, 2, # n.2.$. , 2N − 1}. 1 Case (i) If 2N = 2N , then k = l. %



&

γ (. − λ), γ (. − σ ) = n1

n2



=  = where v(ξ ) = v(2N ξ ) =

 j∈Z



R

e−2πi(λ−σ )ξ γˆ n 1 (ξ )γˆ n 2 (ξ )∗ dξ e−2πi(λ−σ )ξ

[0,N )

[0,N )



γˆ n 1 (ξ + N j)γˆ n 2 (ξ + N j)∗ dξ

j∈Z

e−2πi(λ−σ )ξ v(ξ ) dξ,

γˆ n 1 (ξ + N j)γˆ n 2 (ξ + N j)∗ . Therefore,

γˆ n 1 (2N (ξ + j/2))γˆ n 2 (2N (ξ + j/2))∗

j∈Z

=



n1

n2

Q (k) (ξ + j/2)γˆ [ 2N ] (ξ + j/2)γˆ [ 2N ] (ξ + j/2)∗ Q (l) (ξ + j/2)∗ .

j∈Z

On solving as same as in Theorem 9.5, we obtain v(2N ξ ) = 2

2N −1  j=0

Q (k) (ξ + j/2)Q (l) (ξ + j/2)∗ .

9 Orthogonal Vector-Valued Wavelets on R+

352

Also 2N −1 

v(ξ + p/2) =

p=0



γˆ n 1 (ξ + j/2)γˆ n 2 (ξ + j/2)∗

j∈Z

=2

2N −1  j=0

Q (k)





ξ j + 2N 4N

Q (l)



ξ j + 2N 4N

∗

.

If λ = 2m 1 and σ = 2m 2 where m 1 , m 2 ∈ Z, we obtain % n & γ (. − λ), γ n (. − σ ) =

 [0,N )

e−4πi(m 1 −m 2 )ξ v(ξ ) dξ

 =

[0,1/2)

 =2

⎤

⎡

2N −1  e−4πi(m 1 −m 2 )ξ ⎣ v(ξ + p/2)⎦ dξ

[0,1/2)

p=0

e−4πi(m 1 −m 2 )ξ

2N −1 

Q (k)



j=0

ξ j + 2N 4N



Q (l)



ξ j ∗ + dξ 2N 4N

= δm 1 ,m 2 δk,l Is = δλ,σ δk,l Is .

If λ = 2m 1 + r/N and σ = 2m 2 where m 1 , m 2 ∈ Z, we obtain by using (9.79) & % n γ (. − λ), γ n (. − σ ) =

 [0,N )

e−4πi(m 1 −m 2 )ξ e−2πiξr/N v(ξ ) dξ ⎡

 =

[0,1/2)

e

−4πi(m 1 −m 2 )ξ −2πiξr/N

e

⎣

2N −1 

⎤ e

−πi pr/N

v(ξ + p/2)⎦ dξ

p=0

= 0.

 'n ( 'n ( 'n ( 'n ( [n 1 /2N ] 1 2 1 2 = then take = 2N + k1 and = Case (ii) If 2N 2N 2N 2N 2N  [n 2 /2N ] 2N + l1 where k1 , l1 ∈ {0, 1, 2, . . . , 2N − 1}. 2N  'n ( 'n ( [n 1 /2N ] 1 2 Let = 2N p1 + k1 and = 2N q1 + l1 where p1 = and 2N 2N  2N [n 2 /2N ] q1 = . 2N If p1 = q1 , then the result follows from Case (i). ( ' pimmediately 'q ( 1 1 + k2 = 2N p2 + k2 and q1 = 2N + If p1 = q1 , then take p1 = 2N 2N 2N l2 = 2N q2 + l2 where k2 , l2 ∈ {0, 1, 2, . . . , 2N − 1}. If p2 = q2 then the result follows from Case (i).

9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties

353

If p2 = q2 then apply the above procedure. After performing a finite number of steps, we have pm−1 = 2N pm + km and qm−1 = 2N qm + lm where km , lm ∈ {0, 1, 2, . . . , 2N − 1} and pm , qm ∈ {0, 1, 2, . . . , 2N − 1}. Case I If pm = qm Case II If pm = qm . For Case I, the result follows from Case (i). For Case II, we have & γ n 1 (. − λ), γ n 2 (. − σ )  e−2πi(λ−σ )ξ γˆ n 1 (ξ )γˆ n 2 (ξ )∗ dξ = R      n1 n2 ξ ξ ξ ∗ (l1 ) ξ ∗ γˆ [ 2N ] γˆ [ 2N ] e−2πi(λ−σ )ξ Q (k1 ) Q dξ = 2N 2N 2N 2N R      ξ ξ ξ ξ pm (km ) ˆ . . . Q γ Q (k2 ) e−2πi(λ−σ )ξ Q (k1 ) = 2N (2N )2 (2N )m (2N )m R  ∗  ∗  ∗  ∗ ξ ξ ξ ξ (lm ) (l2 ) (l1 ) γˆ qm Q . . . Q Q dξ (2N )m (2N )m (2N )2 (2N ) )m  *    ξ ξ pm −2πi(λ−σ )ξ (kn ) ˆ e Q = γ (2N )n (2N )m R n=1  ∗ )  *∗ m ξ ξ qm (ln ) γˆ Q dξ (2N )m (2N )n n=1 %

)

 =

e

−2πi(λ−σ )ξ

[0,N )

m 

Q

(kn )

n=1

 γˆ

qm



ξ (2N )n

⎡ *   pm ⎣ γˆ

ξ + Nj (2N )m

j∈Z

∗ )  m n=1

ξ + Nj (2N )m

Q

(ln )



ξ (2N )n



*∗ dξ

= 0, which completes the proof.



Corollary 9.1 If {γ n (t), n ∈ Z+ } is a vector-valued nonuniform wavelet packet with respect to orthogonal vector-valued nonuniform scaling function φ(t), then ∀ n ∈ Z+ , and k, l ∈ {0, 1, . . . , 2N − 1}, we have < γ 2N n+k (. − λ), γ 2N n+l (. − σ ) >= δλ,σ δk,l Is , λ, σ ∈ Λ. Proof We have

(9.82)

9 Orthogonal Vector-Valued Wavelets on R+

354

< γ 2N n+k (. − λ), γ 2N n+l (. − σ ) >  e−2πi(λ−σ )ξ γˆ 2N n+k (ξ )γˆ 2N n+l (ξ )∗ dξ = R   2N n+k = γˆ e−2πi(λ−σ )ξ (ξ + N j)γˆ 2N n+l (ξ + N j)∗ dξ. [0,N )

j∈Z

Let vn (ξ ) =



γˆ 2N n+k (ξ + N j)γˆ 2N n+l (ξ + N j)∗ .

j∈Z

Therefore, on solving the equation as same as in Theorem 9.5, we have vn (2N ξ ) =



γˆ 2N n+k (2N (ξ + j/2)) γˆ 2N n+l (2N (ξ + j/2))∗

j∈Z

=



Q (k) (ξ + j/2) γˆ n (ξ + j/2) γˆ n (ξ + j/2)∗ Q (l) (ξ + j/2)∗

j∈Z

=2

2N −1 

Q (k) (ξ + j/2) Q (l) (ξ + j/2)∗ .

