Methods of Fourier Analysis and Approximation Theory [1st ed.] 3319274651, 978-3-319-27465-2, 978-3-319-27466-9

Table of contents :
Front Matter....Pages i-viii
Some Problems in Fourier Analysis and Approximation Theory....Pages 1-19
Front Matter....Pages 21-21
Parseval Frames with n + 1 Vectors in \(\mathbb{R}^{n}\) ....Pages 23-32
Hyperbolic Hardy Classes and Logarithmic Bloch Spaces....Pages 33-42
Multidimensional Extremal Logan’s and Bohman’s Problems....Pages 43-58
Weighted Estimates for the Discrete Hilbert Transform....Pages 59-69
Q-Measures on the Dyadic Group and Uniqueness Sets for Haar Series....Pages 71-83
Off-Diagonal and Pointwise Estimates for Compact Calderón-Zygmund Operators....Pages 85-112
Front Matter....Pages 113-113
Elementary Proofs of Embedding Theorems for Potential Spaces of Radial Functions....Pages 115-138
On Leray’s Formula....Pages 139-146
Front Matter....Pages 147-147
Order of Approximation of Besov Classes in the Metric of Anisotropic Lorentz Spaces....Pages 149-159
Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness....Pages 161-179
Reconstruction Operator of Functions from the Sobolev Space....Pages 181-191
Front Matter....Pages 193-193
Laplace–Borel Transformation of Functions Holomorphic in the Torus and Equivalent to Entire Functions....Pages 195-209
Optimization Control Problems for Systems Described by Elliptic Variational Inequalities with State Constraints....Pages 211-224
Two Approximation Methods of the Functional Gradient for a Distributed Optimization Control Problem....Pages 225-235
Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods....Pages 237-258

Citation preview

Applied and Numerical Harmonic Analysis

Michael Ruzhansky Sergey Tikhonov Editors

Methods of Fourier Analysis and Approximation Theory

Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA Editorial Advisory Board Akram Aldroubi Vanderbilt University Nashville, TN, USA

Gitta Kutyniok Technische Universität Berlin Berlin, Germany

Douglas Cochran Arizona State University Phoenix, AZ, USA

Mauro Maggioni Duke University Durham, NC, USA

Hans G. Feichtinger University of Vienna Vienna, Austria

Zuowei Shen National University of Singapore Singapore, Singapore

Christopher Heil Georgia Institute of Technology Atlanta, GA, USA

Thomas Strohmer University of California Davis, CA, USA

Stéphane Jaffard University of Paris XII Paris, France

Yang Wang Michigan State University East Lansing, MI, USA

Jelena Kovaˇcevi´c Carnegie Mellon University Pittsburgh, PA, USA More information about this series at http://www.springer.com/series/4968

Michael Ruzhansky • Sergey Tikhonov Editors

Methods of Fourier Analysis and Approximation Theory

Editors Michael Ruzhansky Department of Mathematics Imperial College London London, United Kingdom

Sergey Tikhonov ICREA Research Professor Centre de Recerca MatemJatica Barcelona, Spain

ISSN 2296-5009 ISSN 2296-5017 (electronic) Applied and Numerical Harmonic Analysis ISBN 978-3-319-27465-2 ISBN 978-3-319-27466-9 (eBook) DOI 10.1007/978-3-319-27466-9 Library of Congress Control Number: 2016932897 Mathematics Subject Classification (2010): 41-XX, 42-XX 32A-XX, 65C-XX, 49K-XX © Springer International Publishing Switzerland 2016 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. Printed on acid-free paper This book is published under the trade name Birkhäuser. The registered company is Springer International Publishing AG Switzerland (www.birkhauser-science.com)

Preface

This volume consists of a collection of papers originating from two events devoted to areas of harmonic analysis and their interplay with the approximation theory. The first event took place during the 9th ISAAC Congress in Krakow, Poland, 5–9 August 2013, at the section “Approximation Theory and Fourier Analysis”. The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4–8 November 2013, organized by the authors. It continued the successful tradition of workshop series: Osaka University 2008, Imperial College London 2008, Nagoya University 2009, University of Göttingen 2010, ICREA Conference 2011 in CRM, and Aalto University 2012. The topics of the conference include: Fourier analysis, function spaces, pseudodifferential operators, microlocal and time-frequency analysis, partial differential equations, and their links to modern developments in the approximation theory. London, UK Barcelona, Spain November 2015

Michael Ruzhansky Sergey Tikhonov

v

Contents

Some Problems in Fourier Analysis and Approximation Theory.. . . . . . . . . . Michael Ruzhansky and Sergey Tikhonov Part I

1

Fourier Analysis

Parseval Frames with n C 1 Vectors in Rn . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Laura De Carli and Zhongyuan Hu

23

Hyperbolic Hardy Classes and Logarithmic Bloch Spaces .. . . . . . . . . . . . . . . . . Evgueni Doubtsov

33

Multidimensional Extremal Logan’s and Bohman’s Problems . . . . . . . . . . . . . D.V. Gorbachev

43

Weighted Estimates for the Discrete Hilbert Transform.. . . . . . . . . . . . . . . . . . . . E. Liflyand

59

Q-Measures on the Dyadic Group and Uniqueness Sets for Haar Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Mikhail G. Plotnikov

71

Off-Diagonal and Pointwise Estimates for Compact Calderón-Zygmund Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Paco Villarroya

85

Part II

Function Spaces of Radial Functions

Elementary Proofs of Embedding Theorems for Potential Spaces of Radial Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 115 Pablo L. De Nápoli and Irene Drelichman On Leray’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139 E. Liflyand and S. Samko

vii

viii

Part III

Contents

Approximation Theory

Order of Approximation of Besov Classes in the Metric of Anisotropic Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 149 K.A. Bekmaganbetov Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness . . . . . . 161 M.K. Potapov and B.V. Simonov Reconstruction Operator of Functions from the Sobolev Space . . . . . . . . . . . . 181 N.T. Tleukhanova Part IV

Optimization Theory and Related Topics

Laplace–Borel Transformation of Functions Holomorphic in the Torus and Equivalent to Entire Functions . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195 L.S. Maergoiz Optimization Control Problems for Systems Described by Elliptic Variational Inequalities with State Constraints.. . . . . . . . . . . . . . . . . 211 Simon Serovajsky Two Approximation Methods of the Functional Gradient for a Distributed Optimization Control Problem .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 225 Ilyas Shakenov Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237 Kanat Shakenov

Some Problems in Fourier Analysis and Approximation Theory Michael Ruzhansky and Sergey Tikhonov

Abstract We give a short overview of some questions and methods of Fourier analysis, approximation theory, and optimization theory that constitute an area of current research. Keywords Approximation theory • Fourier analysis • Harmonic analysis • Optimization theory

Mathematics Subject Classification 42-06, 42-02, 41-02, 41-06

1 Introduction In this review we present a short survey of topics related to the two conferences (see the preface to the volume), with brief introduction to more extensive presentations contained in the papers of this collection.

2 Fourier Analysis 2.1 Parseval Frames Chapter 1 by L. De Carli and Z. Hu is “Parseval Frames with n C 1 Vectors in Rn ”. A frame in a finite-dimensional vector space is a set of vectors that

M. Ruzhansky () Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK e-mail: [email protected] S. Tikhonov ICREA, Centre de Recerca Matemàtica, and UAB, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_1

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M. Ruzhansky and S. Tikhonov

contains a basis. Frames (or redundant systems) have become increasingly popular in applied mathematics, computer science and engineering; representing signals in terms of a set of a redundant set of basic frequencies ensures resilience against noise, quantization errors and erasures in signal transmissions. Appropriate frame decomposition may reveal hidden signal characteristics, and have been employed as detection devices. The references are too many to cite, but see [9], the recent book [10] and the references cited there. Parseval frames are nontrivial generalisations of orthonormal bases and have important applications. Vectors in a Parseval frame are not necessarily orthogonal or linearly independent, and do not necessarily have the same length, but the Parseval identities vD

N X

hv; vj ivj

jD1

and jjvjj2 D

N X hv; vj i2 jD1

still hold for every v 2 Rn . In recent years, several inquiries about tight frames have been raised. In particular: how to characterise Parseval frames with N elements in Rn and whether it is possible to scale the vectors of a given frame so that the resulting frame is Parseval. In the paper by De Carli and Hu, the authors give a method for constructing Parseval frames with n C 1 vectors in Rn that contain a given vector w with jjwjj < 1. Precisely, they construct a Parseval triangular frames (i.e., a frame B D fv1 ; : : : ; vn ; wg  Rn such that the matrix with columns v1 , . . . , vn is right triangular) and they show that B is essentially unique, in the sense that if B 0 D fv10 ; : : : ; vn0 ; wg is another Parseval triangular frame that contains w, then vj0 D ˙vj . Scalable frames with n C 1 vectors in Rn are also discussed. The method can be used to construct Parseval frames with n C k vectors in Rn that contain given vectors w1 ; : : : ; wk (this work is still in progress).

2.2 Hyperbolic Hardy Classes and Logarithmic Bloch Spaces Chapter 2 is the paper “Hyperbolic Hardy Classes and Logarithmic Bloch Spaces” by E. Doubtsov. Let H.Bn / denote the space of holomorphic functions on the unit ball Bn of Cn , n  1. Given a holomorphic mapping ' W Bn ! Bm , the composition operator C' W H.Bm / ! H.Bn / is defined as

Some Problems in Fourier Analysis and Approximation Theory

.C' f /.z/ D f .'.z//;

3

z 2 Bn :

As indicated in the monograph [34], the main problem in a study of composition operators is to relate the operator-theoretic properties of C' and the functiontheoretic properties of the symbol '. So, Ahern and Rudin [1] posed the problem of characterising those holomorphic mappings ' W Bn ! Bm , m D 1, for which the composition operator C' maps the Bloch space ( B.Bm/ D

) 2

f 2 H.Bm / W sup jrf .w/j.1  jwj / < 1 z2Bm

into the classical Hardy space H p .Bn /, p > 0, or into BMOA.Bn /, the space of holomorphic functions having bounded mean oscillation. First major results related to the Bloch-to-BMOA problem were obtained by Ramey and Ullrich [32]. In particular, quite surprisingly, every Lipschitz symbol ' has the required property for m D 1 and n  2. Also, Ramey and Ullrich indicated that the Bloch-to-BMOA property is related to BMOAh .Bn ; Bm /, the hyperbolic BMOA class introduced much earlier by S. Yamashita for n D m D 1. In fact Yamashita used the following formal rule: replace a function f 2 H.B1 / and the Euclidean metric on C by a holomorphic mapping ' W B1 ! B1 and the hyperbolic (Bergman) metric on the disk B1 , respectively. He investigated hyperbolic analogs of various classical spaces, in particular, those of the Hardy spaces H p .B1 /; see [42]. For m D 1, a solution of the Bloch-to-Hardy problem in terms of the hyperbolic Hardy classes was obtained by E.G. Kwon in a series of publications (see, e.g., [16]). The paper by E. Doubtsov is motivated by the above problems for m  2. Standard arguments are not applicable in this case, so the first technical idea is to use test functions with appropriate sharp lower estimates of their moduli instead of lower estimates of their derivatives. It turns out that such derivative-free approach gives a complete theoretical solution of the Bloch-to-BMOA problem for all m; n  1; see [13]. Also, Kwon [16] argued that various hyperbolic classes introduced by Yamashita naturally arise in the descriptions of related bounded composition operators defined on the Bloch space B.B1 /. So, the second idea is to consider logarithmic perturbations of the standard weight 1  jwj2 and to show that the Bloch space B.Bm / is not an exception. Namely, given ˛ < 12 and 0 < p  1, the hyperbolic Hardy class with parameter .1  2˛/p is related to the bounded composition operators from the logarithmic Bloch space L˛ B.Bm / to the Hardy space H 2p .Bn /.

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2.3 Logan’s and Bohman’s Extremal Problems “Multidimensional Extremal Logan’s and Bohman’s Problems” by D. Gorbachev is Chap. 3. A.M. Odlyzko formulated the extremal problem related to zeros of Dedekind zeta functions. Let f be an arbitrary positive definite function such that supp fO  Œ1; 1, fO .0/ D 0, f .0/ D 1, where fO .s/ D

Z R

f .t/e2ist dt

is the Fourier transform of f . The Odlyzko problem is to determine the quantity T D inf T provided that f .t/  0 for jtj  T. This problem was solved by Logan in [22]. He proved that f ; t2 f 2 L1 .R/, T D 3=2, and f .t/ D

cos2 .t/ .1  .2t/2 /.1  .2t=3/2 /

is an extremiser. The periodic case of Logan’s problem was considered by Chernykh (1967, 1979). He proved sharp Jackson’s inequality in the space L2 .T/ with the optimal argument. A generalisation of the Chernykh results for the multidimensional case was given by Yudin (1981). For an arbitrary convex 0-centrally symmetric body C  Rn , Yudin constructed a positive definite function FC 2 L1 .Rn / such that supp b F C  C, b FC .0/ D 0, FC .0/ D 1, and FC .x/  0 for 1 .C/=, where 12 .C/ is the first eigenvalue of the Laplacian  in C. Using properties of the function FC , Yudin also obtained several results in analytic number theory and in algebraic lattice theory (1996). In particular, he got the asymptotic estimate R.L/1 .L / 

 n  1 C O.n2=3 / ; as n ! 1; 2

where L is a lattice in Rn , L is the dual lattice to L, R.L/ is the covering radius of L, 1 .L / is the first successive minimum of L . By a more complicated argument Banaszczyk (1993) also derived this result. The multidimensional analogue of Logan’s problem was stated by Berdysheva [6]. Suppose that U; V  Rn are convex bodies, and let f 2 L1 .Rn / be an arbitrary positive definite function such that supp fO  V, fO .0/  0, and f .0/ D 1. The problem is to determine the quantity T D inf T provided that f .x/  0 for kxkU  T. Berdysheva solved multidimensional Logan’s problem for the lnp -ball U D Bnp and V D Bn1 , where 1  p  2,  > 0, and Bn1 n is a cube. p For  D 1, it was shown that an extremiser is Yudin’s function FB1 and T D n=2. To prove this fact, the multidimensional Poisson summation formula

Some Problems in Fourier Analysis and Approximation Theory

5

was used. Using one-dimensional Bessel quadrature formulas, Gorbachev (2000) solved the problem for the Euclidean balls U D V D Bn2 . It turns out that Yudin’s function FV is a unique extremiser and T D qn=21 =, where qn=21 is the smallest positive zero of the Bessel function Jn=21 . Logan’s problem is closely related to extremal Delsarte’s problem on finding of inf f .0/ provided that f 2 L1 .Rn /, fO  0, fO .0/ D 1, and f .x/  0 for jxj  2. Using the solution of Delsarte’s problem, one can derive the upper bound for the sphere packing density of Rn , as shown by Gorbachev (2000) and by Elkies and Cohn (2001). Using an approximate solution of Delsarte’s problem and analogue of first Logan’s problem, Cohn and Kumar (2004) proved the optimality and the uniqueness of the Leech lattice in R24 . The main goal of the paper by Gorbachev is to generalise the methods used by Logan and Berdysheva. Theorem 2.1 in Chap. 3 asserts that if 1 .L /V D 1, then R.L/U  T   1 1 .V/ sup kxkU : jxjD1

The proof is based on the generalised Poisson summation formula and Yudin’s function FV . The above estimates are sharp in the case of L D Zn . Logan’s problem is closely related to Turan’s, Bohman’s and others extremal problems of harmonic analysis. Turan’s problems were investigated, for example, by D. Gorbachev, C.L. Siegel, V.V. Arestov, E.E. Berdysheva, V.I. Ivanov, M.N. Kolountzakis, Sz.Gy. Révész. In the second part of his paper, Gorbachev solves Bohman’s extremal problem. The problem is to determine a minimum of the second moment Z M D inf

Rn

jxj2 f .x/ dx

for a nonnegative function f 2 L1 .Rn / such that supp fO  V and fO .0/ D 1. For n D cos2 .t/ 1, Bohman [7] showed that M D 1=4 and f .t/ D  28.1.2t/ 2 /2 . Bohman proved the optimality of some numerical integration methods of probability density functions. Later, Papoulis (1972) also proved this result by means of variational methods in connection with a determination of minimum-bias windows for high-resolution spectral estimates. Periodic Bohman’s problem for applications to approximation theory was studied by Korovkin (1958), Yudin (1976), and Ivanov (1994). Yudin constructed the admissible function GV which is a generalisation of one-dimensional Bohman’s function. In the case of the Euclidean ball V D Bn2 , Bohman’s problem was solved by Ehm et al. [14]: M D .qn=21 =/2 and f D GBn2 . Theorem 2.2 in Chap. 3 asserts that if 1 .L /V D 1, then R2 .L/  M  .1 .V/=/2 : In the case of L D Zn and V D Bn1 , one has M D n=4 and f D GBn1 .