j=0

Also

2N −1 

vn (ξ + p/2) =

p=0



γˆ 2N n+k (ξ + j/2)γˆ 2N n+l (ξ + j/2)∗ .

j∈Z

Therefore, we have 

γˆ 2N n+k (ξ + j/2)γˆ 2N n+l (ξ + j/2)∗ = 2

2N −1  j=0

j∈Z

Q (k)



ξ j + 2N 4N



Q (l)



ξ j + 2N 4N

∗

When λ = 2m 1 and σ = 2m 2 , where m 1 , m 2 ∈ Z, by using (9.78) we obtain < γ 2N n+k (. − λ), γ 2N n+l (. − σ ) >  e−2πi(λ−σ )ξ vn (ξ ) dξ = [0,N )  e−4πi(m 1 −m 2 )ξ vn (ξ ) dξ = [0,N ) ⎤ ⎡  2N −1  e−4πi(m 1 −m 2 )ξ ⎣ vn (ξ + p/2)⎦ dξ = [0,N )

 =2

[0,1/2)

p=0

e−4πi(m 1 −m 2 )ξ

2N −1  j=0

Q (k)



ξ j + 2N 4N



Q (l)



ξ j + 2N 4N

∗ dξ

.

9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties

 =2

[0,1/2)

355

e−4πi(m 1 −m 2 )ξ δk,l Is dξ

= δm 1 ,m 2 δk,l Is = δλ,σ δk,l Is . When λ = 2m 1 , σ = 2m 2 + r/N where m 1 , m 2 ∈ Z, we obtain by using (9.79) < γ 2N n+k (. − λ), γ 2N n+l (. − σ ) >  = e−4πi(m 1 −m 2 )ξ e−2πiξr/N vn (ξ ) dξ [0,N ) ⎞ ⎛  2N −1  e−4πi(m 1 −m 2 )ξ e−2πiξr/N ⎝ e−πi pr/N vn (ξ + p/2)⎠ dξ = [0,1/2)

p=0

= 0, 

which completes the proof.

9.6.1 Vector-Valued Nonuniform Wavelet Bases In this subsection, we shall construct orthogonal vector-valued nonuniform wavelet bases of L 2 (R, Cs ) by using orthogonal vector-valued nonuniform wavelet packets. Define a dilation operator by (DF)(t) = F(2N t) for F ∈ L 2 (R, Cs ). ∀Ω ⊂ L 2 (R, Cs ) and ∀ n ∈ Z+ , denote DΩ = {D F : F ∈ Ω} and Ωn = {F(t) : F(t) =



Mλ γ n (t − λ), {Mλ }λ∈Λ ∈ l 2 (Λ)s×s }.

(9.83)

λ∈Λ

Then Ω0 = V0 and Ω1 ⊕ Ω2 ⊕ Ω3 ⊕ · · · ⊕ Ω2N −1 = W0 , where ⊕ denotes the orthogonal direct sum. Lemma 9.5 If {γ n (t), n = 0, 1, 2, . . .} is vector-valued wavelet packet with respect to the orthogonal vector-valued scaling function γ 0 (t), then for every n ∈ Z+ , 2N γ n (2N t − k) =

2N −1   l=0 λ∈Λ

∗ 2N n+l (Q (l) (t − λ), k ∈ Λ. k−2N λ ) γ

Proof Consider 2N −1 1   (l) (Q )∗ γ 2N n+l (t − λ) 2N l=0 λ∈Λ k−2N λ

=

2N −1  (l) 1   (l) (Q k−2N λ )∗ Q λ1 γ n (2N t − 2N λ − λ1 ) 2N l=0 λ∈Λ λ ∈Λ 1

9 Orthogonal Vector-Valued Wavelets on R+

356

1  = 2N λ ∈Λ 2

* )2N −1   (l) (l) ∗ (Q k−2N λ ) (Q λ2 −2N λ ) γ n (2N t − λ2 ) l=0 λ∈Λ

1  = δk,λ2 Is γ n (2N t − λ2 ) 2N λ ∈Λ 2

= γ n (2N t − k), 

which completes the proof.

Lemma 9.6 For n ∈ Z+ , the space DΩn can be orthogonally decomposed into the spaces Ω2N n+k , k = 0, 1, 2, . . . , 2N − 1, i.e., DΩn = Ω2n ⊕ Ω2n+1 ⊕ · · · ⊕ Ω2n+(2N −1) . Proof By Eqs. (9.73) and (9.83), we have Ω2n ⊕ Ω2n+1 ⊕ · · · ⊕ Ω2n+(2N −1) ⊂ Ωn ∀n ∈ Z+ . But Ω2n , Ω2n+1 , . . . , Ω2n+(2N −1) are orthogonal to each other by Theorem 9.6. By Lemma 9.5, we have 2N γ (2N t − k) = n

2N −1   l=0 λ∈Λ

∗ 2N n+l (Q (l) (t − λ), k ∈ Λ. k−2N λ ) γ

Therefore, the basis of the space Ωn can be linearly represented by the basis of the space Ω2n , Ω2n+1 , . . . , Ω2n+(2N −1) . Thus Ωn ⊂ Ω2n ⊕ Ω2n+1 ⊕ · · · ⊕ Ω2n+(2N −1) . Therefore DΩn = Ω2n ⊕ Ω2n+1 ⊕ · · · ⊕ Ω2n+(2N −1) , i.e.,

−1 DΩn = ⊕2N k=0 Ω2n+k .

 Let X, Y ⊂ R and a X = {ax : x ∈ X } for a ∈ R. X + Y = {x + y : x ∈ X and y ∈ Y }, X − Y = {x − y : x ∈ X and y ∈ Y }. For a fixed positive integer r denote U˜ r =

r  i=0

(2N )i {0, 1, 2, . . . , 2N − 1} and Ur = U˜ r \U˜ r −1 .

9.6 Vector-Valued Nonuniform Wavelet Packets and Its Properties

357

Theorem 9.9 The set of vector-valued functions {γ n (. − λ), n ∈ Ur , λ ∈ Λ} forms an orthonormal basis of Dr W0 . In particular, the collection {γ n (. − λ), n ∈ Z+ , λ ∈ Λ} constitute an orthonormal basis of L 2 (R, Cs ). Proof By Lemma 9.6, we have −1 DΩ0 = Ω0 ⊕ Ω1 ⊕ · · · ⊕ Ω2N −1 = ⊕2N k=0 Ωk ,

i.e., DV0 = V0 ⊕ W0 . By using induction and by using Theorem 9.6 and Lemma 9.6, it can be proved that Dr V0 = ⊕k∈U˜ r Ωk and Dr V0 ⊕ Dr W0 = Dr +1 V0 , i.e., Dr W0 = ⊕k∈Ur Ωk . Therefore, the set {γ n (. − λ), n ∈ Ur , λ ∈ Λ} forms an orthonormal basis of Dr W0 . Moreover,  L 2 (R, Cs ) = V0 ⊕ ⊕0≤r Dr W0   = Ω0 ⊕ ⊕0≤r ⊕k∈Ur Ωk = ⊕k∈Z+ Ωk . Therefore, the set {γ n (. − λ), n ∈ Z+ , λ ∈ Λ} forms an orthonormal basis of  L 2 (R, Cs ). Theorem 9.10 For each r ∈ Z+ \{0}, the family of vector-valued functions {γ n ((2N ) j t − λ), n ∈ Ur , j ∈ Z, λ ∈ Λ} forms an orthonormal basis of L 2 (R, Cs ). Proof By Theorem 9.7, {γ n (. − λ), n ∈ Ur , λ ∈ Λ} forms an orthonormal basis of Dr W0 . Then for each j ∈ Z, {γ n ((2N ) j t. − λ), n ∈ Ur , λ ∈ Λ} forms an orthonormal basis of Dr W0 . Hence, for each r ∈ Z+ \{0}, ⊕ j∈Z D j Dr W0 = ⊕ j∈Z D j+r W0 = ⊕ j∈Z D j W0 . Thus the family {γ n ((2N ) j t − λ), n ∈ Ur , j ∈ Z, λ ∈ Λ} forms an orthonormal  basis of L 2 (R, Cs ).

9.7 Exercise • Prove Theorems 9.1 and 9.2. • Prove the necessary and sufficient condition for the existence of associated wavelets with VMRA on R+ .

358

9 Orthogonal Vector-Valued Wavelets on R+

• Prove that the system of wavelets Ψ0 , Ψ1 , . . . , Ψ2N −1 in V1 as defined in Theorem 9.5 forms an orthonormal basis for L 2 (R, Cs ). • Construct vector-valued nonuniform wavelets associated with φ(t) ∈ L 2 (R, C2 ) with supp φ(t) = [0, 2].

References 1. Farkov, Yu. A. (2005). Orthogonal p-wavelets on R + . In Proceedings of the International Conference on Wavelets and Splines (p. 426). Saint Petersburg: St. Petersburg University Press. (Saint Petersburg, Russia, July 3–8). 2. Lang, W. C. (1996). Orthogonal wavelets on the cantor dyadic group. SIAM Journal on Mathematical Analysis, 27(1), 305–312. 3. Lang, W. C. (1998). Fractal multiwavelets related to the cantor dyadic group. International Journal of Mathematics and Mathematical Sciences, 21, 307–317. 4. Lang, W. C. (1998). Wavelet analysis on the cantor dyadic group. Houston Journal of Mathematics, 24, 533–544. 5. Farkov, Yu. A. (2005). Orthogonal Wavelets with compact support on locally compact abelian groups. Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 193–220. (English Translate, Izvestia: Mathematics, 69(3), 623–650.) 6. Protasov, V. Yu., & Farkov, Yu. A. (2006). Dyadic Wavelet and refinable Functions on a half line. Sboinik: Mathematics, 197(10), 1529–1558. 7. Farkov, Yu. A. Maksimov A. Yu. & Stroganov S. A. (2011). On biorthogonal wavelets related to the walsh functions. International Journal of Wavelets Multiresolution and Information Processing, 9(3), 485–499. 8. Manchanda, P., Meenakshi, M., & Siddiqi, A. H. (2014). Construction of vector-valued nonuniform wavelets and wavelet packets. Special Issue of Indian Journal of Industrial and Applied Mathematics. 9. Meenakshi, M., Manchanda, P., & Siddiqi, A. H. (2014). Wavelets associated with vector valued non-uniform multiresolution analysis. Applicable Analysis: An International Journal, 93(1), 84–104. 10. Meenakshi, M., & Manchanda, P. (2017). Vector-valued nonuniform wavelet packets. Numerical Functional Analysis and Optimization, Taylor and Francis. Published Online, https://doi. org/10.1080/01630563.2017.1355814 11. Manchanda, P., & Sharma V. (2012). Orthogonal vector valued wavelets on R+. International Journal of Pure and Applied Mathematics, 75(4), 493–510. 12. Xia, X. G., & Suter, B. W. (1996). Vector-valued wavelets and vector filter banks. IEEE Transactions on Signal Processing, 44(3), 508–518. 13. Chen, Q., & Cheng, Z. (2007). A study of compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons and Fractals, 31(4), 1024–1034.

Appendix

A.1 Basic Results of Functional Analysis Definition A.1 1. Let X be a set, and d a real function on X × X , with the properties (a) d (x, y) ≥ 0 ∀ x, y ∈ X , d (x, y) = 0 if and only if x = y. (b) d (x, y) = d (y, x) ∀ x, y ∈ X (symmetry property). (c) d (x, y) ≤ d (x, z) + d (z, y) (the triangle inequality). Then d is called a metric, or a metric function on or over or in X . (X , d ) is called a metric space. 2. Sr (x) = {y/d (x, y) < r} is called an open sphere with radius r and center x in X. 3. Let A and B be subsets of a metric space (X , d ). Then (a) (b) (c) (d)

d (A, B) = supa∈A inf b∈B d (a, b). The diameter of A, which is denoted by δ(A), is the number supa,b∈A d (a, b). S(A, ε) = {x/d (A, x) < ε is called the ε− neighborhood of A. A is bounded if δ(A) < ∞.

4. A sequence in a metric space is a function whose domain is the set of positive integers and range is a subset of the metric space. 5. A sequence {An } of subsets of a metric space is called a decreasing sequence if A1 ⊇ A2 ⊇ A3 ⊇ . . .. Theorem A.1 1. The collection of open spheres in a metric space (X , d ) forms a base for the metric topology. 2. Every metric space is normal and hence a Hausdorff topological space. Definition A.2 1. A sequence {xn } in a metric space (X , d ) is said to be convergent to an element x ∈ X if limn→∞ d (xn , x) = 0. {xn } is called a Cauchy sequence if limm,n→∞ d (xm , xn ) = 0. © Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2