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2.4 Weighted Estimates for the Hilbert Transform In Chap. 4, “Weighted Estimates for the Discrete Hilbert Transform”, E. Liflyand deals with the classical topic of boundedness properties of the Hilbert transform. Being somewhat apart of the mainstream, the Paley-Wiener theorem asserts that for an odd and monotone decreasing on RC function g 2 L1 its Hilbert transform is also integrable, i.e., g is in the (real) Hardy space H 1 .R/. The oddness of g is 1 Ressential, since by Kober’s result, if g 2 H .R/, then the cancelation property holds R g.t/ dt D 0: In [19], the monotonicity assumption has been relaxed in the Paley-Wiener theorem, and in [18] weighted versions have been obtained for both the sine and cosine Fourier transforms of functions more general than monotone ones. The periodic case has also been covered in [18] in the same manner. One can find historical background relevant to these problems in [18]. The goal of Liflyand’s note is to prove the weighted analogues of the PaleyWiener theorem for odd and even sequences, that is, for the discrete Hilbert transform. More precisely, the estimates are given in the weighted `1 spaces, the space of sequences a D fak g1 kD0 endowed with the norm kak`.w/ D

1 X

jak jwk < 1;

kD0

where w D fwk g; k D 0; 1; 2; : : : ; is a non-negative sequence, or a weight. The author uses the definition of general monotone sequences [38, 39]. A null sequence a, that is, vanishing at infinity, is said to be general monotone if it satisfies the conditions Œcn X jak j ; jak  akC1 j  C k kDn

2n X

n D 1; 2; : : : ;

kDŒn=c

where C > 1 and c > 1 are independent of n. See also the survey [19]. In particular the author shows that for a weight wk D k˛ , the discrete Hilbert transform is bounded in `.w/, when 1 < ˛ < 1 provided that a is an odd and general monotone on ZC sequence, or, when 2 < ˛ < 0 provided that a is even and general monotone on RC . Thus, assuming general monotonicity of a allows one to obtain analogues of the results of [18] to sequences; recall that the estimates in [18] were, in turn, generalisations of Flett’s, Hardy-Littlewood’s, and Andersen’s results to the case p D 1. These results can also be considered as generalisations of the known results for sequences in [3].

Some Problems in Fourier Analysis and Approximation Theory

7

2.5 Q-Measures and Uniqueness Sets for Haar Series Chapter 5 by M. Plotnikov is “Q-Measures on the Dyadic Group and Uniqueness Sets for Haar Series”. In 1870 G. Cantor proved the following result: if a trigonometric series 1

a0 X C .TS/ D an cos.nx/ C bn sin.nx/ 2 nD1 converges to zero everywhere on Œ0; 2/ except possibly on a finite set, then .TS/ is the trivial series, i.e., all an D 0, bn D 0. Cantor’s theorem starts the theory of uniqueness for orthogonal series, which nowadays is well-developed (G. Cantor, P. du Bois-Reymond, A. Lebesgue, Ch.J. Vallée Poussin, W.H. Young, F. Bernstein, D.E. Men’shov, A. Rajchman, N.K. Barí, J. Marcinkiewicz, R. Salem, A. Zygmund, I.I. Pyatetckii-Shapiro, etc.). A set of uniqueness is one of the basic notions of the theory of uniqueness. Let f fn g be a system of functions on Œa; b. We recall that a set A  Œa; b is called a set of uniqueness, or U-set, for series X

cn fn .x/;

cn 2 R or cn 2 C;

(1)

n

if the only series (1) converging to zero everywhere on Œa; b n A is the trivial series. The Cantor theorem means that every finite set A  Œ0; 2/ is a U-set for trigonometric series. The next theorem is one of the most remarkable results in this direction: let F be the Cantor symmetric set with a constant ratio ; then F is a U-set if and only if  1 is a Pisot number (N.K. Barí, R. Salem, I.I. PyatetckiiShapiro, R. Salem and A. Zygmund, 1937–1955). In particular, the Cantor middle thirds set F1=3 is a set of uniqueness. So there is a fascinating connection between sets of uniqueness of trigonometric series and number theory. In general, the problem of a characterisation of U-sets is open even for closed sets. Moreover, in 1980s A.S. Kechris shows that this problem can not be solved constructively. The theory of uniqueness of trigonometric series is treated for example in [15]. Nontrigonometric sets of uniqueness were systematically investigated since 1960s. The Haar system fHn .x/g1 nD0 is widely used in harmonic and functional analysis. Note that every series on the Haar system can be represented as some finitely additive set function (so called quasi-measure). Moreover, the set of all Haar series and the set of all quasi-measures are linear isomorphic. It was shown in for example [27, 36]) that convergence and behavior of the partial sums of any Haar series is associated with differentiability, continuity, and smoothness of an appropriate quasi–measure.

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In contrast to trigonometric series the empty set is the only set of uniqueness for Haar series. Some authors (Skvortsov, Wade, Talalyan, Mushegyan, Gevorkyan, Tetunashvili, Plotnikov) have studied sets of uniqueness for various subclasses of Haar series (conditional sets of uniqueness). A relationship between such sets and Hausdorff measures was established in [26]: A set A  Œ0; 1 is a U-set for Haar series satisfying an D O .np1=2 /;

0 < p  1;

(2)

if and only if A contains no perfect subsets of positive Hausdorff .1  p/-measure. Let F be the Cantor symmetric set with a constant ratio . Then F is a U-set for 2 Haar series under condition (2) if and only if p < 1 C log log  . In particular, the Cantor middle third set F1=3 is a U-set for Haar series under condition (2) if and only if 2 p < 1  log log 3 . So such sets of uniqueness can be described only in terms of metric characteristics. In Plotnikov’s paper, he extends the family of subclasses of Haar series and uses so called Q-measures for description of appropriate sets of uniqueness.

2.6 Off-Diagonal Estimates for Calderón-Zygmund Operators Villarroya’s Chap. 6, “Off-Diagonal and Pointwise Estimates for Compact Calderón-Zygmund Operators”, deals with operators T satisfying an off-diagonal estimate on Lp .Rn /, i.e., when there exists a function G W Œ0; 1/ ! Œ0; 1/ vanishing at infinity such that kT.f E / F kLp .Rn / . G.dist.E; F//kf kLp .Rn / ; for all Borel sets E; F  Rn and all f 2 Lp .Rn /. These inequalities play an important role in analysis because they enclose all the orthogonality properties of the operator required to prove its boundedness. Their interest in harmonic analysis renewed with the proof of the T(1) Theorem based on wavelet decompositions [11]. This approach involved estimates of the form jhT.

I /;

J ij

.

 jJj  12 C nı  dist.I; J/ .nCı/ 1C k 1 jIj jIj n

I kL2 .Rn / k J kL2 .Rn / ;

for all cubes I; J  Rn , with jJj  jIj, under the appropriate hypothesis on the operator T, the parameter ı > 0 and the functions I ; J . Similar bounds were also crucial in the solution of the famous Kato’s conjecture [4] about boundedness of square root of elliptic operators and are nowadays extensively used in the study of boundedness of second order elliptic operators (see [5] for example).

Some Problems in Fourier Analysis and Approximation Theory

9

In [41], Villarroya developed a T.1/ Theorem to characterize not only boundedness but also compactness of Calderón-Zygmund operators. Now, the paper Off-diagonal and pointwise estimates for compact Calderón-Zygmund operators provides off-diagonal bounds for singular integral operators that extend compactly on Lp .Rn /. Under the right hypotheses, the estimates can be stated as follows: jhT.

I /;

J ij

.

 jJj  12 C ın  dist.I; J/ .nCı/ 1C F.I; J/ k 1 jIj jIj n

I kL2 .Rn / k J kL2 .Rn / ;

for appropriate bump functions I ; J well localized on cubes I; J  Rn . The last factor F.I; J/ is a new element, completely absent in the analog inequalities for bounded singular integrals, which encodes the extra rate of decay obtained as a consequence of the compactness properties of the operator. Therefore, the focus of the work is placed on obtaining a sharp and detailed description of this new extra factor in terms of the size and location of the cubes I and J.

3

Function Spaces of Radial Functions

3.1 Potential Spaces of Radial Functions Chapter 7 by P. De Napoli and I. Drelichman is “Elementary Proofs of Embedding Theorems for Potential Spaces of Radial Functions”. Embedding theorems play a central role in the theory of nonlinear partial differential equations, in particular when applying variational methods for proving existence of solutions. In many cases, the problem and the associated functional are symmetric under the action of the orthogonal group, hence one is led to the study of radial solutions. Moreover, since the pioneering works of Ni [24] and Strauss [37], it is well known that better embedding theorems can be obtained when restricting to subspaces of radially symmetric functions. Indeed, Strauss’ inequality implies that a radial function in the Sobolev space H 1 .Rn / necessarily has a decay at infinity like a negative power of jxj, and Ni’s inequality gives a bound for the behaviour near the origin of such a function. As a consequence, one can recover compactness of the embeddings, which is an essential feature for the success of variational methods and which does not hold for arbitrary functions in Rn due to the translation invariance of the Sobolev norm. These results were generalised by Lions [21] in several directions, including Sobolev spaces of fractional order. The paper by De Napoli and Drelichman surveys some known results in the theory of radial Sobolev and Bessel potential spaces. The authors give two simple proofs of a version of Strauss’ inequality for potential spaces, one for p D 2 (using the Fourier transform) and another one for general p (using a weighted convolution theorem for radial functions by the authors [12]). Another section of the paper is devoted to embedding theorems with power weights for radial functions,

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M. Ruzhansky and S. Tikhonov

which include a generalisation of Ni’s inequality in Rn . Then they analyse the compactness of the embeddings, both in the unweighted case (giving an alternative proof of a theorem by Lions, avoiding the use of complex interpolation) and in the weighted case. Finally, they discuss a generalisation of Ni’s inequality and embedding theorems for potential spaces in a ball. The proofs presented are elementary in the sense that they avoid the use of interpolation theory and sophisticated tools such as atomic or wavelet decompositions.

3.2 On Leray’s Formula In Chap. 8, “On Leray’s Formula”, E. Liflyand and S. Samko suggest a new direction in a study of the multidimensional Fourier transform of a radial functions. It is well known that if f is an integrable radial function, that is, f .x/ D f0 .jxj/, the R n-dimensional Fourier integral fO .x/ D Rn f .u/ eixu du; u; x 2 Rn ; reduces to the one-dimensional integral: n fO .x/ D fO0 .jxj/ D .2/ 2

Z

1 0

f0 .t/.jxjt/ 2 1 J 2n 1 .jxjt/tn1 dt; n

where J is the Bessel function of first type and order : This makes sense also for 2n radial Lp -functions with 1 < p < nC1 W in that case fO .x/ is everywhere continuous away from the origin. However, working with Bessel functions is quite tedious business sometimes, and certain attempts are known to simplify this formula towards obtaining a “genuine” one-dimensional Fourier transform of some function related to f0 ; let us mention [28] and [20]. Leray (see [17, 33]) proved that if Z

1 0

tn1 .1 C t/

1n 2

jf0 .t/j dt < 1;

the following relation holds n1 fO .x/ D 2 2

Z

1

I n1 f0 .t/ cos jxjt dt; 2

0

(3)

where the fractional integral I˛ is defined by 2 I˛ f0 .t/ D .˛/

Z

1

sf0 .s/.s2  t2 /˛1 ds:

t

This formula seems to be very attractive, however much is hidden in the fractional integral of f0 . Liflyand and Samko change (3) in such a way that it reduces to the one-dimensional Fourier transform of a function “closer” to f0 .

Some Problems in Fourier Analysis and Approximation Theory

11

4 Approximation Theory 4.1 Approximation Order of Besov Classes “Order of Approximation of Besov Classes in the Metric of Anisotropic Lorentz Spaces” by K. Bekmaganbetov is Chap. 9. In this paper the author deals with the anisotropic Lorentz spaces L q .Td / endowed with a norm kf kLpq .Td / D 0 @

Z

.2/dn 0

Z :::

0

.2/d1

1 q1n ! qq2  1 q1 1 1 dt dt 1 nA p t1 1 : : : tnpn f 1 ;:::;n .t1 ; : : : ; tn / ::: < 1; t1 tn

where f  .t/ D f 1 ;:::;n .t1 ; : : : ; tn /. Using this definition, he defines the Besov space as follows: n o  .˛;s/  2 D k .f /k kf kB˛ d/ d/ s L .T .T pr  pr s2Zn

C

   < 1;  l

where k  kl is a norm of the discrete Lebesgue spaces with mixed metric l and s .f ; x/ D

X

ak .f /ei.k;x/ :

k2.s/

Here fak .f /gk2Zd are the Fourier coefficients of the function f with respect to the multiple trigonometric system and

.s/ D fk D .k1 ; : : : ; kn / 2 Zd W 2si 1  max jkji j < 2si ; i D 1; : : : ; ng: jD1;:::;di

Bekmaganbetov obtains sharp order of approximation of functions from the d Besov classes B˛ pr .T / in the Lorentz space metric, that is, he finds two-sided estimates for sup kf kB˛ .Td / 1

E 0 d;N .f /L q ;

p

where E d;N .f /Lpr is the best approximation of f 2 Lpr .Td / by polynomials with harmonics in hyperbolic crosses. The obtained results are related to the previous investigations by K.I. Babenko, Ya.S. Bugrov, E.M. Galeev, Dinh Dung, V.N. Temlyakov, and the recent results by Akishev [2] and Nursultanov and Tleukhanova [25].

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M. Ruzhansky and S. Tikhonov

4.2 Ulyanov Inequalities for Moduli of Smoothness Chapter 10 by M.K. Potapov and B.V. Simonov is “Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness”. The authors study an important problem in approximation theory on sharp inequalities between moduli of smoothness in different metrics .Lp ; Lq /, 1  p < q  1. In the one-dimensional case, if f 2 Lp .T/; 1  p  1; the modulus of smoothness of fractional order is given by 1 ˇˇ X ˇˇ   ˇˇ ˇˇ .1/ ˛ f .x C .˛  /h/ˇˇ : !˛ .f ; t/p D sup ˇˇ jhjt

p

D0

Sharp Ulyanov inequality [35] provides an optimal estimate for moduli of smoothness in .Lp ; Lq /, 1 < p < q < 1:  Zı !˛ .f ; ı/q  C 0

 . 1  1 / q dt t p q !˛C 1  1 .f ; t/p p q t

 1q

:

This inequality also holds if p D 1 and q D 1 but not if p D 1, q < 1 or 1 < p, q D 1. In these cases, one has [40]: for p D 1; 1 < q < 1,  Zı !˛ .f ; ı/q  C 0

 . 1  1 / q dt t p q .ln 2=t/1=q !˛C 1  1 .f ; t/p p q t

 1q

and for 1 < p < 1; q D 1, Zı !˛ .f ; ı/q  C

1

1

1

t. p  q / .ln 2=t/1 p !˛C 1  1 .f ; t/p p

q

0

dt : t

For periodic functions on T2 , the mixed modulus of smoothness of a function f 2 Lp .T2 / of orders ˛1 > 0 and ˛2 > 0 with respect to the variables x and y; respectively, is defined as follows: !˛1 ;˛2 .f ; ı1 ; ı2 /p D

sup jhi jıi ;iD1;2

  jj˛h11 ˛h22 .f / jjp ;

where ˛h11 .f / is the difference of order ˛1 > 0 with respect to the variable x and ˛h22 .f / is the difference of order ˛2 > 0 with respect to the variable y. For the main properties of the mixed moduli of smoothness see [30]. Sharp Ulyanov inequality for the mixed moduli was obtained in [31] for 1 < p < q < 1.

Some Problems in Fourier Analysis and Approximation Theory

13

In their paper, Potapov and Simonov prove a sharp Ulyanov inequality for the mixed moduli in the case of p D 1 and q D 1.