359

360

Appendix

2. A metric space is called complete if every Cauchy sequence in it is convergent to an element of this space. 3. Let (X , d1 ) and (Y , d2 ) be metric spaces, and f is a mapping of X into Y . f is called continuous at a point x0 in X if either of the following equivalent conditions is satisfied: (a) For every ε > 0, there exists a δ > 0 such that d1 (x, x0 ) < δ implies d2 (f (x), f (x0 )) < ε. (b) For each open sphere Sε (f (x0 )) centered at f (x0 ), there exists an open sphere Sδ (x0 ) centered at x0 such that f (Sδ (x0 )) ⊆ Sε (f (x0 )). f is called continuous if it is continuous at each point of its domain. 4. In (3), it can be seen that the choice of δ depends not only on ε but also on the point x0 . The concept of continuity, in which for each ε > 0, a δ > 0 can be found which works uniformly over the entire metric space X , is called uniform continuity. Theorem A.2 1. If a convergent sequence in a metric space has infinitely many distinct points, then its limit is a limit point of the set of points of the sequence. 2. Let X and Y be metric spaces and f is a mapping of X into Y . Then f is continuous if and only if {xn } in X converges to x ∈ X implies that f (xn ) in Y converges to f (x) ∈ Y . 3. A point x is in the closure of a set A in a metric space if and only if there is a sequence {xn } of points of A converging to x. Definition A.3 A norm on a vector space X denoted by ||.||is a function defined on X satisfying the following conditions: (i) x > 0; x = 0 if and only if x = 0; (ii) αx = |α| x , α is an arbitrary scalar. (iii) x + y ≤ x + y for all x, y ∈ X . Every normed space (X , . ) is a metric space with d (x, y) = x − y . A complete normed space is called a Banach space. The vector space Rn of all n-tuples x = (x1 , x2 , . . . , xn ) of real numbers is a normed space, infact Banach space with respect to norms  (i) x 1 = nk=1 |xk |, (ii) x 2 = (nk=1 |xk |2 )1/2 , (iii) x 3 = ( nk=1 |xk |p )1/p , where 1 ≤ p < ∞, and (iv) x 4 = max{|x1 |, |x2 |, . . . , |xk |}.. Spaces set of all sequences x = {x1 , x2 , . . . , xn , . . .} of real numbers such that ∞ The p |x | < ∞, for 1 ≤ p < ∞ denoted by lp , in the case p = 2, it is called the n n=1 infinite-dimensional Euclidean space. For a, b ∈ R and a < b, [a, b] = {x ∈ R/a ≤ x ≤ b} (a, b) = {x ∈ R/a < x < b} and (a, b] = {x ∈ R/a < x ≤ b} are called, respectively the closed, open, semiclosed (semi-open) intervals of R. R is a metric space with the metric d (x, y) =

Appendix

361

|x − y| and therefore a closed and bounded subset of R can be defined. As a special case, a real- valued function f (x) on [a, b] is called bounded if there exists a real number k such that |f (x)| ≤ k ∀ x ∈ [a, b]. It is said to be continuous at x0 ∈ [a, b] if, for ε > 0, there exists δ > 0 such that |f (x) − f (x0 ))| < ε whenever |x − x0 | < δ. It is called uniformly continuous on [a, b] if the choice of δ does not depend on the point x0 in the definition of continuity. We can define a continuous function in a similar fashion on an arbitrary subset of R. C[a, b] denotes the set of all continuous real functions defined on [a, b]. It is called the space of continuous functions on [a, b]. Similarly, C(T ), where T ⊆ R, called the space of continuous functions on T . If X is an arbitrary topological space, then C(X ) is called the space of continuous real functions on the topological space X . p(x) = a0 + a1 x + . . . + an xn for [a, b], where a0 , a1 , . . . , an are real numbers, is called a polynomial of degree n. It is a continuous real function. P[a, b] denotes the set of all polynomials over [a, b].Pn [a, b] denotes the set of all polynomials of degree less than or equal to n over [a, b]. Cn [0, π ] denotes the set of all functions on [0, π ] of the form f (x) = b1 cos x + b2 cos 2x + . . . + bn cos nx. Let f (x) be a real-valued function defined on [a, b]. The limits 

f− (x) =

lim

f (y) − f (x) y−x

lim

f (y) − f (x) y−x

y→x,yx

are called left and right derivatives, respectively. The function f is said to be differentiable if the right and left derivatives at x exist and their values are equal. This value is . If the first derivative called the first derivative of f at x and is denoted byf  or f 1 or df dx exists at every point of [a, b], then it is said to be differentiable over [a, b]. If the first derivative of f is also differentiable, then f is called twice differentiable. In general, the nth derivative of the function f is the derivative of the (n − 1)th derivative. f is called n times differentiable over [a, b] if nth derivative exists at every point of [a, b]. C (n) [a, b] denotes the set of all functions which have continuous derivatives upto and including the nth order over [a, b]. If derivatives of all order on [a, b] exist, then f is called infinitely differentiable. C ∞ [a, b] denotes the class of all infinitely differentiable functions over [a, b]. Let f (x) be a real-valued function defined on [a, b]. Let P : x0 ≤ x1 ≤ x2 ≤ . . . , ≤  xn = b be a partition of [a, b], ba (f ) = supP ni=0 |f (xi ) − f (xi−1 )| is a variation of  f (x) over [a, b]. f (x) is called a function of bounded variation on [a, b] if ba (x) < ∞. BV [a, b] denotes the space of all functions of bounded variation over [a, b]. A real-valued function f (x) on [a, b] is called absolutely continuous on [a, b] if, for ε > 0, there exists a δ > 0 such that for any collection {(ai , bi )}n1 of disjoint open

362

Appendix

 ∞ subintervals of [a, b], ∞ i=1 |f (bi ) − f (ai )| < ε holds whenever i=1 (bi − ai ) < δ.AC[a, b] denotes the class of all absolutely continuous function on [a, b]. A real function f (x) defined on [a, b] is said to satisfy a Hölder condition of exponent α over [a, b] or to be Hölder continuous or Lipschitz class if |f (x) − f (y)| 0 such that |g(x)|

f (x) f (x) = o(g(x)) if g(x) → 0. These relationships are valid when x → ∞, for x → −∞ or x → x0 , where x0 is some fixed number. If bn > 0, n = 0, 1, 2, . . . and abnn → 0 as n → ∞, then we write an = o(bn ). If abnn is bounded, then we write an = O(bn ).

Remark A.1 1. a. Every function of the Lipschitz class over [a, b] belongs to AC[a, b]. b. AC[a, b] ⊂ C[a, b] c. AC[a, b] ⊂ BV [a, b] 2. All continuous differentiable functions over [a, b] are absolutely continuously over [a, b]. Examples (i) lp , 1 ≤ p < ∞ is Banach space with respect to norm

x p =

∞ 

1/p |xk |

p

k=1

is a Banach space (ii) If A is any measurable subset of Rn , in particular Rn itself, then Lp (A) with 1 ≤ p < ∞ of all (classes of equal a.e.) measurable functions such that 

f p =

1/p |f (x)|p dx

defined on X × X into the underlying field, is called an inner product of any two elements x and y of X if the following conditions are satisfied: 1. 2. 3. 4.

< x + x , y >=< x, y > + < x , y > < αx, y >= α < x, y >, α belongs to the underlying field < x, y > =< y, x > < x, x >≥ 0, ∀x ∈ X ; and < x, x >= 0 if and only if x = 0.