4.3 Approximation Order of Besov Classes Tleukhanova’s Chap. 11 is “Reconstruction Operator of Functions from the Sobolev Space”. In this work, she deals with the reconstruction problem of periodic functions from the spaces Wp˛ with dominant mixed derivative by values of a function at a given number of nodes. Let X D Wp˛ Œ0; 1n and Y D Lq Œ0; 1n . The author studies the decay order of the orthogonal diameter given by ? dM .X; Y/ D inf

sup kf 

fgj gM jD1 kf kX D1

M X .f ; gj /gj kY ; jD1

n where the infimum is taken over all orthogonal systems fgj gM jD1 from L1 Œ0; 1 . It is shown that ? sup kf  Fm .f /kLq  dM .Wp˛ ; Lq /;

kf kWp˛ D1

for 1 < p  2  q  1 and ˛ > 1p ; where Fm .f / is given by Fm .f I x/ D

X 1 X r1 rn r1 rn f . k ; : : : ; k /k;r .x1 C k ; : : : ; x1 C k /; m 1 n 1 2 2 2 2 2n k jkjDm k2Nn

0r h.I f /g;

 2 .; :

The system of Laplace integrals Z 1.arg tD / G .z/ D f .t/etz dt ; 0

z 2 … I

 2 .; 

Some Problems in Fourier Analysis and Approximation Theory

15

determines an analytic extension of the series G given by (6) to the set D D S … j 2 .; : Moreover, K D C n D D fz 2 C W zN 2 Ih g .the conjugate diagram of f / is the smallest compact convex set in C outside of which the series F admits analytic continuation. Here Ih is the indicator diagram of the entire function f (see (5)). These results by Borel and Polya as well as their one-dimensional and multidimensional analogues can be considered as a kind of a bridge between methods of the theory of entire functions and various other sections of complex analysis, in particular, summation theory of the Laurent series. The paper by Maergoiz is devoted to investigations of the Laplace-Borel integral transformation of functions equivalent to entire functions of several variables.

5.2 Optimization Control Problems Chapter 13 is Serovaisky’s “Optimization Control Problems for Systems Described by Elliptic Variational Inequalities with State Constraints”. The optimization control theory has many practical applications. Among other things, it is applicable for different mathematical models. One of the most important and difficult classes of mathematical models are the variational inequalities. Different optimization control problems of systems described by variational inequalities were considered by V. Barbu, D. Tiba, J. Bonnans, E. Casas, and other. The most difficult class of the optimization control problems are problems with state constraints. Special optimization control problems for systems described by variational inequalities have been analysed by He Zheng-Xu, G. Wang, Y. Zhao, W. Li, J. Bonnans, E. Casas. In his paper, Serovajsky considers an analogous optimization control problem with state constraint in the form of the general inclusion for the elliptic case. His analysis is based on the Warga’s concept of the search of minimizing sequences, but not optimal controls. He proposes a double regularization of the optimization control problem. At first the variational inequality, which defines the state of the system, is approximated by a nonlinear equation by using the penalty method. Then he obtains an optimization control problem for a nonlinear elliptic equation with state constraints. This problem is approximated by a minimization problem for a penalty functional on the set of admissible “state-control” pairs. J.L. Lions used the analogous technique for distributed singular systems without state constraints. However, the system is regular, and state constraints are present at this case. Lions applied penalty method for obtaining necessary conditions of optimality for the initial solvable optimization problem. Nevertheless, this problem can be unsolvable. The author uses the idea of finding minimizing sequences but not optimal control here. Warga proposed this method for the extension of unsolvable optimization controls problems. However, the author proves the solvability of his problem, and this method is applied for finding an optimal control. Thus, an approximate solution of the initial optimization control problem is chosen as the optimal control for approximate problem for a large enough step of the algorithm. Then the necessary

16

M. Ruzhansky and S. Tikhonov

conditions of optimality for the approximate optimization control problem are obtained in the standard form.

5.3 Optimization Control Problems for Parabolic Equation Chapter 14 is Ilyas Shakenov’s “Two Approximation Methods of the Functional Gradient for a Distributed Optimization Control Problem”. Practical solving of optimization control problems uses different approximation methods. Therefore, there arises a serious problem: One can use standard optimization methods (gradient methods, necessary conditions of optimality, etc.) with finite difference approximation of the state and adjoint systems. However, one can also first approximate the system and then solve the obtained discrete optimization problem. The question is: what will be more efficient, to use the approximation before or after the application of optimization methods? A. Karchevsky, V. Zubov, B. Rysbaiuly considered this question for inverse problems of mathematical physics. Here the author considers corresponding practical computing aspects of the order of approximation and optimization for an optimization control problem. The author solves an optimization control problem for the system described by a one-dimensional parabolic equation. He uses two different methods of numerical analysis for finding an optimal control. He approximates functional gradient and calculates the gradient of functional approximation. He discusses the results of computer experiments for different values of the system parameters and compares the application of different forms of the approximation.

5.4 Numerical Modeling of the Linear Filtration Chapter 15 is “Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods” by Kanat Shakenov. The numerical methods for the relaxational filtration phenomenon is an important branch of mathematics directed at practical solutions of problems of mathematical physics. Problems of relaxational filtration have been considered by Yu.M. Molokovich, P.P. Osipov, A.V. Kazhihov, V.M. Monahov, etc. For example, in the book [23], a process of non stationary filtration flow of uniform dropletcompressible monophase fluid in isotropic weakly-deformable porous environment has been considered, etc. A. Haji-Sheikh, E.M. Sparrow, S.M. Ermakov, V.V. Nekrutkin, G.A. Mihailov, and others solved a number of problems of mathematical physics by Monte Carlo methods. H. Kushner used probability difference methods for solving boundary problems for elliptic and parabolic systems. The author previously considered a process of nonstationary filtration flow of uniform droplet-compressible monophase fluid in isotropic weakly-deformable

Some Problems in Fourier Analysis and Approximation Theory

17

porous environment. He analysed four models of linear “relaxational” filtration in a one-dimensional domain. However, these models are not applicable for the practical situations. Now the author considers these four models of linear “relaxational” filtration process in a three-dimensional domain. These are the model of classical elastic filtration, the simplest model of filtration with a constant speed of disturbance spread, the filtration model in relaxationaly-compressed porous environment realised by the linear Darcy law, and the model of filtration by the simplest unbalanced law in elastic porous environment. Dirichlet, Neumann and mixed problems for these models have been solved. The author approaches these problems by using “random walk on spheres”, “random walk on balls” and “random walk on lattices” algorithms by Monte Carlo methods and by probability difference methods. He obtains and discusses the results of the computer experiments. Acknowledgements The first author “Michael Ruzhansky” was supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02. The second author “Sergey Tikhonov” was partially supported by MTM2014-59174-P, 2014 SGR 289, and RFFI 13-01-00043. We thank the authors for the assistance in writing this survey.

References 1. P. Ahern, W. Rudin, Bloch functions, BMO, and boundary zeros. Indiana Univ. Math. J. 36(1), 131–148 (1987) 2. G.A. Akishev, Approximation of function classes in spaces with mixed norm. Sb. Math. 197(7– 8), 1121–1144 (2006) 3. K. Andersen, Inequalities with weights for discrete Hilbert transforms. Can. Math. Bull. 20, 9–16 (1977) 4. P. Ausher, S. Hofmann, M. Lacey, A. McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn . Ann. Math. 2, 633–654 (2002) 5. A. Axelson, S. Keith, A. McIntosh, Quadratic estimates and functional calculi of perturbated dirac operators. Invent. Math. 163(3), 455–497 (2006) 6. E.E. Berdysheva, Two related extremal problems for entire functions of several variables. Math. Notes 66(3), 271–282 (1999) 7. H. Bohman, Approximate Fourier analysis of distribution functions. Ark. Mat. 4, 99–157 (1960) 8. E. Borel, Leçons sur les séries divergentes (Gaunter-Villars, Paris, 1901) 9. P.G. Casazza, M. Fickus, J. Kovacevic, M.T. Leon, J.C. Tremain, A physical interpretation of finite frames. Appl. Numer. Harmon. Anal. 2–3, 51–76 (2006) 10. P. Casazza, G. Kutyniok, Finite Frames: Theory and Applications (Birkhauser, New York, 2013) 11. R.R. Coifman, Y. Meyer, Wavelets, Calderon-Zygmund and Multilinear Operators (Cambridge University Press, Cambridge, 1997) 12. P. De Nápoli, I. Drelichman, Weighted convolution inequalities for radial functions. Ann. Mat. Pura Appl. 194, 167–181 (2015) 13. E. Doubtsov, Bloch-to-BMOA compositions on complex balls. Proc. Am. Math. Soc. 140(12), 4217–4225 (2012) 14. W. Ehm, T. Gneiting, D. Richards, Convolution roots of radial positive definite functions with compact support. Trans. Am. Math. Soc. 356, 4655–4685 (2004)

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15. A.S. Kechris, A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness. London Mathematical Society lecture series, vol. 128 (Cambridge University Press, Cambridge, 1987) 16. E.G. Kwon, Hyperbolic mean growth of bounded holomorphic functions in the ball. Trans. Am. Math. Soc. 355(3), 1269–1294 (2003) 17. J. Leray, Hyperbolic Differential Equations (Institute for Advanced Study, Princeton, 1953) 18. E. Liflyand, S. Tikhonov, Weighted Paley-Wiener theorem on the Hilbert transform. C.R. Acad. Sci. Paris, Ser. I 348, 1253–1258 (2010) 19. E. Liflyand, S. Tikhonov, A concept of general monotonicity and applications. Math. Nachr. 284, 1083–1098 (2011) 20. E. Liflyand, W. Trebels, On asymptotics for a class of radial Fourier transforms. Z. Anal. Anwen. 17, 103–114 (1998) 21. P.L. Lions, Symétrie e compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982) 22. B.F. Logan, Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral. SIAM J. Math. Anal. 14(2), 249–252 (1983) 23. Y.M. Molokovich, P.P. Osipov, Basics of Relaxation Filtration Theory (Kazan University, Kazan, 1987) 24. W.M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31(6), 801–807 (1982) 25. E.D. Nursultanov, N.T. Tleukhanova, On the approximate computation of integrals for functions in the spaces Wp˛ .Œ0; 1n /. Russ. Math. Surv. 55(6), 1165–1167 (2000) 26. M.G. Plotnikov, Quasi-measures, Hausdorff p-measures and Walsh and Haar Series. Izv. RAN: Ser. Mat. 74, 157–188 (2010); Engl. Transl. Izvestia: Math. 74, 819–848 (2010) 27. M. Plotnikov, V. Skvortsov, On various types of continuity of multiple dyadic integrals, Acta Math. Acad. Paedagog. Nyházi (N. S.) (2015, to appear) 28. A.N. Podkorytov, Linear means of spherical Fourier sums, in Operator Theory and Function Theory, ed. by M.Z. Solomyak, vol. 1 (Leningrad University, Leningrad) (1983), pp. 171–177 (Russian) 29. G. Polya, Untersuchungen über Lücken and Singularitäten von Potenzsrihen. Math. Zeits. Bd. 29, 549–640 (1929) 30. M.K. Potapov, B.V. Simonov, S.Yu. Tikhonov, Mixed moduli of smoothness in Lp ; 1 < p < 1: a survey. Surv. Approx. Theory 8, 1–57 (2013) 31. M.K. Potapov, B.V. Simonov, S.Yu. Tikhonov, Relations between the mixed moduli of smoothness and embedding theorems for Nikol’skii classes, in Proceeding of the Steklov Institute of Mathematics, vol. 269 (2010), pp. 197–207; translation from Russian: Trudy Matem. Inst. V.A. Steklova 269, 204–214 (2010) 32. W. Ramey, D. Ullrich, Bounded mean oscillation of Bloch pull-backs. Math. Ann. 291(4), 591–606 (1991) 33. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, New York, 1993) 34. J.H. Shapiro, Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics (Springer, New York, 1993) 35. B. Simonov, S. Tikhonov, Sharp Ul’yanov-type inequalities using fractional smoothness. J. Approx. Theory 162, 1654–1684 (2010) 36. V.A. Skvortsov, Henstock-Kurzweil type integrals in P -adic harmonic analysis. Acta Math. Acad. Paedagog. Nyházi (N. S.) 20, 207–224 (2004) 37. W.A. Strauss, Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977) 38. S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326, 721–735 (2007)

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39. S. Tikhonov, Best approximation and moduli of smoothness: computation and equivalence theorems. J. Approx. Theory 153, 19–39 (2008) 40. S. Tikhonov, Weak type inequalities for moduli of smoothness: the case of limit value parameters. J. Fourier Anal. Appl. 16(4), 590–608 (2010) 41. P. Villarroya, A characterization of compactness for singular integrals. J. Math. Pure Appl. (arXiv:1211.0672) (to appear) 42. S. Yamashita, Hyperbolic Hardy class H 1 . Math. Scand. 45(2), 261–266 (1979)

Part I

Fourier Analysis

Parseval Frames with n C 1 Vectors in Rn Laura De Carli and Zhongyuan Hu

Abstract We prove a uniqueness theorem for triangular Parseval frame with n C 1 vectors in Rn . We also provide a characterization of unit-norm frames that can be scaled to Parseval frames. Keywords Parseval frames • Scalable frames

Mathematics Subject Classification (2000). Primary 42C15, Secondary 46C99

1 Introduction Let B D fv1 ; : : : ; vN g be a set of vectors in Rn . We say that B is a frame if it contains a P basis of Rn , or equivalently, if there exist constants A; B > 0 for which 2 Ajjvjj  NjD1 < v; vj >2  Bjjvjj2 for every v 2 Rn . Here and throughout the paper, < ; > and jj jj are the usual scalar product and norm in Rn . In general A < B, but we say that a frame is tight if A D B, and is Parseval if A D B D 1. Parseval frames are nontrivial generalizations of orthonormal bases. Vectors in a Parseval frame are not necessarily orthogonal or linearly independent, and do not P necessarily have the same length, but the Parseval identities v D NjD1 < v; vj > vj PN 2 and jjvjj2 D jD1 < v; vj > still hold. In the applications, frames are more useful than bases because they are resilient against the corruptions of additive noise and quantization, while providing numerically stable reconstructions [6, 7, 9]. Appropriate frame decomposition may reveal hidden signal characteristics, and have been employed as detection devices. Specific types of finite tight frames have been

L. De Carli () Department of Mathematics, Florida International University, Miami, FL 33199, USA e-mail: [email protected] Z. Hu Department of Economics, Florida International University, Miami, FL 33199, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_2

23

24

L. De Carli and Z. Hu

studied to solve problems in information theory. The references are too many to cite, but see [4], the recent book [2] and the references cited there. In recent years, several inquiries about tight frames have been raised. In particular: how to characterize Parseval frames with N elements in Rn (or Parseval N frames), and whether it is possible to scale a given frame so that the resulting frame is Parseval. Following [1, 11], we say that a frame B D fv1 ; : : : ; vN g is scalable if there exists positive constants `1 , . . . , `N such that f`1 v1 ,. . . , `N vN g is a Parseval frame. Two Parseval Nframes are equivalent if one can be transformed into the other with a rotation of coordinates and the reflection of one or more vectors. A frame is nontrivial if no two vectors are parallel. In the rest of the paper, when we say “unique” we will always mean “unique up to an equivalence”, and we will often assume without saying that frames are nontrivial. It is well known that Parseval nframes are orthonormal (see also Corollary 3.3). Consequently, for given unit vector w, there is a unique Parseval nframe that contains w. If jjwjj ¤ 1, no Parseval nframe contains w. When N > n and jjwjj  1, there are infinitely many non-equivalent Parseval N-frames that contain w.1 By the main theorem in [3], it is possible to construct a Parseval frame fv1 ; : : : ; vN g with vectors of prescribed lengths 0 < `1 , . . . , `N  1 that satisfy `11 C : : : C `2N D n. We can let `N D jjwjj and, after a rotation of coordinates, assume that vN D w, thus proving that the Parseval frames that contain w are as many as the sets of constants `1 , . . . `N1 . But when N D n C 1, there is a class of Parseval frames that can be uniquely constructed from a given vector: precisely, all triangular frames, that is, frames fv1 ; : : : ; vN g such that the matrix .v1 ; : : : vn / whose columns are v1 , . . . , vn is right triangular. We recall that a matrix fai;j g1i; jn is right-triangular if ai;j D 0 if i > j. The following theorem will be proved in Sect. 3. Theorem 1.1 Let B D fv1 ; : : : ; vn ; wg be a triangular Parseval frame, with jjwjj < 1. Then B is unique, in the sense that if B 0 D fv10 ; : : : ; vn0 ; wg is another triangular Parseval frame, then vj0 D ˙vj . Every frame is equivalent, through a rotation of coordinates , to a triangular frame, and so Theorem 1.1 implies that every Parseval .n C 1/-frame that contains a given vector w is equivalent to one which is uniquely determined by .w/. However, that does not imply that the frame itself is uniquely determined by w because the rotation depends also on the other vectors of the frame. We also study the problem of determining whether a given frame B D fv1 ; : : : ; vn ; vnC1 g  Rn is scalable or not. Assume jjvj jj D 1, and let i;j 2 Œ0; / be the angle between vi and vj . If B contains an orthonormal basis, then the problem has no solution, so we assume that this not the case. We prove the following

1

We are indebted to P. Casazza for this remark.

Parseval Frames in Rn

25

Theorem 1.2 B is scalable if and only there exist constants `1 , . . . , `nC1 such that for every i ¤ j .1  `2i /.1  `2j / D `2i `2j cos i;j2 :

(1)

The identity (1) has several interesting consequences (see corollary 3.2). First of all, it shows that `2j  1; if `i D 1 for some i, we also have cos i;j2 D 0 for every j, and so vi is orthogonal to all other vectors. This interesting fact is also true for other Parseval frames, and is a consequence of the following. Theorem 1.3 Let B D fv1 ; : : : ; vN g be a Parseval frame. Let jjvj jj D `j . Then N X

`2j cos2 ij D 1

jDi

N X

`2j sin2 ij D n  1:

(2)

jDi

The identities (2) are probably known, but we did not find a reference in the literature. It is worthwhile to remark that from (2) follows that N X

`2j cos2 ij 

jDi

N X jDi

`2j sin2 ij D

N X

`2j cos.2ij / D 2  n:

jDi

When n D 2, this identity is proved in Proposition 2.1. Another consequence of Theorem 1.1 is the following Corollary 1.4 If B D fv1 ; : : : ; vn ; vnC1 g is a scalable frame, then, for every 1  i  n C 1, and every j ¤ k ¤ i and k0 ¤ j0 ¤ i, j cos k0 ;j0 j j cos k;j j D : j cos k;j j C j cos k;i cos j;i j j cos k0 ;j0 j C j cos k0 ;i cos j0 ;i j

(3)

We prove Theorems 1.1, 1.2 and 1.3 and their corollaries in Sect. 3. In Sect. 2 we prove some preliminary results and lemmas.

2 Preliminaries We refer to [2] or to [10] for the definitions and basic properties of finite frames. n We recall that B D fv1 ; : : : ; vN g is a Parseval frames in RP if and only if the N 2 rows of the matrix .v1 ; : : : ; vN / are orthonormal. Consequently, iD1 jjvi jj D n: If p the vectors in B have all the same length, then jjvi jj D n=N. See e.g. [2]. We will often let Ee1 D .1; 0; : : : ; 0/, . . . Een D .0; : : : ; 0; 1/, and we will denote by .v1 ; : : : ; vO k ; : : : vN / the matrix with the column vk removed.

26

L. De Carli and Z. Hu

To the best of our knowledge, the following proposition is due to P. Casazza (unpublished, 2000) but can also be found in [8] and in the recent preprint [5]. Proposition 2.1 B D fv1 ; : : : ; vN g  R2 is a tight frame if and only if for some index i  N, N X

jjvj jj2 e2ii;j D 0:

(4)

jD1

It is easy to verify that if (4) is valid for some index i, then it is valid for all other i’s. Proof Let `j D jjvj jj. After a rotation, we can let vi D v1 D .`1 ; 0/ and 1;j D j , so that vj D .`j cos j ; `j sin j /. B is a tight frame with frame constant A if and only if the rows of the matrix .v1 ; : : : ; vN / are orthogonal and have length A. That implies N X

`2j cos2 j D

jD1

N X

`2j sin2 j D A

(5)

jD1

and N X

`2j cos j sin j D 0:

(6)

jD1

P PN 2 From (5) follows that NjD1 `2j .cos2 j  sin2 j / D jD1 `j cos.2j / D 0, and PN 2 from (6) that jD1 `j sin.2j / D 0, and so we have proved (4). If (4) holds, then (5) and (6) hold as well, and from these identities follows that B is a tight frame. t u Corollary 2.2 Let B D fv1 ; v2 ; v3 g  R2 be a tight frame. Assume that the vi ’s have all the same length. Then, 1;2 D =3, and 1;3 D 2=3. frame B p is equivalent to a dilation of the “Mercedes-Benz frame” n So, every such o p 3 3 1 1 .1; 0/; . 2 ; 2 /; . 2 ;  2 / : Proof Let v1 D .1; 0/, and 1;i D i for simplicity. By Proposition 2.1, 1 C cos.22 / C cos.23 / D 0, and sin.22 / C sin.23 / D 0. It is easy to verify that these equations are satisfied only when 2 D 3 and 3 D 2 t u 3 or viceversa. The following simple proposition is a special case of Theorem 1.1, and will be a necessary step in the proof. Lemma 2.3 Let w D .˛1 ; ˛2 / be given. Assume jjwjj < 1. There exists a unique nontrivial Parseval frame fv1 ; v2 ; wg  R2 , with v1 D .a11 ; 0/, v2 D .a1;2 ; a2;2 /, and a1;1 , a2;2 > 0.

Parseval Frames in Rn

27

 Proof We find a1;1 , a1;2 and a2;2 so that the rows of the matrix

a11 ; a12 ; ˛1 0; a22 ; ˛2

 are

orthonormal. That is, ˛12 C a21;1 C a21;2 D 1;

˛22 C a22;2 D 1;

˛1 ˛2 C a1;2 a2;2 D 0:

(7)

q From the second equation, a2;2 D ˙ 1  ˛22 ; if we can chose a2;2 > 0, from the third equation we obtain a1;2 D  p˛1 ˛2 2 and from the first equation 1˛2

a21;1 D 1  ˛12  a212 D 1  ˛12 

˛12 ˛22 1  ˛12  ˛22 D : 1  ˛22 1  ˛22

Note that a21;1 > 0 because jjwjj2 D ˛12 C ˛22 < 1. We can chose then

a1;1

q 1  ˛22  ˛12 D q : 1  ˛22

Note also that v and v2 cannot be parallel; otherwise, ˛22

D1

˛22 ,

a1;2 a2;2

˛1 ˛2 D  1˛ 2 D

which is not possible.

2

˛1 ˛2

” t u

Remark The proof shows that v1 and v2 are uniquely determined by w. It shows also that if jjwjj D 1, then a1;1 D 0, and consequently v1 D 0.

3 Proofs In this section we prove Theorem 1.1 and some of its corollaries. Proof of Theorem 1.1 Let w D .˛1 ; : : : ; ˛n /. We construct a nontrivial Parseval frame M D fv1 ; : : : ; vn ; wg  Rn with the following properties: the matrix .v1 ; : : : ; vn / D fai;j g1i;jn is right triangular, and

aj;j D

8p ˆ < 1  ˛n2

q P 1 nkDj ˛k2 ˆ : q Pn 1 kDjC1 ˛k2

if j D n if 1  j < n:

(8)

The proof will show that M is unique, and also that the assumption that jjwjj < 1 is necessary in the proof. To construct the vectors vj we argue by induction on n. When n D 2 we have already proved the result in Lemma 2.3. We now assume that the lemma is valid in dimension n  1, and we show that it is valid also in dimension n.

28

L. De Carli and Z. Hu

Let wQ D .˛2 ; : : : ; ˛n /. By assumptions, there exist vectors vQ 2 ; : : : ; vQn such f D fvQ 2 ; : : : ; vQ n ; wg that the set M Q  Rn1 is a Parseval frame, and the matrix .vQ 2 ; : : : ; vQn ; w/ Q is right triangular and invertible. If we assume that the elements of the diagonal are positive, the vQ j ’s are uniquely determined by w. We let vQ j D .a2;j ; : : : ; an;j /, with ak;j D 0 if k < j and aj;j > 0. f is the projection on Rn1 of a Parseval frame in Rn D R  Rn1 We show that M that satisfies the assumption of the theorem. To this aim, we prove that there exist scalars x1 ,. . . , xn so that the vectors fv1 ; : : : ; vnC1 g which are defined by v1 D .x1 ; 0; : : : ; 0/;

vj D .xj ; vQj / if 2  j  n; vnC1 D w

(9)

form a Parseval frame of Rn . The proof is in various steps: first, we construct a unit vector .y2 ; : : : ; ynC1 / which is orthogonal to the rows of the matrix .vQ 2 ; : : : ; vQ n ; w/. Q Then,p we show that there exists 1 <  < 1 so that ynC1 D ˛1 . Finally, we chose x1 D 1  2 , xj D yj , and we prove that the vectors v1 ; : : : ; vnC1 defined in (9) form a Parseval frame that satisfies the assumption of the lemma. First of all, we observe that fv1 ; : : : ; vnC1 g is a Parseval frame if and only if Ex D .x1 ; x2 ; : : : ; xn ; ˛1 / is a unit vector that satisfies the orthogonality conditions: 0 a22 B0 B Q xDB : .vQ 2 ; : : : vQn ; w/E @ :: 0

a23 a33 :: :

::: ::: :: :

a2;n a3;n :: :

10 1 0 1 x2 0 ˛2 B :: C B0C ˛3 C CB : C B C :: C B CD B :: C : : A @x A @ : A

(10)

n

0 : : : an;n ˛n

˛1

0

By a well known formula of linear algebra, the vector 0

Ey D y2 Ee2 C : : : C EenC1 ynC1

eE2 Ba B 2;2 B B 0 D det B B :: B : B @ 0 0

Ee3 a2;3 a3;3 :: :

: : : Een : : : a2;n : : : a3;n :: :: : : 0 : : : an1;n 0 : : : an;n

1 eEnC1 ˛2 C C C ˛3 C :: C C : C C ˛n1 A ˛n

(11)

is orthogonal to the rows of the matrix in (10), and so it is a constant multiple of Ex. That is, Ex D Ey for some  2 R. Let us prove that jjEyjj D 1. The rows of the matrix .vQ 2 ; : : : ; vQ n ; w/ Q are orthonormal, and so after a rotation 0

1 0 0C C :: C : :C C 0 : : : 1 0A 0 0 ::: 0 1

0 B0 B B .vQ 2 ; : : : ; vQn ; w/ Q D B ::: B @0

1 ::: 0 ::: :: :: : :

0 0 :: :

(12)

Parseval Frames in Rn

29

The formula in (11) applied with the matrix in (12) produces the vector Ee1 D .1; 0; : : : ; 0/. Thus, Ey in (11) is a rotation of Ee1 , and so it is a unit vector as well. We now prove that jj < 1. From Ex D .x2 ; : : : ; xn ; ˛1 / D .y2 ; : : : ; yn ; ynC1 /, we obtain  D ˛1 =ynC1 : By (11), 0 ynC1 D .1/

nC1

a2;2 0 :: :

B B B det B B @ 0 0

Recalling that by (8), aj;j D

ynC1 D .1/nC1 q

: : : a2;n1 : : : a3;n1 :: :: : : 0 : : : an1;n1 0 ::: 0

a2;3 a3;3 :: :

q P 1 nkDj ˛k2 q P , 1 nkDjC1 ˛k2

n Y

a2;n a3;n :: :

1

C n C Y C aj;j : C D .1/nC1 C jD2 A an1;n an;n

we can see at once that

q 2 aj;j D .1/nC1 1  ˛22  : : :  ˛n1  ˛n2

jD2

D.1/nC1 1  jjwjj2 C ˛12 :

(13)

In view of ynC1 D ˛1 , we obtain  D .1/nC1 q

˛1 1  jjwjj2 C ˛12

:

Clearly, jj < 1 because jjwjj < 1. We now let p x 1 D 1  2 D q

p

1  jjwjj2

1  jjwjj2 C ˛12

;

(14)

and p we define thepvj ’s as in (9). The first rows of the matrix .v1 ; : : : ; vnC1 / is . 1  2 ; Ex/ D . 1  2 ; Ey/, and so it is unitary and perpendicular to the other rows. Therefore, the fvj g form a tight frame that satisfies the assumption of the theorem. t u The proof of Theorem 1.1 shows the following interesting fact: By (13) and (14) det.v1 ; : : : ; vn / D

n Y jD1

ajj D x1

n Y jD2

ajj D

p 1  jjwjj2 :

30

L. De Carli and Z. Hu

This formula does not depend on the fact that .v1 ; : : : ; vn / is right triangular, because every n  n matrix can be reduced in this form with a rotation that does not alter its determinant and does not alter the norm of w. This observation proves the following Corollary 3.1 Let fw1 ; : : : ; wnC1 g be a Parseval frame. Then, q det.w1 ; : : : ; wO j ; : : : ; wnC1 / D ˙ 1  jjwj jj2 : Proof of Theorem 1.2 Let `j D jjvj jj. If fv1 ; : : : ; vnC1 g is a Parseval frame, then the rows of the matrix B D .v1 ; : : : ; vnC1 / are orthonormal. While proving Theorem 1.1, we have constructed a vector Ex D .x1 ; : : : ; xnC1 /, with xj D .1/jC1  det.v1 ; q : : : ; vO j ; : : : ; vnC1 /, which is perpendicular to the rows of B. By Corollary 3.1,

xj D ˙ 1  `2j . Since B is a Parseval frame, `21 C : : : C `2nC1 D n, and so jjExjj2 D x21 C : : : C x2nC1 D .1  `21 / C : : : C .1  `2nC1 / D 1:

So, the .n C 1/  .n C 1/ matrix BQ which is obtained from B with the addition of the row Ey, is unitary, and therefore also the columns of BQ are orthonormal. For every i; j  n C 1, hvi ; vj i ˙

q

q q q 1  `2i 1  `2j D `i `j cos ij ˙ 1  `2i 1  `2j D 0

(15)

which implies (1).

q Conversely, suppose that (1) holds. By (15), the vectors vQj D .˙ 1  `2j ; vj / are orthonormal for some choice of the sign ˙; therefore, the columns of the matrix BQ are orthonormal, and so also the rows are orthonormal, and B is a Parseval frame. t u Corollary 3.2 Let B D fv1 ; : : : vnC1 g be a nontrivial Parseval frame. Then, `2j for every j. Moreover, for all j with the possible exception of one, 12 < `2j .

1 nC1




1 . nC1

t u

Parseval Frames in Rn

31

Proof of Theorem 1.3 After a rotation, we can assume vi D v1 D .`1 ; 0; : : : ; 0/. We let 1;j D j for simplicity. With this rotation vj D .`j cos j ; `j sin j wj / where wj is a unitary vector in Rn1 . The rows of the matrix .v1 ; : : : vN / are orthonormal, and so the norm of the first row is X `2j cos2 j C `21 D 1 (17) j1

The projections of v2 , . . . . vN over a hyperplane that is orthogonal to v1 form a tight frame on this hyperplane. That is to say that f`2 sin 2 w2 ; : : : ; `N sin N wN g is a tight frame in Rn1 , and so it satisfies `22 sin 22 jjw2 jj22 C : : : C `2N sin N2 jjwN jj22 D `22 sin 22 C : : : C `2N sin N2 D n  1:

(18) t u

Corollary 3.3 fv1 ; v2 ; : : : ; vn g is a Parseval frame in Rn if and only if the vi ’s are orthonormal. Proof By (17), all vectors in a Parseval frame have length  1. By (18) n X

`2j sin2 i;j D n  1

jD1

which implies that `j D 1 and sin ij D 1 for every j ¤ i, and so all vectors are orthonormal. u t Proof of Corollary 1.4 Assume that B is scalable; fix i < nC1, and chose j ¤ k ¤ i. By 1 .1  `2i /.1  `2j / D `2i `2j cos i;j2 ; 2 ; .1  `2i /.1  `2k / D `2i `2k cos i;k 2 : .1  `2k /.1  `2j / D `2k `2j cos k;j

These equations are easily solvable for `21 ; `2j and `2k ; we obtain `2i D

j cos k;j j : j cos k;j j C j cos k;i cos j;i j

This expression for `i must be independent of the choice of j and k, and so (3) is proved. u t

32

L. De Carli and Z. Hu

Acknowledgements We wish to thank Prof. P. Casazza and Dr. J. Cahill for stimulating conversations.