If the inner product < ., . > is defined for every pair of elements (x, y), x and y ∈ X , then the vector space X together with the inner product < ., > is called an inner product space or pre-Hilbert space, usually denoted by (X , < ., . >) or simply X . Pre-Hilbert space is a normed space. It is called a Hilbert space if it is complete. Rn , l2 , L2 (a, b) are Hilbert spaces. Definition A.7 Two normed spaces (X , . 1 ) and (Y , . 2 ) are said to be isometric and isomorphic to one another if there exists a one-to-one mapping T of X onto Y such that d (Tu1 , Tu2 ) = Tu1 − Tu2 2 = u1 − u2 1 = d (u1 , u2 ) or Tx 2 = x 1 and T is linear, that is, T (x + y) = Tx + Ty, ∀x, y ∈ X T (αx) = αTx, ∀x ∈ X , α ∈ R or C

366

Appendix

This definition means that there exists a mapping of X into Y which is one-one, onto, norm-preserving and linear. If such mapping exists, we write X = Y . Definition A.8(a) Two normed spaces (X , . 1 ) and (X , . 2 ) are called topologically equivalent or two norms ( . 1 and . 2 ) are called equivalent if there exist constants k1 and k2 such that k1 x 1 ≤ x 2 ≤ k2 x 1 . (b) A normed space is called finite-dimensional if the underlying vector space is finite-dimensional; otherwise it is infinite-dimensional. Bounded and Unbounded Operators Definitions and Examples Definition A.9 Let U and V be two normed spaces. Then 1. A mapping T from U into V is called an operator or a transformation. The value of T at x ∈ U is denoted by T (x) or Tx. 2. T is called a linear operator or a linear transformation if the following conditions are satisfied: (a) T (x + y) = Tx + Ty, ∀ x, y ∈ U . (b) T (αx) = αT (x), ∀ x ∈ U and real α. 3. The operator T is called bounded if there exists a real k > 0 such that Tx ≤ k x ∀ x ∈ U . 4. T is called continuous at a point x0 ∈ U if, given ε > 0, there exists a δ > 0, depending upon ε and x0 such that Tx − Tx0 < ε, whenever x − x0 < ε. T is called continuous on X if it is continuous at every point of X . 5. T is called uniformly continuous if, for ε > 0, there exists a δ > 0, independent of x0 , such that for any x0 and x ∈ X with x − x0 < δ, we have Tx − Tx0 < ε. /x = 0} is called the norm of the bounded operator T (For an 6. T = sup { Tx

x

unbounded operator, the sup may not exist). 7. If Y = R, the normed space of real numbers, then T is called a functional and it is usually denoted by F. 8. For the operator T , the set R = {Tx ∈ U/x ∈ V } and the set N = {x ∈ U/T (x) = 0} are called the range and null spaces, respectively. 9. X ∗ = F : X → R, FlinearboundedonnlsX is called dual of X . Banach–Alagolu Theorem Theorem A.3 Suppose X is a normed space and X ∗ is its dual. Then the closed unit sphere S1∗ = {f ∈ X ∗ / f ≤ 1} is compact with respect to weak topology.

Appendix

367

Principle of Uniform Boundedness Theorem A.4 Let X be a Banach space, Y a normed space and Ti a sequence of bounded linear operators over X into Y such that {Ti (x)} is a bounded subset of Y for all x ∈ X . Then { Ti } is a bounded subset of real numbers, i.e., {Ti } is a bounded sequence in the normed space B[X , Y ].

A.2 Vilenkin Systems Topological group is a group G together with a topology on G such that the group’s binary operation and the group’s inverse function are continuous with respect to the topology. A topological group is a mathematical object with an algebraic structure and a topological structure. In mathematical terms a topological group G, is a topological space, which is a also a group such that the group operation of product: G × G −→ G : (x, y) −→ xy and taking inverses

G −→ G : x −→ x−1

are continuous. Here G × G is viewed as a topological space with the product topology. A topological group G is called compact if G is a compact topological space. G is called abelian topological group if G is abelian group. The Vilenkin systems were introduced in 1947 by Vilenkin. They include as a special case the Walsh system. We below give a brief introduction to the Vilenkin systems and relate them to other types of orthonormal systems which have come up in the course of our narrative. Let m = (mk , k ∈ N) be a sequence of natural numbers such that mk ≥ 2 (k ∈ N). We shall construct a compact abelian group G m for each such sequence m. Let Zmk (k ∈ N) be the mk th discrete cyclic group, i.e., Zmk can be represented by the set {0, 1, 2, . . . , mk − 1}, where the group operation is mod mk addition, and every subset is open. Haar measure on Zmk can be generated by insisting that the measure of singleton is 1/mk . The group G m is defined as the complete discrete product of the compact groups Zmk (k ∈ N). Thus on G m we use coordinate-wise addition as the group operation, the product topology and the product measure. Thus G m is a compact abelian group. The group G m is metrizable. Let M0 = 1 and Mk = mk−1 Mk−1 (k ∈ P). Define the distance between the elements (xk , k ∈ N) ∈ G m and (yk , k ∈ N) ∈ G m by

368

Appendix

ρ(x, y) =

∞  |xk − yk | . Mk+1 k=0

It is obvious that the topology induced by this metric coincides with that of G m . It is easy to give a base for the neighborhoods of G m I0 (x) = G m In (x) = {y = (yi , i ∈ N ∈ G m : yi = xi for i = 0, 1, . . . , n − 1)} for x ∈ G m , n ∈ P. Moreover, if 0 denotes the null element of G m , then the topology induced by the sets In (0) (n ∈ N) (the base of neighborhoods of 0) coincides with the topology of G m . Furthermore, it can be shown that the product measure introduced in G m is the normalized Haar measure of G m . Clearly, if mk = 2 (k ∈ N) then G m coincides with the dyadic group. Thus no confusion will arise if we use some of the same notation introduced in the study of the dyadic groups for these Vilenkin groups G m . Accordingly, for each n ∈ N set ρn (x) = exp

2π ixn mn

for x = (xk , k ∈ N) ∈ G m , n ∈ N. Clearly, each ρn is acharacter of G m . Infact, every complex-valued function defined on G m whose values depend only on one coordinate the element x is continuous on G m in the topology of G m . To enumerate finite products of the functions ρn (n ∈ N), write each n ∈ N uniquely with the help of the sequence (Mn , n ∈ N) in the form n=

∞ 

nk Mk (0 ≤ nk < mk , nk ∈ N).

(A.2)

k=0

As in the dyadic case, it can be shown that the set Gˆm consists of the functions ψn =

∞ 

ρknk (n ∈ N),

k=0

where the sequence (nk , k ∈ N) has been defined in A1 . The system Gˆm = (ψn , n ∈ N) is called a Vilenkin system. A group operation ⊕ corresponding to that introduced earlier in the set N can be defined as follows: ∞  (nk + jk (mod mk ))Mk n⊕j = k=0

Appendix

369

where n=

∞ 

nk Mk and j =

k=0

∞ 

jk Mk .

k=0

It is clear, then, that ψn⊕j = ψn ψj (n, j ∈ N) and Gˆm isomorphic to N.

A.3 Vilenkin-Pontrjagin Class of Functions Let G be a compact Abelian group satisfying the second axiom of countability. It is well known that we can associate with G a countable set of functions which are continuous on G and satisfy the equation φ(x + y) = φ(x)φ(y), (Pontrjagin, [2]). These functions form a complete orthonormal system on G. Such a class of functions was introduced by Pontrjagin and its Fourier expansion was studied by Vilenkin [3]. This system of functions includes Walsh functions as a special case when the Group G is a topological direct sum of cyclic groups of order two. For the details of this class one can see [1–3] .

A.3.1

The Dyadic Group G

The dyadic group G is the set of all sequences x¯ = {xn }, xn = 0, 1 for n = 1, 2, 3, . . ., the operation in G, denoted by +∗ , is the addition modulo 2 in each coordinate. Corresponding to each element x¯ = {xn } of G, there is a real number, λ(¯x) =

x2 x3 x1 + 2 + 2 + ... 2 2 3

(A.3)

lying in the closed interval [0, 1]. At the dyadic rationals x, 0 < x < 1, we have two representations in the dyadic scale and therefore, the mapping λ is not one-one but it is onto. If μ(x) is the inverse of λ, for all real x, we have, λ(μ(x)) = x − [x], . . .

(A.4)

the finite expansion in G being associated under μ with the dyadic rationals. Thus μ(λ¯x) = x¯ , provided λ(¯x) is not a dyadic rational. Let χn (¯x) = χn ({x1 , x3 , . . . , xn , . . .)

1 xn = 0 = −1 xn = 1

370

Appendix

It is clear that χn is a character of G. Fine [4] has established that φn−1 (x) = χn (μ(x)), n ≥ 1. All characters of G may be obtained by taking finite products of these basic characters. Thus each Walsh function can be expressed in the form ψn (x) = χj1 (μ(x))χj2 (μ(x)) . . . χjt (μ(x)), and therefore Walsh functions are identified with the whole set of characters of G. ˙ where y and Notation: Henceforth λ(μ(y) +∗ μ(z)) would be abbreviated to y+z, z are reals. There is a natural topology of G which is obtained by taking the set of points {x1 , x2 , . . . , xn , dn+1 , . . .} as neighborhoods, where x1 , x2 , . . . , xn are fixed and dn+1 vary independently. These neighborhoods, denoted by N (x1 , x2 , . . . , xn ), form a basis for the topology of G. Proposition A.1 λ defines a metric on G. This metric is continuous on G, and the topology induced on G by this metric is equivalent to the original topology of G. Proposition A.2 |λ(¯y) − λ(¯z )| ≤ λ(¯y  +¯z ), y¯ , z¯ ∈ G. Proposition A.3 For any two real numbers x and h ˙ − (x − [x])| ≤ h − [h], (a) |(x+h) ˙ − x| ≤ h if 0 ≤ x < 1 and 0 ≤ h < 1. (b) |(x+h) Theorem A.5 (a) ψn (y + z) = ψn (y)ψn (z) where y + z is not a dyadic rational, its exception is a denumerable event for each fixed x. (b) Let A is ameasurable set, x be a fixed element and ˙ y ∈ A} Tx (A) = {(x+y), then the Lebesgue measure of Tx (A) is equal to the Lebesgue measure of A. (c) if f is integrable then for every fixed x, 

1 0

˙ dy = f (x+y)



1

f (y) dy.

0

The following definition forms a basis for discussing the relationship between two given classes of functions one on G and another one on [0, 1]:

Appendix

371

Definition A.10 For each real-valued function g(x) of period 1, there corresponds a function g(¯ ¯ x) on G defined by

g(¯ ¯ x) =

g(x) μ(x) = x¯ for some x ∈ [0, 1] ¯ y) μ(x) = x¯ for any x ∈ [0, 1] limy¯ →¯x supg(¯

(A.5)

where the lim sup is taken over those y¯ ’s which correspond to the dyadic rationals. Characteristic functions of neighborhoods of G are continuous on G because each neighborhood is both open and closed. Finite linear combinations of such characteristic functions are then continuous on G. Classes of functions: The discussion of the Fourier properties of the Walsh system may be proceeded from two points of view (a) Characters χn (x) of G and their properties, Haar measure of G, Various classes of functions on G, (b) Functions ψn (x) on [0, 1], Lebesgue measure on [0, 1]. Definition A.11 A function f (x) of period 1 is said to Continuous (W) at a point x, 0 ≤ x < 1 if for every ε > 0, there exists δ > 0 depending upon ε and x such that ˙ − f (x)| < ε |f (x+y) for all y for which 0 ≤ y < δ. Theorem A.6 Every continuous function is Continuous (W) but converse is not true. Theorem A.7 If f (x) is a continuous (W) on 0 ≤ x < 1 and if f (x − 0) exists and is finite at each dyadic rational x, 0 ≤ x < 1, then f¯ (¯x), the G-extension of f (x), is continuous on G. If f¯ (¯x) is continuous on G, then the extension f (x) = f¯ (μ(x)) is continuous (W) on 0 ≤ x < 1 and f (x − 0) exists and is finite at each dyadic rational x, 0 ≤ x < 1. Definition A.12 A function f¯ (¯x) is said to belong to the class of Lipschitz α on G, 0 ≤ α ≤ 1 if there exists a constant C > 0 such that |f¯ (¯x) − f¯ (¯y)| < Cλ(¯x  +¯y)α , for all x, y ∈ G. Definition A.13 A function f (x) defined on [0, 1) is said to belong to the class Lipschitz α(W). 0 < α ≤ 1 if there is a constant C > 0 such that ˙ − f (x)| < Cyα , |f (x+y)

372

Appendix

˙ = μ(x)  +μ(y), that is, except when μ(x)  for 0 ≤ x < 1, 0 ≤ y < 1 and μ(x+y) +μ(y) ends in a sequence of 1’s. The following theorem of Morgenthaler, see for example Siddiqi [5], gives the relationship between Lipschtz α functions on G and Lipschitz α(W) functions on [0, 1): Theorem A.8 If f¯ (¯x) is a Lipschitz function on G, 0 < α ≤ 1, then f (x) = f¯ (μ(x)) is a Lipschitz α(W) function on [0, 1), and f (x0 − 0) exists, is finite at each dyadic rational x, 0 ≤ x < 1. If f (x) is a Lipschitz α(W) function on [0, 1), 0 ≤ α < 1 and f (x0 − 0) exists, is finite at each dyadic rational x, 0 ≤ x < 1, then the G-extension of f , f¯ (¯x) is a Lipschitz function on G. Remark A.3 (i) If f (x) is a function of class Lipschitz α on [0, 1) then f (x) is also a Lipschitz α(W) function. (ii) If f (x) is a function of class Lipschitz α(W) on [0, 1), then f (x) is continuous (W) on [0, 1). (iii) The G-extension of the Walsh functions are Lipschitz 1 functions on G. (iv) The Walsh functions ψn (x) are Lipschitz α(W) functions on [0, 1). (v) If we define w(δ, ¯ f¯ ) = max |f¯ (¯x) − f¯ (¯y)| for all x¯ , y¯ ∈ G, with λ(x  +y) < δ, then w(δ, ¯ f¯ ) ≤ Cδ α if f¯ belongs to the class Lipschitz α on G. (vi) If ˙ − f (x)| WW (δ, f ) = max0≤x N . Note: The notation N = N (x, ε) means that the number N depends on the choice of x and ε. Uniform Convergence: Let D be a sequence of real-valued functions defined on D. Then fn converges uniformly to f if given any ε > 0, there exists a natural number N = N (ε) such that |fn (x) − f (x)| < ε for every n > N and for every x in D. Note: In the above definition the number N depends on ε but not on x. It is clear that Uniform Convergence implies pointwise convergence. But the converse need not be true. A series ∞ if the sequence < sn > of n=1 fn (x) is called pointwise convergent  (x) = f1 (x) + f2 (x) + ... + fn (x) or ni=1 fi (x) is pointwise convergent. partial sums sn The series ∞ n=1 fn (x) is a called uniformly convergent if the sequence of partial sums is uniformly convergent. Convergence almost everywhere: A sequence of function fn n∈N is said to converge almost everywhere on D to f if μ{x ∈ D|fn (x) does not converge to f (x)}=0 where μ is measure of the set namely there exists a set E with μ(E) = 0 such that limn→∞ fn (x) = f (x) for all x ∈ D \ E. In case of Lebsegue measure, the measure of an interval will be length of the interval. See for example Appendix of [5]. Let f (t) be a periodic function with period T and Lebesgue integrable continuous function having at most finite discontinuous over (−T /2, T /2). Then the fourier series of f (t) is the trigonometric series ∞   2π k 2π k 1 Ak cos A0 + x + Bk sin x 2 T T