References 1. J. Cahill, X. Chen, A note on scalable frames. ArXiv:1301.7292v1 (2013) 2. P. Casazza, G. Kutyniok, Finite Frames: Theory and Applications (Birkhauser, New York, 2013) 3. P. Casazza, M.T. Leon, Existence and construction of finite tight frames. J. Concr. Appl. Math. 4(2), 277–289 (2006) 4. P.G. Casazza, M. Fickus, J. Kovacevic, M.T. Leon, J.C. Tremain, A physical interpretation of finite frames. Appl. Numer. Harmon. Anal. 2(3), 51–76 (2006) 5. M. Copenhaver, Y. Kim, C. Logan, K. Mayfield, S.K. Narayan, M. Petro, J. Sheperd, Diagram vectors and tight frame scaling in finite dimensions. Oper. Matrices 8(1), 73–88 (2014) 6. Z. Cvetkovic, Resilience properties of redundant expansions under additive noise and quantization. IEEE Trans. Inf. Thesis (2002) 7. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992) 8. V.K. Goyal, J. Kovacevic, Quantized frames expansion with erasures. Appl. Comput. Harmon. Anal. 10, 203–233 (2001) 9. V.K Goyal, M. Vetterli, N.T. Thao, Quantized overcomplete expansions in RN : analysis, synthesis, and algorithms. IEEE Trans. Inf. Theory 44(1), 16–31 (1998) 10. D. Han, K. Kornelson, D. Larson, E. Weber, Frames for Undergraduates. Student Mathematical Library Series (American Mathematical Society, Providence, 2007) 11. G. Kutyniok, K. Okoudjou, F. Philipp, E. Tuley, Scalable frames. Linear Algebra Appl. 438(5), 2225–2238 (2013)

Hyperbolic Hardy Classes and Logarithmic Bloch Spaces Evgueni Doubtsov

Abstract Let ' be a holomorphic mapping between complex unit balls. We use the composition operators C' W f 7! f ı ' to relate the hyperbolic Hardy classes and the logarithmic Bloch spaces. Keywords Hardy space • Logarithmic Bloch space

Mathematics Subject Classification (2000). Primary 32A35, Secondary 32A18, 47B33

1 Introduction 1.1 Hyperbolic Hardy Classes Let H.Bn / denote the space of holomorphic functions on the unit ball Bn of Cn , n 2 N. For 0 < p < 1, the Hardy space H p .Bn / consists of f 2 H.Bn / such that Z p

k f kH p .Bn / D sup

0r 0, the hyperbolic Hardy class Hh D Hh .Bn ; Bm / consists of those holomorphic mappings ' W Bn ! Bm for which Z sup

0r 0 such that jhR'.z/; '.z/ij  jR'.z/jj'.z/j

when s < j'.z/j < 1:

(2)

The above definition was used in [6] to study composition operators on B.Bm /; a very close notion was earlier introduced in the Bloch-to-BMOA setting (see [3]). In fact, we show that the scheme applied in [6] is adaptable to the logarithmic Bloch spaces. The main result of the present paper is the following theorem. Theorem 1.2 Let ˛ < 12 , 0 < p  1 and let ' W Bn ! Bm be a regular holomorphic mapping. Then the following properties are equivalent: C' W L˛ B.Bm/ ! H 2p .Bn / is a bounded operatorI .12˛/p

.Bn ; Bm /I ' 2 Hh  2˛ Z 1 e jR'.r/j2 .1  r/ dr 2 Lp .@Bn /: log 2 /2 2 .1  j'.r/j .1  j'.r/j 0

(3) (4) (5)

Since every holomorphic mapping ' W Bn ! B1 is regular, we obtain, as a corollary, the following direct extension of Theorem 1.1 for 0 < p  1: Corollary 1.3 Let ˛ < 12 , 0 < p  1 and let ' W Bn ! B1 be a holomorphic mapping. Then the following properties are equivalent: C' W L˛ B.B1/ ! H 2p .Bn / is a bounded operatorI .12˛/p

.Bn ; B1 /I ' 2 Hh  2˛ Z 1 e jR'.r/j2 log .1  r/ dr 2 Lp .@Bn /: 2 2 .1  j'.r/j2 0 .1  j'.r/j /

(6) (7) (8)

36

E. Doubtsov

1.4 Comments 1.4.1 Property (7) indicates that the parameter ˛ D 12 is critical. In fact, if ˛ > 12 , then L˛ B.B1 /  H q .B1 / for all q > 0. In particular, property (6) with ˛ > 12 holds for all holomorphic mappings ' W Bn ! B1 . The case ˛ D 12 is related to the hyperbolic Nevanlinna class (cf. [4]). 1.4.2 Let ˛ < 12 and let 0 < p  1. If the product .1  2˛/p is fixed, then Corollary 1.3 guarantees that the operators C' W L˛ B.B1 / ! H 2p .Bn / are bounded for the .12˛/p same collection of mappings '. Remark that the property ' 2 Hh .Bn ; B1 / also characterizes the bounded composition operators C' between appropriate pairs of growth and Hardy spaces (see [1]).

1.5 Notation We denote by C an absolute positive constant whose value may change from line to line. The notation C.p; q; : : : / indicates that the constant depends on the parameters p; q; : : : . The symbol C' is reserved for a composition operator.

1.6 Organization of the Paper Section 2 contains preliminary technical results related to reverse estimates in the logarithmic Bloch spaces, properties of regular mappings and applications of Green’s formula. Theorem 1.2 is proved in Sect. 3.

2 Auxiliary Results 2.1 Reverse Estimates The key technical tool of the present paper is the following lemma which provides reverse estimates, that is, integral estimates from below for appropriate test functions. Lemma 2.1 ([5, 9]) Let m 2 N, ˛ < 12 and let 0 < q < 1. Then there exist functions Fx 2 L˛ B.Bm /, 0  x  1, such that kFx kL˛ B.Bm /  1 and Z

1 0

 jFx .w/jq dx   log

. 12 ˛/q 1 ; 1  jwj2

w 2 Bm ;

(9)

Hyperbolic Hardy Classes

37

for a constant  D .m; ˛; q/ > 0.

  It is shown in [5] that the exponent 12  ˛ q in (9) is sharp. Corollary 1.3 also indicates that Lemma 2.1 is sharp for 0 < q  2, since estimate (9) guarantees that (6) implies (7); see Sect. 3.

2.2 Hardy Spaces The classical Littlewood–Paley characterization of H p .B1 / extends to H p .Bn / for all n 2 N. Theorem 2.2 (cf. [2, Theorem 3.1]) Let q > 0 and let n 2 N. Then there exist constants C1 ; C2 > 0 such that Z q C1 k f kH q .Bn /

 jf .0/j C

Z

1

q

@Bn

0

2

jRf .r/j .1  r/ dr

 q2

q

dn ./  C2 k f kH q .Bn /

for all f 2 H.Bn /.

2.3 Estimates for Regular Holomorphic Mappings Given a holomorphic mapping ' W Bn ! Bm , put ˆ.z/ D log

e ; 1  j'.z/j2

z 2 Bn :

Lemma 2.3 Let 0 < q < 1 and let ' W Bn ! Bm be a regular holomorphic mapping. Then  ˆq  C.s; ; q/ log

e 1  j'j2

q1

jR'j2 ; .1  j'j2 /2

where s 2 .0; 1/ and  > 0 are the constants provided by property (2).

(10)

38

E. Doubtsov

Proof For k D 1; : : : ; n, we have 2

1 @ˆ q @zk @zk q

 D .q  1/ log  C log

e 1  j'j2

e 1  j'j2

q1

ˇ2 ˇ q2 ˇˇh @' ; 'iˇˇ @zk .1  j'j2 /2 ˇ ˇ2 ˇ ˇ2 ˇ @' ˇ ˇ @' ˇ ˇ @zk ˇ .1  j'j2 / C ˇh @zk ; 'iˇ .1  j'j2 /2

D Sk;1 C Sk;2 : Also, observe that ˇ n ˇ X ˇ @' ˇ2 2 ˇ ˇ ˇ @z ˇ  jR'j I kD1

(11)

k

ˇ2 X ˇ2 n ˇ n ˇ 2 X ˇ @' ˇ @' ˇ ˇ ˇh ; 'iˇ  ˇhzk ˇ  jhR'; 'ij : ; 'i ˇ @z ˇ @z ˇ ˇ n k k kD1 kD1

(12)

First, assume that 1  q < 1. Let the constants s 2 .0; 1/ and  > 0 be those provided by property (2). To estimate ˆq from below, we replace Sk;1 C Sk;2 by Sk;2 . So, if s < j'.z/j < 1, then (2), (11) and (12) imply (10). Clearly, estimate (10) holds when j'.z/j  s. Hence, the proof of the lemma is finished for 1  q < 1. Second, assume that 0 < q < 1. To estimate ˆq from below, we replace Sk;1 by  .q  1/ log

e 1  j'j2

ˇ2 ˇ q1 ˇˇh @' ; 'iˇˇ @zk .1  j'j2 /2

:

Therefore, 2

@ ˆ @zk @zk q

  q log

e 1  j'j2

ˇ ˇ2 ˇ ˇ2 q1 ˇˇ @' ˇˇ .1  j'j2 / C q ˇˇh @' ; 'iˇˇ @zk @zk .1  j'j2 /2

:

Hence, repeating the arguments used in the case 1  q < 1, we obtain (10).

t u

2.4 A Maximal Function Estimate Lemma 2.3 guarantees that ˆq is a subharmonic function for any q > 0. So, we have the following lemma:

Hyperbolic Hardy Classes

39

Lemma 2.4 ([6, Lemma 4.7]) Let 0 < q < 1. For 0 < < 1 and a holomorphic mapping ' W Bn ! Bm , consider the radial maximal function Mˆ ./ D sup ˆ .r/ D sup ˆ. r/; 0r 0 9 A" > 0W

j f .z/j  A" e2.1C"/kzkCı

8 z 2 Cn :

On the other hand, if the restriction of an entire function of exponential type Cı to Rn is a real-valued function belonging to L1 .Rn /, then f 2 Fn .C/. This inverse result follows from [25, Theorem 4.9]. Functions of type Bn with  > 0 are also called entire functions of exponential spherical type 2 [23]. One example of such functions is the normalized Bessel

Multidimensional Extremal Logan’s and Bohman’s Problems

45

function [5, Chap. VII] j˛ .2jxj/ D

Z

1 !n1

e.xs/ ds; Sn1

x 2 Rn ;

˛ D n=2  1;

where !n1 D 2 n=2 = .n=2/ is the surface area of Sn1 , j˛ .u/ D .˛ C 1/.2=u/˛ J˛ .u/: We denote by q˛ the smallest positive zero of j˛ .u/. For n D 1, we have ˛ D 1=2, j1=2 .u/ D cos u and q1=2 D =2. 1. Let us formulate extremal Logan’s problem. Suppose U; V 2 Cn . Let us introduce the quantity l. f /U D supfkxkU W f .x/ > 0g: This quantity is equal to the radius of the smallest ball in the norm of the body U outside which the function f is nonpositive. For brevity, define l. f /p D l. f /Bnp , l. f / D l. f /2 . Consider the following classes of functions: Ln .V/ D ff 2 Fn .V/W f .0/ D 1; fO .0/  0g; O LC n .V/ D ff 2 Ln .V/W f  0g  Ln .V/: From the definition of the class LC n .V/ it follows that it coincides with the class 1 n ı of positive-definite entire functions R R f 2 L .R / of exponential type V such that supp fO  V, Rn f .x/ dx  0 and V fO .s/ ds D 1. The class LC n .V/ is nonempty and contains functions f for which l. f /U < 1 (see, for example, Yudin’s function (14)). Extremal Logan’s problem consists in the calculation of the quantity C LC n .V/U D inffl. f /U W f 2 Ln .V/g:

Similarly, we can define the quantity Ln .V/U . Obviously, Ln .V/U  LC n .V/U :

(2)

The quantities Ln .V/U and LC n .V/U are homogeneous. Indeed, if f .x/ D f .x/,  > 0, then fO .s/ D  n fO . 1 s/ [25, Chap. I], l. f /U D  1 l. f /U D l. f / 1 U ;

supp fO D  supp fO :

46

D.V. Gorbachev

Thus, for example, Ln .V/U D  1 Ln .V/U D Ln .V/ 1 U :

(3)

Let f .x/  0 for x … rBnC . If kxkC  r, then kxkC0  Mn kxkC  rMn , where Mn D Mn .C0 /C . So rBnC  rMn BnC0 and f .x/  0 for x … rMn BnC0 . Consequently, l. f /C0  Mn l. f /C and, for example, 0 C LC n .V/C0  Mn .C /C Ln .V/C :

(4)

For n D 1, Logan [22] proved that  LC 1 .Œ; / D l. f / D

1 ; 2

 > 0;

where f  .x/ D

cos2 .x/ 2 LC 1 .Œ; /; 1  .2x/2

fO .s/ D

 jsj   sin Œ;  .s/: 4 

(5)

This result also follows from [3, 8]. In the multidimensional case, Logan’s problem was studied by E. Berdysheva and the author. Let n 2 N,  > 0. Theorem 1.1 ([6]) Let 1  p  2. If V D Bn1 and U D Bnp , then n n Ln .Bn1 /p D LC n .B1 /p D l.F B1 / D

p n ; 2

where F

 Bn1

n  2 2 jxj2  Y cos.xj / 2 n .x/ D 1  2 LC n .B1 /: 2 n 1  .2x / j jD1

Theorem 1.2 ([13]) If V D Bn and U D Bn , then n Ln .Bn / D LC n .B / D l.F Bn / D

q˛ ; 

where F Bn .x/ D

j2˛ .jxj/ n 2 LC n .B /: 1  .jxj=q˛ /2

In both cases, the extremal function is Yudin’s function FC 2 LC n .C/ (see Sect. 3, (14)). Note that l.FC / D 1 .C/=, where 1 .C/ is the first eigenvalue of

Multidimensional Extremal Logan’s and Bohman’s Problems

47

the eigenvalue problem [11]  u.x/ D  2 u.x/;

x 2 C;

uj@C D 0:

(6)

Here  D @2 =@x21 C    C @2 =@x2n is the Laplacian. To prove the desired lower estimates, the authors use the multidimensional Poisson summation formula in the case of Theorem 1.1 and the one-dimensional Gaussian quadrature formula based on the zeros of the Bessel function in the case of Theorem 1.2. We shall discuss the following result of Yudin [28]: 2R.L / .L/ 

 qn=21 n  D 1 C O.n2=3 / ;  2

n ! 1;

(7)

where L  Rn is a lattice of rank n, .L/ is the packing radius of L, R.L / is the covering radius of the dual lattice L . In particular, we shall show the connection between inequalities of type (7) and Logan’s problem. Logan’s problem allows us to get sharp Jackson’s inequality in the space L2 .Rn / [27]. By the best approximation of a function f 2 L2 .Rn / we mean the quantity  ˚ EU . f / D inf k f  gkL2 .Rn / W 8 g 2 L2 .Rn /; supp gO  BnU : The modulus of continuity of a function f 2 L2 .Rn / is defined by the equality  ˚ !.V; f / D sup k f .x C t/  f .x/kL2 .Rn / W t 2 V : It follows from [6] that EU . f /  21=2 !.LC n .V/U ; f /

8 f 2 L2 .Rn /;

where 21=2 is a sharp constant and argument in the modulus of continuity is optimal. Thus, in two specific situations Yudin’s problem [27] to determine the optimal argument in the sharp Jackson inequality in the space L2 .Rn / can be solved by Theorems 1.1 and 1.2. Logan’s problem is closely related to the Turán, Delsarte, and other extremal problems [1, 2, 9, 12–15, 18–20, 24]. 2. To solve extremal Bohman’s problem, we can apply the methods used to solve Logan’s problem. Let us formulate the Bohman’s problem. Let V 2 Cn , Bn .V/ D ff 2 Fn .V/W jxj2 f 2 L1 .Rn /; fO .0/ D 1; f  0g; BnC .V/ D ff 2 Bn .V/W fO  0g  Bn .V/:

48

D.V. Gorbachev

In terms of probability theory, the class Bn .V/ coincides with the set of probability density functions f such that ' D fO 2 C2 .Rn / and supp '  V. Recall that ' is called a characteristic function [12]. The quantity Z b. f / D

Rn

jxj2 f .x/ dx D .2/2 fO .0/

is called the second moment of probability density function f [12]. Bohman’s problem consists in the determination of the quantity Bn .V/ D inffb. f /W f 2 Bn .V/g: C Similarly, we can define the quantity BC n .V/. Obviously, Bn .V/  Bn .V/. Setting f .x/ D  n f .x/, we have fO .s/ D fO . 1 s/, b. f / D  2 b. f /, and, for example,

Bn .V/ D  2 Bn .V/:

(8)

For n D 1, Bohman [7] proved that B1 .Œ; / D b.g / D

 1 2 ; 2

 > 0;

where 8  cos .x/ 2 2 B1C .Œ; /;  2 1  .2x/2  jsj   jsj o n 1 jsj  gO .s/ D 1  cos C sin Œ;  .s/:     g .x/ D

(9)

It follows that B1 .Œ; / D BC 1 .Œ; /. W. Ehm, T. Gneiting and D. Richards solved multidimensional Bohman’s problem for the Euclidean ball Bn . Theorem 1.3 ([12]) For n 2 N and  > 0, n Bn .Bn / D BC n .B / D b.G Bn / D

 q 2 ˛



;

where the extremal function is of the form G Bn .x/ D

2 !n1  n  j˛ .jxj/ 2 BnC .Bn /: 2n1 q2˛ 1  .jxj=q˛ /2

Multidimensional Extremal Logan’s and Bohman’s Problems

49

We shall prove that n n Bn .Bn1 / D BC n .B1 / D b.G B1 / D

n ; 4 2

 > 0;

where GC is a function such that GC 2 BnC .C/ (16), l.GC / D .1 .C/=/2 . The function GC is closely related to the function FC (see (17)). For any C  Cn , the functions GC and FC were constructed by Yudin in [26, 27], respectively. Note that the function G Bn is used in the proof of sharp Jackson’s inequalities in the spaces Lp .Tn / for 1  p < 2 [17].

2 Main Results Suppose ƒn is the set of lattices L  Rn of rank n [10], L D fl D v1 1 C    C vn n D MW  2 Zn g; where fv1 ; : : : ; vn g  Rn is a basis of L, M D .v1 : : : vn / 2 Rnn is a generator matrix of L, det L D .det M/2 is the lattice determinant. The quantity 1 .L/ D inffjljW l 2 L; l ¤ 0g is called the first successive minimum of L [21]. We have 1 .L/ D 2 .L/, where .L/ is the packing radius of L. Recall that the packing radius of L is defined by Conway and Sloane [10] .L/ D supfr > 0W .l C rBn / \ .l0 C rBn / D ; 8 l; l0 2 L; l ¤ l0 g: Suppose R.L/ D supx2Rn infl2L jxlj is the covering radius of L. A vector x0 2 Rn such that jx0 j D R.L/ is called a deep hole of L. For C 2 Cn , we define [21] 1 .L/C D inffklkC W l 2 L; l ¤ 0g D inffr > 0W .L n f0g/ \ rC ¤ ;g; R.L/C D sup inf kx  lkC D inffr > 0W L C rC D Rn g: x2Rn l2L

We denote by L the dual lattice to L: L D fl 2 Rn W l l 2 Z 8 l 2 Lg D M  Zn ; where M  D .M 1 /T is a generator matrix of L , det L D .det L/1 is the determinant of L . The problem of estimate of R.L/U 1 .L /V , U; V 2 Cn , is studied in lattice theory, especially in the case U D V D Bn , V D U ı [4, 21]. Using the following theorem, we can obtain some useful estimates.

50

D.V. Gorbachev

Theorem 2.1 If n 2 N, L 2 ƒn , U; V 2 Cn , and 1 .L /V D 1, then C R.L/U  Ln .V/U  LC n .V/U  Mn .U/Ln .V/  Mn .U/l.FV / D

Mn .U/1 .V/ ; 

where FV 2 LC n .V/ is the function (14). The stated theorem is one of the main results of the paper. In the case when L D Zn , U D Bnp , 1  p  2, and V D Bn1 , this can be deduced using the p results of [6]. In this situation, we have 1 .Bn1 / D 1, 1 .Bn1 / D  n=2, R.Zn /p D jx0 jp D n1=p =2, where x0 D .1=2; : : : ; 1=2/ 2 Rn is a deep hole of Zn (for any p  1), Mn .Bnp / D n1=p1=2 . Thus, p n1=p n1=p n1=p1=2  n=2 n  Ln .Bn1 /p  LC D : .B /  n 1 p 2  2 Hence, using the relation (3), we obtain Theorem 1.1. Theorem 1.1 does not hold for p > 2, since Mn .Bnp / D 1. For such p, the n determination of the quantity LC n .B1 /p is an open problem. n Let L 2 ƒn , U D B , V D 1 .L /Bn . Then 1 .L /V D 1 and it follows from Theorems 2.1, 1.2 that R.L/  Ln .1 .L /Bn / D

q˛ : 1 .L /

Hence R.L/1 .L /  qn=21 =. Thus, Yudin’s inequality (7) is established. In [21] the upper estimate of R.L/1 .L / is equal to n3=2 =2. It was known [21] that there exist self-dual lattices Ln 2 ƒn such that R.Ln /1 .Ln / 

n .1 C o.1//; 4e

n ! 1:

The inequality R.L/1 .L /  n=2 was proved in [4]. Also, it was mentioned without a detailed proof that R.L/1 .L / 

 n  1 C O.n1=2 / : 2

The method of proving Theorem 2.1 can be used to obtain some new case of Bohman’s problem. Theorem 2.2 If n 2 N, L 2 ƒn , V 2 Cn , and 1 .L /V D 1, then R2 .L/  Bn .V/  BC n .V/  b.GV / D where GV 2 BnC .V/ is the function (16).

  .V/ 2 1 ; 

Multidimensional Extremal Logan’s and Bohman’s Problems

51

Suppose L D Zn , V D Bn1 ; then   pn=2 2 n n n  Bn .Bn1 /  BC .B /  D : n 1 4  4 Consequently, by (8) and (18), we obtain n n Bn .Bn1 / D BC n .B1 / D b.G B1 / D

n ; 4 2

where G Bn1 .x/ D

n   8 n Y cos.xj / 2 ;  2 jD1 1  .2xj /2

 > 0:

3 Proof of Theorem 2.1 1. The inequalities C Ln .V/U  LC n .V/U  Mn .U/Ln .V/

follows from (2) and (4). Our first objective is the construction of Yudin’s function FC 2 LC n .C/ [27] such that l.FC / D 1 =, where 1 D 1 .C/. Let n  2 (for n D 1, see (5)). We denote by u1 the first eigenfunction of the problem (6) with  D 1 . So u1 D 12 u1 . It is known [11, Chap. VI] that the function u1 is even and ˇ u1 ˇCn@C > 0;

@u1 ˇˇ ˇ  0: @n @C

(10)

Put ( v.x/ D

u1 .x/; x 2 C; 0;

x … C:

(11)

The function v is continuous, even, nonnegative with support C. Find the Fourier transform v.s/, O s 2 Rn . By Green’s identity, Z C

  v.x/ e.sx/  v.x/e.sx/ dx D

Z  @e.sx/  @v.x/ e.sx/  v.x/ dx: @n @n @C

52

D.V. Gorbachev

Therefore, c C j2sj2 vO D 12 vO C j2sj2 vO D vO n ; v

vO D

12

vO n ;  j2sj2

(12)

where Z vOn .s/ D

@C

@v.x/ e.sx/ dx: @n

Introduce the following continuous, even, nonnegative function: Z w.x/ D

 @v.y/  dy: v.x  y/  @n @C

By (10) and (12), we have supp w  2C, w.0/ D 0, wO D vO vO n D

.vO n /2 ; 12  j2sj2

w.0/ O > 0;

w.s/ O  0;

j2sj  1 :

Then it follows from the results of [25, Chap. I] that Z 0 D w.0/ D lim

"!0 Rn

Thus, Z

2"jsj w.s/e O ds D

Z

Z w.s/ O ds 

j2sj 0:

2. This part of the proof is based on an idea of Logan [22]. Let us prove that R.L/U  Ln .V/U , where U; V 2 Cn , 1 .L /V D 1, L 2 ƒn is the lattice with a generator matrix M. The Poisson summation formula [10] X l2L

f .x  l/ D

1 X O  f .l /e.l x/; det M  

(15)

l 2L

holds for functions f such that j f .x/j C jfO .x/j D O.1 C jxjn" /, " > 0. Also, it holds for functions f 2 Fn .V/ such that l. f /U < 1. This fact can be shown by applying the properties supp fO .x/  V and j f .x/j D f .x/, kxkU  l. f /U , in an obvious manner (see (13)).

54

D.V. Gorbachev

Let f 2 Ln .V/, l. f /U < 1, 1 .L /V D 1. Then f is an analytic function, supp fO  V D BnV , kl kV  1 8 l 2 L , l ¤ 0. Therefore, fO .l / D 0 8 l 2 L n f0g. By (15), we get X

f .x  l/ D .det M/1 fO .0/  0

8 x 2 Rn :

l2L

Suppose R.L/U D kx0 kU , where x0 is a deep hole in the U-norm of the lattice L. Then kx0  lkU  kx0 kU 8 l 2 L. It follows that if l. f /U D kx0 kU  ", " > 0, then there exists a neighborhood N" of x0 such that kx  lkU  l. f /U and f .x  l/  0 8 l 2 L, x 2 N" . Thus, we have X 0 f .x  l/  0 8 x 2 N" ; l2L

which contradicts the analyticity of f . This proves that l. f /U  kx0 kU and Ln .V/U  R.L/U . The theorem is proved. Theorem 2.1 allows us to prove Theorem 1.1. In the case of Theorem 1.2, we need to use the function averaging and the Bessel quadrature formulas [13]. We see that Theorem 2.1 leads to the eigenvalue problem for the Laplacian  [16]. Consider one  example. For n D 2, let L D A2 be the hexagonal  1 1=2 p lattice with M D 0 3=2 . Then 1 .A2 / D 1. The lattice A2 is self-dual. We have   1 0 M  D 1=p3 2=p3 and R.A2 / D 2=3. Suppose V0 D MB21 , V1 D 2C, where p C is the Voronoi cell of L around the origin; then 1 .Vi / D 1, mes .Vi / D 2 3. By Theorem 2.1, 2=3  L2 .Vi /  1 .Vi /=. On the other hand, from the Faber–Krahn inequality [16, Chap. 3] 12 .C/ 

q20 mes .C/

it follows that  1 1 .Vi /  0:728 > 2=3.

4 Proof of Theorem 2.2 1. Let f 2 Bn .V/ and 1 .L /V D 1. Then g.x/ D jxj2 f .x/ 2 L1 .Rn /, gO .s/ D .2/2 fO .s/ 2 C.Rn /;

gO .0/ D b. f /;

gO .s/ D 0;

kskV > 1:

Multidimensional Extremal Logan’s and Bohman’s Problems

55

Therefore, gO .l / D 0 8 l 2 L n f0g. By (15), X

g.x  l/ D .det M/1 gO .0/ 8 x 2 Rn :

l2L

Similarly, we have X

f .x  l/ D .det M/1 fO .0/ D .det M/1 :

l2L

Suppose x0 is a deep hole of the lattice L; then jx0  lj  R.L/ 8 l 2 L. Consequently, using f  0, we conclude that X l2L

g.x0  l/ D

X

jx0  lj2 f .x0  l/  R2 .L/

l2L

X

f .x0  l/ D .det M/1 R2 .L/:

l2L

Thus, .det M/1 gO .0/  .det M/1 R2 .L/ and b. f /  R2 .L/. 2. Construct a function GC 2 BnC .C/ such that b.GC / D .1 =/2 [26]. Let n  2 (for n D 1, see (9)). Let us consider the convolution Z Z z.x/ D .v v/.x/ D v.x  y/v.y/ dy D v.x  y/v.y/ dy; Rn

C

where the function v is defined in (11). This convolution is a continuous, even, nonnegative function with support in 2C. By (12), 2 zO.s/ D .v.s// O D

 12

vOn .s/ 2 0  j2sj2

8 s 2 Rn :

Let za .x/ D z.a1 x/=z.0/, a > 0. Then supp za  2aC;

za .0/ D 1;

zOa .s/ D

an zO.as/ : z.0/

Finally, for x 2 Rn , we set GC .x/ D zO1=2 .x/ D AC

 vO .x=2/=vO .0/ 2 n n ; 1  .jxj=1 /2

AC D GC .0/ D

.vO n .0//2 : 2n 14 z.0/

(16)

bC  C, G bC  0, Then the function GC is continuous, even, nonnegative, supp G bC .0/ D 1. G

56

D.V. Gorbachev

Comparing (14) with (16) yields   2 jxj2  GC .x/: FC .x/ D A1 C 1 12

(17)

It follows that jxj2 GC 2 L1 .Rn / and bC .0/ D A1 0DF C

Z     2 jxj2  2 1 1 G 1  .x/ dx D A b.G / : C C C 12 12 Rn

Consequently, GC 2 BnC .V/ and b.GC / D .1 =/2 . The theorem is proved. In the case C D Bn , we have vOn .0/ D k!n1 , 1 D q˛ , Z z.0/ D jxj1

j2˛ .q˛ jxj/ dx D !n1

Z

1 0

j2˛ .q˛ r/rn1 dr D

!n1 22˛  2 .˛ C 1/ I; q2˛ ˛

where [5, Chap. VII] Z

1

ID 0

J˛2 .q˛ u/u du D

2 q2˛C2 .q˛ / j2˛C1 .q˛ / J˛C1 ˛ D 2˛C3 : 2 2  2 .˛ C 2/

It follows that z.0/ D

!n1 q2˛ j2˛C1 .q˛ / j2˛C1 .q˛ / !n1 22˛  2 .˛ C 1/ q2˛C2 ˛ D : q2˛ 22˛C3  2 .˛ C 2/ 23 .˛ C 1/2 ˛

Thus, A2 D

1  q2˛ j˛C1 .q˛ /!n1 2 23 .˛ C 1/2 !n1 .vO n .0//2 D D n1 2 4 2 n 4 n 2 2 q˛ n 2 q˛ 2 1 z.0/ !n1 q˛ j˛C1 .q˛ /

with vO n .0/ D k!n1 D 

q2˛ j˛C1 .q˛ /!n1 ; n

1 D q˛ :

For  D 1, using (17), we can construct the function G Bn from Theorem 1.3. For C D Bn1 , we obtain GBn1 .x/ D A1

n  Y cos.xj / 2 ; 1  .2xj /2 jD1

A1 D

 8 n : 2

(18)

Multidimensional Extremal Logan’s and Bohman’s Problems

57

Acknowledgements The author “D.V. Gorbachev” thanks the referee and R. Veprintsev for their careful reading and valuable comments. This research was supported by the RFFI (no. 13-0100045), the Ministry of Education and Science of the Russian Federation (no. 5414GZ), and Dmitry Zimin’s Foundation “Dynasty”.

References 1. V.V. Arestov, E.E. Berdysheva, Turán’s problem for positive definite functions with supports in a hexagon. Proc. Steklov Inst. Math. (Suppl.), suppl. 1, S20–S29 (2001) 2. V.V. Arestov, E.E. Berdysheva, The Turán problem for a class of polytopes. East J. Approx. 8(3), 381–388 (2002) 3. V.V. Arestov, N.I. Chernykh, On the L2 -approximation of periodic functions by trigonometric polynomials, in Approximation and Function Spaces: Conference Proceedings, Gdansk (1979) (North-Holland, Amsterdam, 1981), pp. 25–43 4. W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296(4), 625–635 (1993) 5. G. Bateman, A. Erdélyi, et al., Higher Transcendental Functions, II (McGraw Hill, New York, 1953) 6. E.E. Berdysheva, Two related extremal problems for entire functions of several variables. Math. Notes 66(3), 271–282 (1999) 7. H. Bohman, Approximate Fourier analysis of distribution functions. Ark. Mat. 4, 99–157 (1960) 8. N.I. Chernykh, On best approximation of periodic functions by trigonometric polynomials in L2 . Mat. Zametki 2(5), 513–522 (1967) 9. H. Cohn, New upper bounds on sphere packings II. Geom. Topol. 6, 329–353 (2002) 10. J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, 3rd edn. (Springer, New York, 1999) 11. R. Courant, D. Hilbert, Methods of Mathematical Physics I (Interscience, New York, 1953) 12. W. Ehm, T. Gneiting, D. Richards, Convolution roots of radial positive definite functions with compact support. Trans. Am. Math. Soc. 356, 4655–4685 (2004) 13. D.V. Gorbachev, Extremal problems for entire functions of exponential spherical type. Math. Notes 68(1–2), 159–166 (2000) 14. D.V. Gorbachev, An extremal problem for entire functions of exponential spherical type, which is connected with the Levenshtein bound for the density of a packing of Rn by balls (Russian). Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inf. 6(1), 71–78 (2000) 15. D.V. Gorbachev, An extremal problem for periodic functions with support in a ball. Math. Notes 69(3–4), 313–319 (2001) 16. A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics (Birkhäuser, Basel, 2006) 17. V.I. Ivanov, Approximation of functions in spaces Lp . Math. Notes 56(2), 770–789 (1994) 18. A.V. Ivanov, V.I. Ivanov, Jackson’s theorem in the space L2 .Rd / with Power Weight. Math. Notes 88(1), 140–143 (2010) 19. A.V. Ivanov, V.I. Ivanov, Optimal arguments in Jackson’s inequality in the power-weighted space L2 .Rd /. Math. Notes 94(3), 320–329 (2013) 20. M.N. Kolountzakis, Sz.Gy. Révész, On pointwise estimates of positive definite functions with given support. Can. J. Math. 58(2), 401–418 (2006) 21. J.C. Lagarias, H.W. Lenstra Jr., C.P. Schnorr, Korkin–Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10(4), 333–348 (1990) 22. B.F. Logan, Extremal problems for positive-definite bandlimited functions. II. Eventually negative functions. SIAM J. Math. Anal. 14(2), 253–257 (1983)