(A.6)

k=1

where 2 Ak = T A0 =

1 T

Bk =

2 T



T /2

−T /2  T /2 −T /2  T /2 −T /2

f (t) cos

2π kt dt, k = 1, 2, 3, . . . T

f (t)dt f (t) sin

(A.7) (A.8)

2π kt dt, k = 1, 2, 3, . . . T

(A.9)

and we write it as ∞   1 2π k 2π k f ∼ A0 + Ak cos x + Bk sin x 2 T T k=1

Here, we take

(A.10)

374

Appendix

wk =

k , k = 0, 1, 2, 3, . . . T

(A.11)

Very often, we choose T = 2π, Ak and Bk are called cosine Fourier coefficient and sine Fourier coefficient, respectively. The set of triplex (Ak , Bk , wk ) where Ak , Bk , wk are given by Eqs. (A.7), (A.8), and (A.9), respectively, is called the Fourier series, frequency content. The complex form of the Fourier series of f (x) is ∞ 

Cn e2πint/T

−∞

where Cn =

An + iBn , n>0 2

C0 = A0 An − iBn , n>0 Cn = 2 n wn = , n = −2, −1, 0, 1, 2. T Let T = 2π and Sn (f )(x) =

n 

 1 A0 + (Ak cos kx + Bk sin kx) 2 n

Ck eikx =

k=−n

k=1

be the nth partial sum of the Fourier series of f . Then  1 π  1 = π

Sn (f )(x) =

where

2

2

π0 f (x − t)Dn (t)dt π0 f (t)Dn (x − t)dt



n sin n + 21 x 1  Dn (x) = + cos kx = 2 2 sin 2x

(A.12)

(A.13)

k=1

is the “Dirichlet kernel”, and σn (f ) =

S0 (f ) + S1 (f ) + · · · + Sn (f ) 1 = n+1 π



2

π0 f (x − t)Kn (t)dt

(A.14)

Appendix

375

where Kn (x) =

(x) D0 (x) + D1 (x) + · · · + Dn (x) 1 sin2 n+1 2 = n+1 n + 1 2 sin2 2x

(A.15)

is called the “Fejer Kernel”. Theorem A.9 (Bessel’s Inequality) ∞ 

|Ck |2 ≤ f 2L2 (0,2π)

k=−∞

or

∞

1 2  2 A + (A + Bn2 ) ≤ f 2L2 (0,2π) 4 0 n=1 n

This also mean that {Ak } and {Bk } are elements of l2 . Theorem A.10 (Reisz -Fisher Theorem) Let {Ck } ∈ l2 . Then there exists f ∈ L2 (−π, π ) such that {Ck } is the kth Fourier coefficient of f . Furthermore ∞ 

|Ck |2 ≤ f 2L2 (−π,π) .

k=−∞

For {Ak }, {Bk } belonging to l2 , there exists f ∈ L2 (0, 2π ) such that Ak , Bk are, respectively, kth cosine and sine Fourier coefficient of f . Furthermore ∞

1 2  2 A + (A + Bk2 ) = 2 0 n=1 k



π −π

|f |2 dt .

Theorem A.11 Let f ∈ L2 (−π, π ), then lim f − Sn (f ) L2 (−π,π) = 0.

n→∞

Theorem A.12 Let f ∈ C[0, 2π ] such that 

2

π0

w(f , t) dt < ∞, w(f , t) = sup |f (x + t) − f (x)|. t t

Then the Fourier series of f converges uniformly to f , that is lim f − Sn (t) L∞ (−π,π) = 0.

n→∞

If w(f , η) = 0(ηα ), then the condition of the theorem holds.

376

Appendix

Theorem A.13 If f is a function of bounded variation, then Sn (f ) →

f (x+ ) + f (x− ) as n → ∞ 2

where f (x+ ) = lim+ f (x + h) h→0

f (x− ) = lim+ f (x − h) h→0

exists at every x, a < x < b.  Theorem A.14 Let f ∈ L1 (R), then the series k=∞ k=−∞ f (x + 2π k) converges almost everywhere to a function λ(x) ∈ L1 (0, 2π ). Moreover, the Fourier coefficient ck of 1 ˆ λ(x) are given by ck = 2π f (k) lim f (k) (x) = 0 for k = 0, 1, . . . , n − 1

|x|→∞

then

F{f (n) } = (iω)n F(f ).

Definition A.14 Let f , g ∈ L1 (R) then the convolution of f and g is denoted by f ∗ g and is defined by 1 F(f ∗ g)(x) = √ 2π



∞

−∞

f (x − u)g(u)du.

Theorem A.15 For f , g ∈ L1 (R), F(f ∗ g) = F(f )F(g) holds. Theorem A.16 Let f be a continuous function on R vanishing outside a bounded interval. Then f ∈ L2 (R) and

f L2 (R) = f L2 (R) . Definition A.15 (Fourier Transform in L2 (R)) Let f ∈ L2 (R) and {Φn be a sequence of continuous functions with compact support to f in L2 (R), that is, f − φn L2 (R) → 0. The Fourier transform of f is denied by fˆ = lim Φn n→∞

where the limit is with respect to the norm in L2 (R).   Theorem A.17 If f ∈ L2 (R), then (i) f , gL2 = fˆ , gˆ (Parseval’s formula) (ii) f L2 = f L2 (Plancherel formula)

L2

Appendix

377

In physical problems the quantity f L2 is a measure of energy while fˆ L2 represents the power spectrum of f . Theorem A.18 (1) Let f ∈ L2 (R). Then 1 fˆ (ω) = lim √ n→∞ 2π



π

e−iωxdx

−π

(2) If f , g ∈ L2 (R), then 

∞

−∞

 f (x)g(x)dx ˆ =

∞

−∞

fˆ (x)g(x)dx.