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23. S.M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer, Berlin/Heidelberg/New York, 1975) 24. Sz.Gy. Révész, Turan’s extremal problem on locally compact abelian groups. Anal. Math. 37(1), 15–50 (2011) 25. E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971) 26. V.A. Yudin, The multidimensional Jackson theorem. Math. Notes 20(3), 801–804 (1976) 27. V.A. Yudin, Multidimensional Jackson theorem in L2 . Math. Notes 29(2), 158–162 (1981) 28. V.A. Yudin, Two external problems for trigonometric polynomials. Sb. Math. 187(11), 1721– 1736 (1996)

Weighted Estimates for the Discrete Hilbert Transform E. Liflyand

Abstract The Paley-Wiener theorem states that the Hilbert transform of an integrable odd function, which is monotone on RC , is integrable. In this paper we prove weighted analogs of this theorem for sequences and their discrete Hilbert transforms under the assumption of general monotonicity for an even/odd sequence. Keywords Discrete Hilbert transform • General monotone functions and sequences • Hilbert transform • Weighted integrability

Mathematics Subject Classification (2000). Primary 42A50, Secondary 40A99, 26A48, 44A15

1 Introduction Weighted estimates for the Hilbert transform is one of the most popular and important topics in today harmonic analysis. Being somewhat apart of the mainstream, the Paley-Wiener theorem (see [13]) asserts that for an odd and monotone decreasing on RC function g 2 L1 its Hilbert transform is also integrable, i.e., g is in the (real) Hardy space H 1 .R/ (for alternative proofs, see [21] and [15, Chap. IV, 6.2]). The oddness of g is essential, since by Kober’s result [9], if g 2 H 1 .R/, then the cancelation property holds Z R

g.t/ dt D 0:

(1)

In [12], the monotonicity assumption has been relaxed in the Paley-Wiener theorem, and in [11] weighted versions have been obtained for both the sine and

E. Liflyand () Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_5

59

60

E. Liflyand

cosine Fourier transforms of functions more general than monotone ones. The periodic case has also been covered in [11] in the same manner. One can find historical background relevant to these problems in [11]. However, for sequences rather than functions such problems are of similar interest and importance as well. The goal of this note is to prove the weighted analogues of the Paley-Wiener theorem for odd and even sequences, that is, for the discrete Hilbert transform. More precisely, the estimates will be given in the weighted `1 spaces `.w/, the space of sequences a D fak g1 kD0 endowed with the norm kak`.w/ D

1 X

jak jwk < 1;

kD0

where w D fwk g; k D 0; 1; 2; : : : ; is a non-negative sequence, or weight (if the sequence is two-sided, that is, defined (and summed) for 1 < k < 1, then we assume w to be an even sequence; we get a usual `1 sequence if the weight is constant). In particular, we show that for a weight wk D k˛ , the discrete Hilbert transform is bounded in `.w/, when 1 < ˛ < 1 provided that a is an odd and monotone on ZC sequence, or, when 2 < ˛ < 0 provided that a is even and monotone on RC . Thus assuming monotonicity or general monotonicity of a allows us to extend the results of [11] to sequences; recall that the estimates in [11] were, in turn, generalizations of Flett’s [4], Hardy-Littlewood’s [5], and Andersen’s [1] results to the case p D 1. Our results can also be considered as generalizations of the known results for sequences in [2]. We mention that for p D 1 not only that two-weighted estimates (for monotone sequences and power weights) are obtained in [16] but also their sharpness is proved there (for more details for functions, see also [17]). The outline of the paper is as follows. In the next section, we start with the needed prerequisites and then formulate the mentioned results for discrete Hilbert transforms. The proofs in the third section go along the same lines as those in [11], roughly speaking, the integrals and functions are replaced with sums and sequences. In the last section, we present an application of the obtained analog of the PaleyWiener theorem to an old problem on the absolute convergence of Fourier series. To fix certain notation, let . mean  C, with some absolute constant C, while & similarly means  C. Notation is used when simultaneously . and & are valid, of course, with different constants. We denote by Œy the integer part of y.

2 Definitions and Statement of Results We first give some prerequisites for discrete Hilbert transforms, then we formulate the above mentioned results.

Discrete Hilbert Transform

61

2.1 Discrete Hilbert Transforms For the sequence a D fak g 2 `1 ; the discrete Hilbert transform is defined for n 2 Z as (see, e.g., [8, (13.127)]) 1 X

„a.n/ D

kD1 k¤n

ak : nk

(2)

If the sequence a is either even or odd, the corresponding Hilbert transforms „e and „o may be expressed in a special form (see, e.g., [2] or [8, (13.130) and (13.131)]). More precisely, if a is even, with a0 D 0; we have „e .0/ D 0 and for n D 1; 2; : : : 1 X 2nak an „e a.n/ D C : 2  k2 n 2n kD1

(3)

k¤n

If a is odd, with a0 D 0; we have for n D 0; 1; 2; : : : „o a.n/ D

1 X 2kak an  : 2 2 n k 2n kD1

(4)

k¤n

Of course,

a0 0

is considered to be zero.

2.2 Estimates for the Discrete Hilbert Transforms A null sequence a, that is, vanishing at infinity, is said to be general monotone (see [11, 12, 18]), or a 2 GMS, if it satisfies the conditions 2n X kDn

jak j  C

Œcn X jak j ; k

n D 1; 2; : : : ;

(5)

n D 1; 2; : : : ;

(6)

kDŒn=c

and Œn=c X jak j ; jak j  C k kD2n n X

kDŒcn

62

E. Liflyand

with ak D ak  akC1 , where C > 1 and c > 1 are independent of n. If a is even or odd, then both conditions are the same. Note that any monotone or quasi-monotone sequence a is general monotone. Definition 2.1 Let a non-negative sequence, or weight, w D fwk g; k D 1; 2; : : : ; belong to the ! class, written w 2 !, if there exists " > 0 such that wk k1" "

for all

k;

(7)

wk k"1 #

for all

k;

(8)

where " and # mean almost increase and almost decrease. Recall that sequence d is called almost increasing (respectively, decreasing) if dm  Cdn or, equivalently, dm . dn when m < n (respectively, m > n). Theorem 2.2 Let a be an odd sequence from `.w/, with a0 D 0 and weight w. If a 2 GMS and w 2 !, then k„ak`.w/ . kak`.w/ :

(9)

A counterpart of Theorem 2.2 for even sequences reads as follows. Definition 2.3 Let a weight w belong to the !  class, written w 2 !  , if there exists " > 0 such that wk k2" " wk k" #

for all for all

k; k:

(10) (11)

Theorem 2.4 Let a be an even sequence from `.w/, with a0 D 0 and weight w. If a 2 GMS and w 2 !  , then (9) holds. Not posing assumptions of evenness or oddness, we obtain a weighted estimate for the discrete Hilbert transform of a sequence summable on the whole Z: Corollary 2.5 Let w 2 ! \ !  , that is, w satisfies (7) and (11). If a D fak g1 kD1 2 GMS, then (9) holds. In particular, for the weight wk D jkj˛ , 1 < ˛ < 0, the discrete Hilbert transform „a is in `.w/ provided a 2 `.w/ \ GMS:

3 Proofs In this section we present the proofs of the above formulated results. Observe that 1 P wn jann j is trivially dominated by kak`.w/ ; therefore we will not discuss the terms nD1

an ˙ 2n in (3) and (4).

Discrete Hilbert Transform

63

Proof of Theorem 2.2 Let n > 0; calculations for n < 0 are the same. Since a is odd, we obtain ˇ0 ˇ 1 ˇ X ˇ Œ3n=2 1 1 1 1 X X X ˇ 2k ak ˇˇ X ˇ A @ wn ˇ C wn jak j 2 ˇ n  k k  n2 ˇ kDŒ3n=2 kD1 ˇ nD1 kDŒ3n=2 nD1 .

1 X

Œ2k=3C1

kjak j

kD1

X nD1

k2

wn :  n2

(12)

Applying (7), we get Œ2k=3C1

X nD1

wn wk k1" . k 2  n2 k2

Œ2k=3C1

X

n"1 .

nD1

wk : k

By this, the right-hand side of (12) is dominated by kak`.w/ : Similarly, but making use of (8), we get ˇ0 ˇ 1 ˇ Œn=2 ˇ X 1 1 1 X X ˇ X wn a k ˇˇ ˇ A @ . wn ˇ C kja j k ˇ 2 n  k2 ˇ kD1 kDŒn=2 n  k ˇ kD1 nD1 nD2k

1 X

.

1 X

kjak j

kD1

1 1 X X wn n"1 . wk jak j: n2C"1 nD2k kD1

We remark that (7) and (8) imply wn wk ;

ˇn  k  n;

0 < ˇ < ;

i.e., C1 wn  wk  C2 wn , where C1 ; C2 > 0 are independent of k and n. Then ˇ ˇ Œn=21 ˇ X ak wn ˇˇ ˇkDŒ3n=2C1 n  k nD1

1 X

ˇ ˇ X 1 2kC1 1 X X ˇ wn ˇ. ja j wk jak j: . k ˇ n C k kD1 ˇ kD1 nDŒ2k=3

Collecting estimates from above, we obtain ˇ ˇ 1 ˇ X a k ˇ wn ˇ ˇ kD1 n  k nD1 ˇ k¤n

1 X

ˇ ˇ ˇ ˇ 1 ˇ ˇ X ˇ ˇ ˇwn ˇ. ˇ ˇ ˇ nD1 ˇ .IC

1 X kD1

Œ3n=21

X

kDŒn=2C1 k¤n

wk jak j;

ˇ ˇ 1 ak ˇˇ X wk jak j ˇC n  k ˇ kD1 ˇ

(13)

64

E. Liflyand

where ˇ ˇ Œn=2 ˇ X ank  anCk wn ˇˇ ID k ˇ kD1 nD1 1 X

ˇ ˇ ˇ ˇ: ˇ ˇ

We then have I

1 X

wn

nD1



1 X kD1

D

Œn=2  nCk X X kD1

jas j  k sDnk

nCk  X jas j  wn k sDnk nD2k 1 X

1 hX 3k X

jasj

sDk

kD1

sCk X

wn C

nD2k

1 X

jas j

sCk X

wn

nDsk

sD3kC1

i1 k

DW I1 C I2 :

By (13), sCk 4k 1 X 1 X wn  wn wk ; k nD2k k nD2k

k  s  3k

and sCk 1 X wn ws ; k nDsk

s  3k:

Hence, since a 2 GMS, I1 .

1 X

wk

3k X

kD1

.

1 X sD1

Œck  X jas j  jas j . wk s sDk kD1



1 X

sDŒk=c

jas j

 Œc.sC1/ X w  k

kDŒs=c

k



1 X

jas jws :

sD1

Changing the order of summation yields I2 .

1 X 1 X kD1

1 1 2k  X X 1X ws jas j . sws jas j . sws jasj: k sDk sD3k sD1 kD1

Discrete Hilbert Transform

65

Using (13) and general monotonicity of a, we get, as above, I2 .

1 X

wk

2k X

jasj .

sDk

kD1

1 X

jak jwk ;

kD1

t u

which completes the proof.

Proof of Theorem 2.4 The proof goes along the same lines as that of Theorem 2.2. Using the evenness of a, we obtain ˇ0 1 ˇ Œ3n=2 1 X ˇ X a A k wn ˇˇ@ C ˇ kDŒ3n=2 kD1 n  k nD1

1 X

ˇ ˇ 1 X ˇ ˇ du . jak j ˇ ˇ kD1 .

1 X

Œ2k=3C1

X nD1

n wn  n2

k2

wk jak j;

kD1

since, by (10), we have Œ2k=3C1

X nD1

wk k2" n wn . 2 2 k n k2

Œ2k=3C1

X

n"1 . wk :

nD1

Taking into account (11), we get ˇ0 1 ˇ Œn=2 1 X ˇ X A ak wn ˇˇ@ C ˇ kD1 kDŒn=2 n  k nD1

1 X

.

1 X kD1

jak j

ˇ ˇ X 1 1 X ˇ n wn ˇ. ja j k ˇ 2  k2 n ˇ kD1 nD2k

1 1 X X wn n" . wk jak j: n1C" nD2k kD1

Finally, we note that (10) and (11) also imply (13) and we can repeat the rest of the proof of Theorem 2.2. t u Proof of Corollary 2.5 Representing a in a standard way as the sum of its even and odd parts ak D

ak  ak ak C ak C ; 2 2

we apply the same calculations as in the proof of Theorem 2.2 to the odd part and of Theorem 2.4 to the even part. Using then jak ˙ ak j  jak j C jak j and w 2 ! \ !  ; we obtain the required estimate. t u

66

E. Liflyand

4 Absolutely Convergent Fourier Series In 1950s (see, e.g., [6] or in more detail [7, Chaps. II and VI]), the following problem in Fourier Analysis attracted much attention: Let fak g1 kD0 be the sequence of the Fourier coefficients of the absolutely P convergent sine (cosine) Fourier series of a function f W T D Œ; / ! C; that is jak j < 1: Under which conditions on fak g the re-expansion of f .t/ . f .t/  f .0/, respectively) in the cosine (sine) Fourier series will also be absolutely convergent?

The obtained condition is quite simple and is the same in both cases: 1 X

jak j ln.k C 1/ < 1:

(14)

kD1

Analyzing the proof, say, in [6], one can see that in fact more general results are hidden in the proofs. They can be given in terms of the (discrete) Hilbert transform. P Theorem 4.1 In order than the re-expansion bk sin kt of f .t/  f .0/ with the absolutely convergent cosine Fourier series be absolutely convergent, it is necessary and sufficient that the discrete Hilbert transform „a of the sequence a of the cosine Fourier coefficients of f be summable. P Similarly, in order than the re-expansion bk cos kt of f with the absolutely convergent sine Fourier series be absolutely convergent, it is necessary and sufficient that the discrete Hilbert transform „a of the sequence a of the sine Fourier coefficients of f be summable. What, in fact, is proven in the mentioned papers, for b D fbk g1 kD0 ; b D „e a

(15)

b D „o a

(16)

in the first part of Theorem 4.1, and

in the second one. In this case (14) is only a sufficient condition for the summability of the discrete Hilbert transform, though sharp on the whole class. We will use the obtained results to show that (14) is not necessary in the second part of Theorem 4.1. Indeed, Theorem 2.2 includes the non-weighted case, that is, when wn D 1 for all n: This is a direct analog of the Paley-Wiener theorem for monotone sequences. It suffices to take any `1 monotone sequence a such that 1 X kD1

jak j ln.k C 1/ D 1:

Discrete Hilbert Transform

67

This does not work for the cosine Fourier series, since the non-weighted case is excluded. An appropriate counter-example can be built, of course, but it is not related to the present work. An interesting open problem appears in a different topic related both to the absolute convergence of Fourier series and to properties of the discrete Hilbert transform. It is proven by Wiener [20, Sect. 11] that if f is supported on Œ;   "; 0 < " < , then fO 2 L1 .R/ if and only if the 2-periodic extension of f has absolutely convergent Fourier series. This is no more true, in general, if " D 0 (see, e.g., [19, 6.1.1]). Here, two independent conditions are necessary and sufficient for the integrability of the Fourier transform of a function f supported on Œ; : not only the 2-periodic extension of f has absolutely convergent Fourier series but also f1 .t/ D tf .t/ has. Let us show how this follows from [3, Thm. 6]. Let `p , 0 < p < 1; be the space of sequences fdj g satisfying kfdj gk`p D

X 1

jdj jp

 1p

:

jD1

We denote by hp the class of `p sequences whose discrete Hilbert transform is also in `p . Denoting by Ep the class of Lp functions whose (inverse) Fourier transform is supported in Œ; , we have Theorem 6 from [3] in the following form. Theorem 4.2 Let 0 < p  1: If g belongs to Ep ; then f.1/k g.k/g belongs to hp : Conversely, if fak g belongs to hp , there is a unique g 2 Ep such that g.k/ D .1/k ak : In this terminology, we consider fO 2 E1 . If ck . f / D Fourier coefficient of f1 , written cn . f1 /, we have 1 2

Z

 

f .t/teint dt D

1 2

Z



1 X

 kD1

ck eikt teint dt D

1 O 2 f .k/;

1 X kD1 k¤n

ck

1 2

Let us calculate the last integral. Integrating by parts, we obtain ˇ Z  ˇ 1 1 tei.kn/t ˇˇ  ei.kn/t dt i.k  n/ i.k  n/   D

.1/kn 2 cos.k  n/ D 2 : i.k  n/ i.k  n/

then for the n-th Z

 

tei.kn/t dt:

68

E. Liflyand

Therefore, with f .t/ D

1 P

ck eikt ;

kD1 1 1 X .1/k ck int 1 X n e ; .1/ f1 .t/ D i nD1 kn kD1 k¤n

and hence 1 .1/n .1/n X .1/k ck cn . f1 / D D „Qc.n/; i kD1 k  n i k¤n

where c D fck g1 Q D f.1/k ck g1 c.n/ D kD1 , c kD1 and „Q

1 P kD1 k¤n

.1/k ck kn

is the n-th

element of the discrete Hilbert transform of cQ . Belonging of cQ to h1 is just equivalent to the absolute convergence of the Fourier series of f and f1 and, by Theorem 4.2, is equivalent to the integrability of fO .  Since the same argument goes through for any 0 < p < 1, we in fact have the following theorem for f supported in Œ;  (see [19, 6.1.1 b)]). Theorem 4.3 Let 0 < p  1: We have fO 2 Lp .R/ if and only if both f and f1 after 2-periodic extension have the sequences of Fourier coefficients from `p . Recall (see [14]) that if 1 < p < 1 then the Fourier transform of a compactly supported function is in Lp if and only if this function, being extended periodically, has the sequence of Fourier coefficients from `p . Back to Wiener’s result, one may ask whether it is possible that there exists a set, containing both  and  and maybe rather thin near one or both endpoints, for which every function with absolutely convergent Fourier series (after 2-periodic extension) supported on that set has integrable Fourier transform. The author failed to prove in [10] that no such set can exist, or, in other words, that Wiener’s result is sharp. Therefore, the problem of sharpness of Wiener’s result remains open.