Theorem A.19 (Inversion of the Fourier Transform in L2 (R)) Let f ∈ L2 (R). Then 1 f (x) = lim √ n→∞ 2π



π

−π

eiωx fˆ (ω)d ω

where convergence is with respect to the norm in L2 (R). Corollary A.1 (1) If f ∈ L1 (R) ∩ L2 (R), then fˆ (ω) = √12π π (2) fˆ (ω) − √12π −π e−iωx f (x)dx L2 (R) → 0 as n → ∞. π (3) f (x) − √12π −π eiωx fˆ (ω)d ω L2 (R) → 0 as n → ∞. (4) f 2 = fˆ 2 . L2 (R)

∞

−∞

e−iωx f (x)dx.

L2 (R)

(5) The map f → fˆ is an isometry of L2 (R) onto L2 (R).

Example A.1 (1) If f (x) = (1 − x2 )e−x /2 = Second derivative of a Gaussian function, then 2 fˆ (ω) = ω2 e−ω /2 . 2

(2) If the Shannon function is defined by f (x) = then fˆ (ω) =



sin 2π x − sin π x πx

√1 , 2π

0,

if π < |ω| < 2π otherwise

The results given in the appendix are based on references [1, 2, …]

378

Appendix

References 1. Schipp, F., Wade, W. R., & Simon, P. (1990). Walsh series. New York: Adam Hilger. 2. Pontrjagin, L. S. (1939). Topological groups. New Jercy: University Press Princiton. 3. Vilenkin, N. Y. (1947). On a class of complete orthonormal systems, Izv. Akad. Nauk Sssr Ser. Mat. 11:363–400. English Translation, American Mathematical Society Translations 28(Series 2)(1963), 1–35. 4. Fine, N. J. (1949). On the Walsh functions. Transactions of the American Mathematical Society, 65(3), 372–414. 5. Siddiqi, A. H. (1978). Walsh function. Aligarh: AMU. 6. Schoenenberg, I. J. (1946). Contribution to the problem of approximation of equidistant data by analytic functions. Quarterly of Applied Mathematics, 4(1), 45–99. 7. Selesnick, I. W. (1998). Multiwavelet bases with extra approximation properties. IEEE Transactions on Signal Processing, 46(11), 2898–2908. 8. Wade, W. R. (1995). A Walsh system for polar coordinates. Computers & Mathematics with Applications, 30(3–6), 221–227. 9. Weide, B. W., Andrews, L. T., & Iannone, A. M. (1978). Real-line analysis of EEG using Walsh transforms. Computers in Biology and Medicine, 8(4), 255–263. 10. Welstead, S. (2000). Fractal and wavelet image compression techniques. Washington: SPIE Optical Engineering Press. 11. Siddiqi, A. H. (2018). Functional Analysis and Applications, Springer.

Notational Index

Symbols CW , H p , Lp , BMO and VMO, 86 Dn (x), 34 G p , 171 H ∗ , G ∗ , 11 H2n +r (x), 235 I (p, ω), 8 Jk (x), 3, 4  f (ω), 6 Ja f (ω), 6 Kk (x, n), 40 Kn (t), 49 (α) Kn (x), 41 2 L (R+ , CN ), 326 Ln (t), 49 MRA, 16 Pa (f ), 247 Pj f , 110 Sn f , 34 Tψ f (a, b), 14 ˆ φ(ω), 123 λ(x), 172

φ(x), 114 ψa,b (t), 14 ψj,k (t), 15 aα , 177 bs , 177 dj,k , 15 h2n +k (x), 10 m0 (ω), 114, 177, 193 pk (t), 237 tn (f , x), 47 W (δ, f ), 3 Wp (f , δ), 46 x ⊕ y, 3 Lipj(t), 44 Ml , 226 MRA in L2 (G), 190 MRA in L2 (R+ ), 103 NUMRA, 284, 285 PNSR, 277 VNUMRA, 334 [x], xi x ⊕p y, xi Z, Z+ , N , C, x

© Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2

379

Index

A Approximation by Walsh–Fourier series, 40 Approximation in dyadic homogeneous Banach spaces, 57 B Banach space, 360 Bessel’s inequality, 375 Biorthogonal wavelets, 253 construction, 259 Vilenkin group, 263, 269 Bounded fluctuation, 29 C Classes of functions, 371 Cohen’s criterion, 270 Convergence absolute, 372 almost everywhere, 373 pointwise, 372 uniform, 373 D Data compression, 73 Discrete Haar Transform (DHT), 84 Discrete wavelet system, 15 Dyadic convolution, 364 Dyadic group G, 369 Dyadic wavelets approximation properties, 158 E ECG analysis, 74

EEG analysis, 75 F Fejer Kernel, 375 Fourier transform, 376 inversion, 377 Fractional functions, 226 G Generalized Haar wavelets, 93 H Haar filters, 11 Haar Fourier series, 82 Haar System, 79, 86, 235 Haar–Vilenkin coefficients, 248 Haar–Vilenkin scaling function, 237 Haar–Vilenkin wavelet, 235, 236 approximation, 243 convergence, 246 orthonormality, 240 Haar wavelet, 13 applications boundary value, 95 initial value, 94 integral equations, 96 Hadamard–Paley matrix, 9 Hadamard transform self adjoint, 9 I Image digitization, 71 Image enhancement, 72 Image transmission, 71

© Springer Nature Singapore Pte Ltd. 2019 Yu. A. Farkov et al., Construction of Wavelets Through Walsh Functions, Industrial and Applied Mathematics, https://doi.org/10.1007/978-981-13-6370-2

381

382 L Lipj(t), 44 Lipschitz class, 362 Locally compact Cantor group, 171

M Mask of scaling equation, 112 Mask of the refinable function, 265 Modified Strang–Fix condition, 256 Multiresolution Analysis (MRA), 16, 267

N Non-uniform Haar wavelets, 88 Non-uniform MRA (NUMRA), 91, 283 Nörlund Kernel, 47

O Orthogonal wavelets in L2 (R+ ), 103, 114 P Pattern recognition, 76 p-wavelets, 194

R Reisz -Fisher Theorem, 375 Riemann–Lebesgue Theorem, 28 Riesz system, 118

S Scaling function, 104 σ − algebra, 362 Speech processing, 76

U Uniformly W -continuous, 101

Index V Vector-valued MRA, 318 Vector-valued MRA p-analysis, 321 Vector-valued Nonuniform MRA, 334 Vector-valued nonuniform wavelet packets, 347 Vilenkin–Christenson transform periodic wavelets, 211 Vilenkin group, 171 periodic wavelets, 205 Vilenkin-Pontrjagin, 369 Vilenkin systems, 367

W Walsh–Dirichlet Kernel, 220, 363 Walsh–Fourier coefficients, 27 transformation, 31 Walsh–Fourier series applications image representation, 70 approximation in Lp spaces, 45 convergence, 34 Walsh–Fourier transform, 5, 101 relation with Fourier transform, 7 Walsh function, 363 integration, 3 relations with Haar function, 8, 83 Walsh polynomials, 2 Walsh-type wavelet packets, 10 Wavelet coefficients, 16 Wavelet representation, 16 Wavelets, 12 compact wavelets, 17 construction, 16 Wavelet series, 16 Wavelet transform continuous, 12 W -compact, 122 W -continuous, 6