References 1. K. Andersen, Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions. Proc. Am. Math. Soc. 56(1), 99–107 (1976) 2. K. Andersen, Inequalities with weights for discrete Hilbert transforms. Can. Math. Bull. 20, 9–16 (1977) 3. C. Eoff, The discrete nature of the Paley-Wiener spaces. Proc. Am. Math. Soc. 123, 505–512 (1995) 4. T.M. Flett, Some theorems on odd and even functions. Proc. Lond. Math. Soc. (3) 8, 135–148 (1958)

Discrete Hilbert Transform

69

5. G.H. Hardy, J.E. Littlewood, Some more theorems concerning Fourier series and Fourier power series. Duke Math. J. 2, 354–382 (1936) 6. S.I. Izumi, T. Tsuchikura, Absolute convergence of trigonometric expansions. Tôhoku Math. J. 7, 243–251 (1955) 7. J.-P. Kahane, Séries de Fourier Absolument Convergentes (Springer, Berlin, 1970) 8. F.W. King, Hilbert Transforms. Encyclopedia of Mathematics and Its Applications, vol. 1 (Cambridge University Press, Cambridge, 2009) 9. H. Kober, A note on Hilbert’s operator. Bull. Am. Math. Soc. 48(1), 421–426 (1942) 10. E. Liflyand, Sharpness of Wiener’s theorem on the Fourier transform of a compactly supported function, in Theory of Mappings and Approximation of Functions (Naukova dumka, Kiev, 1989), pp. 87–95 (Russian) 11. E. Liflyand, S. Tikhonov, Weighted Paley-Wiener theorem on the Hilbert transform. C.R. Acad. Sci. Paris, Ser. I 348, 1253–1258 (2010) 12. E. Liflyand, S. Tikhonov, A concept of general monotonicity and applications. Math. Nachr. 284, 1083–1098 (2011) 13. R.E.A.C. Paley, N. Wiener, Notes on the theory and application of Fourier transform, note II. Trans. Am. Math. Soc. 35, 354–355 (1933) 14. M. Plancherel, G. Pólya, Fonctions entières et intégrales de Fourier multiples. Commun. Math. Helvetici I 9, 224–248 (1937); II 10, 110–163 (1938) 15. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, 1993) 16. V. Stepanov, S. Tikhonov, Two-weight inequalities for the Hilbert transform on monotone functions. Doklady RAN 437, 606–608 (2011) (Russian). Engl. Transl. Dokl. Math. 83, 241– 242 (2011) 17. V. Stepanov, S. Tikhonov, Two power-weight inequalities for the Hilbert transform on the cones of monotone functions. Complex Var. Ellipt. Equat. 56, 1039–1047 (2011) 18. S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326, 721–735 (2007) 19. R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions (Kluwer, London, 2004) 20. N. Wiener, The Fourier Integral and Certain of Its Applications (Dover, New York, 1932) 21. A. Zygmund, Some points in the theory of trigonometric and power series. Trans. Am. Math. Soc. 36, 586–617 (1934)

Q-Measures on the Dyadic Group and Uniqueness Sets for Haar Series Mikhail G. Plotnikov

Abstract The aim of article is to describe, in terms of the Q-measures, uniqueness sets for Haar series with non-decreasing partial sum’s major sequences. Keywords Dyadic group • Haar series • Hausdorff measures • Partial sum’s major sequence • Q-measures • Uniqueness sets

Mathematics Subject Classification (2000). Primary 42C10, Secondary 28A12

1 Introduction In 1870 G. Cantor proved the following theorem (see [1, Chap. 1], [15, vol. 1, Chap. 9]): if a trigonometric series 1

a0 X C an cos.nx/ C bn sin.nx/ 2 nD1 converges to zero everywhere on Œ0; 2/ except possibly on a finite set, then this series is identically zero, that is, all its coefficients are equal to zero. The branched theory of uniqueness of representation of functions by orthogonal series originates with the Cantor Theorem. Definition 1.1 Let f fn g be a system of functions on some set X, and let X

cn fn .x/

(1)

n

M.G. Plotnikov () Vologda Vereshchagin Academy, Schmidt Street 2, 160555 Molochnoe, Vologda, Russia e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_6

71

72

M.G. Plotnikov

be a series with respect to this system. A set A  X is called uniqueness set (or U-set) for series (1) if only identically zero series of the form (1) converges to zero on X n A. From the viewpoint of the last definition the Cantor Theorem means that every finite subset of Œ0; 2/ is a uniqueness set for trigonometric series. The description of more complicated uniqueness sets for trigonometric series requires of involvement of arithmetic characteristics of sets, not only of metric ones [1, Chap. 14]. In present work we study uniqueness problems for Haar series. A. Haar proved (1910) that the empty set is a U-set for Haar series 1 X

an Hn .x/;

nD0

but the proof contained an error. A correct proof was obtained (1964), independently, by F.G. Arutunyan, and by F.G. Arutunyan & A.A. Talalyan, and by M.B. Petrovskaya, and by V.A. Skvortsov. In other side, every one-point set is not a U-set for Haar series (G. Faber, 1910, for the set f1=2g; J. McLaughlin & J. Price, 1969, for an arbitrary one-point set A  Œ0; 1). Thus, in contrast of the case of trigonometric series, only ; is a uniqueness set for Haar ones. See [3] about mentioned results. In several works uniqueness sets for some subclasses of Haar series (conditional uniqueness sets) are studied. Mushegyan proved [4] that a Borel set A is a U-set for Haar series whose coefficients are satisfy the Arutyunyan–Talalyan condition an Hn .x/ D ox .n/; if and only if A is at most countable. W.R. Wade explored [12] properties of uniqueness sets for Haar series such that an D o .np1=2 /;

0  p  1:

In [6] an attempt was made to describe U-sets for Haar series from Wade classes, with a help of some dimension characteristics. In [7] the following theorem was established: a set A 2 Œ0; 1 is a U-set for Haar series such that an D O .np1=2 /;

0 < p  1;

if and only if A contains no perfect subset of positive Hausdorff .1  p/-measure. In [7] a similar result for multidimensional Haar series on the group Gm also was proved. The aim of this work is to describe, in terms of the Q-measures, uniqueness sets for Haar series with non-decreasing partial sum’s majorants.

Q-Measures on the Dyadic Group

73

2 The Group G: Main We shall denote the set of positive integers by P, the set of non-negative integers by N, the set of real numbers by R. We use the convenient notation [8] x D .xn ; n 2 N/ for the sequence x D .x0 ; x1 ; x2 ; : : :/. The set of all 0–1 sequences x D .xj ; j 2 N/ is called the dyadic group G. The zero element of G is the sequence 0 WD .xj D 0; j 2 N/ and the group operation ˚ is given by x ˚ y D .jxj  yj j; j 2 N/ for every x D .xj ; j 2 N/ 2 G, y D .yj ; j 2 N/ 2 G. Let k; p 2 N, pD

k1 X

pj 2j ;

.pj 2 f0; 1g;

0  j  k  1/I

jD0

then the set p

k WD fx D .xj ; j 2 N/ 2 G W xj D pk1j ; j D 0; : : : ; k  1g

(2)

is called the dyadic interval of rank k on G. If t 2 G, then we shall write k;t for the uniquely determined dyadic interval of rank k containing the point t. p p Let k be an arbitrary dyadic interval of the form (2) with k  1. Denote by k the uniquely determined dyadic interval of rank k such that p

p

q

k t k D k1

(3)

for some q 2 f0; : : : ; 2k1  1g. The topology on G is generated by the collection of all dyadic intervals. Obviously, each dyadic interval simultaneously open and closed in this topology. The dyadic group is metrizable (we omit details, see [8, Introduction]). The dyadic group .G; ˚/ is a compact Abelian group. Denote by the normed Haar measure on G. Thus is a translation invariant Borel measure on G such that

.G/ D 1. It is well-known [8, Introduction] that

.k / D 2k

(4)

for each dyadic interval k of rank k. Lemma 2.1 Let ˆ W Œ1; C1/ ! Œ1; C1/ be a function such that ˆ.qx/  q ˆ.x/

(5)

74

M.G. Plotnikov

for all q > 1 and x  1. Then the function Q W .0; 1 ! .0; 1 defined by   1 Q.x/ D x ˆ x

(6)

is non-decreasing. Proof Choose x, y 2 .0; 1, x < y. Then x D y=q for some q > 1. We have       y q (5) q  y 1 (6) 1 (6) Q.x/ D x ˆ D ˆ ˆ  D Q.y/: x q y q y

(7) t u

The statement of Lemma is a corollary of (7).

3 Haar Series on the Group G: Main Haar functions Hn (n 2 N) on the group G are defined by the following way. H0  1 on G. If n D 2k C p (k D 0; 1; : : :, p D 0; : : : ; 2k  1), then 8 2p 2k=2 for t 2 kC1 , ˆ ˆ < Hn .t/ D 2k=2 for t 2 2pC1 kC1 , ˆ ˆ : p 0 for t 2 G n k . A Haar series on G is defined as 1 X

an Hn .t/;

t 2 G:

(8)

nD0

Let N 2 P. Then the NthPrectangular partial sum SN .t/ of the series (8) at the N1 point t is given by SN .t/ WD nD0 an Hn .t/.

4 Quasi-Measures on the Group G: Main Let B be the family of all dyadic intervals (2). Functions  W B ! R are called B-functions. A B-function  is said to be B-superadditive (B-subadditive) [5] if for Fp p any disjunct collection fi giD1 of dyadic intervals such that iD1 i 2 B we have the inequality p X iD1

.i /  

G p iD1

 i

 X p iD1

.i /  

G p iD1

 : i

Q-Measures on the Dyadic Group

75

We denote by the symbol AB (respectively, AB ) the set of all B-superadditive (respectively, B-subadditive) functions. A B-function  2 A WD AB \ AB is called B-additive. Additive B-functions are called in other words quasi-measures, [8, Chap. 7], [13]. The upper derivative of a B-function  at a point t 2 G with respect to the family B is defined as DB .t/ D lim

./ ;

./

./ ! 0;

 2 B;

t 2 ;

 2 B;

t 2 :

see [11]. Similarly, the lower derivative is defined as DB .t/ D lim

./ ;

./

./ ! 0;

If DB .t/ D DB .t/ ¤ ˙1, we say that  is B-differentiable at t and denote the B-derivative by DB .t/. Let Q W .0; 1 ! .0; 1 be a non-decreasing function. We write HQ for the class of all B-functions  such that the following analogue of Hölder’s condition holds for some M > 0: j./j  M  Q . .// ;

 2 B:

(9)

Let HQ;  be the class of all B-functions  satisfying the following condition everywhere on G for some Mt > 0: j .k;t /j  Mt  Q . .k;t // ;

k;t 2 B:

(10)

Lemma 4.1 Let F  G be a closed set. Suppose that  is a non-negative and subadditive B-function satisfying .G/ > 0: Then there is a non-trivial quasi-measure 0

./  ./

(11)

such that for each  2 B:

(12)

p

Proof Consider an arbitrary dyadic interval k of the form (2) with k  1. Then p there exists the uniquely determined collection Ik D fps s gk1 sD0 of dyadic intervals of the form (2), such that p

ps s k

for every s D 0; : : : ; k  1:

Easily, p

p

Ik D Ik :

(13)

76

M.G. Plotnikov

We set p

k Y

p

.k / WD .k /  Q0 sD1

;

(14)

:

(15)

p

p

sD1

where

p

s1 .s1 /

.s s / C .s s /

WD 1. It follows from (3), (13), and (14), that p

p

.k / D .k / 

k Y

p

s1 .s1 /

p

p

sD1

.s s / C .s s /

Combining (11) and (14), we get .G/ D .G/ > 0. Consequently, the Bfunction is non-trivial. Since  is non-negative, we see from (14) that is nonnegative, too. Since  is subadditive, every multiplier at product in (14) no more than 1. Therefore k Y

p

s1 .s1 /

p

p

sD1 p

.s s / C .s s /

1

p

for all s, and .k /  .k / for all k  1, p D 0; : : : ; 2k  1. Finally, we establish that is a quasi-measure. It is sufficient to prove that p

p

.k / C

q

.k / D

.k1 /

(16) q

for all k  1, p D 0; : : : ; 2k  1, where the dyadic interval k1 has introduced by (3). We have p

p

.k / C .k /

(14); (15)

D

k   Y p p .k / C .k /  sD1

  p p D .k / C .k / 

p

k1 D .k1 /

Thus,

k1 Y

p

k1 .k1 /

p

p

.k k / C .k k / p

s1 .s1 /

ps sD1 .s /

C

p .s s /

is a quasi-measure. The lemma is proved.



k1 Y

p

s1 .s1 /

p

p

.s s / C .s s / p

s1 .s1 /

p

p

sD1

.s s / C .s s /

(3); (14)

D

q

.k1 /: t u

Remark 4.2 There exists another way to prove Lemma 4.1, using the Doob expansion, the well-known in the theory of martingales result [9], [14]. We see the similar way would be long in the format of our article.

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5 Haar Series and Quasi-Measures It is well-known (see, for example, [11]) that the study of general Haar series is equivalent in some sense to the study of quasi-measures. For every series (8) we define a B-function by .k;t / D S2k .t/ .k;t /:

(17)

The value of .k;t / defined by (17) is independent of the choice of t 2 . It is well known [10, 11] that is a quasi-measure and that the correspondence between the set of all series (8) and the set of all quasi-measures, defined by (17), becomes an isomorphism if we endow each of these sets with the natural structure of a vector space. In what follows, when speaking of the quasi-measure isomorphic to a series (8), we mean the quasi-measure defined by (17). Lemma 5.1 (See [10, 11]) Suppose we are given t 2 G and a series .S/ of the form (8). Consider the quasi-measure isomorphic to .S/. Then .S/ converges to a finite sum A at t if and only if DB .t/ D A. Lemma 5.2 Let .S/ be a series of the form (8), be the quasi-measure isomorphic to the series .S/, t 2 G. Consider an arbitrary non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/ satisfying (5), and the function Q W .0; 1 ! .0; 1 defined by (6). Assume that for some Mt > 0 the partial sums SN .t/ satisfy jSN .t/j  Mt  ˆ.N/;

N D 1; 2; : : : :

Then we have the inequality (10) for the quasi-measure

(18)

.

Proof Choose an arbitrary dyadic interval k;t 2 B. We have j .k;t /j

(18)   2k jS2k .t/j  Mt  2k ˆ 2k   1 (4) (6) D Mt  .k;t /  ˆ D Mt  Q . .k;t // ;

.k;t / (4); (17)

D

and the formula (10) follows from (19). The lemma is proved.

(19)

t u

Lemma 5.3 Let .S/ be a series of the form (8), be the quasi-measure isomorphic to the series .S/. Consider an arbitrary non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/ satisfying (5), and the function Q W .0; 1 ! .0; 1 defined by (6). Suppose that for some M > 0 the quasi-measure satisfies the inequality (9). Then the Nth partial sums of the series .S/ satisfy jSN .t/j  2 M  ˆ.N/;

N D 1; 2; : : : ;

t 2 G:

(20)

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M.G. Plotnikov

Proof Choose an arbitrary N 2 P and t 2 G. It follows from the structure of Haar series that SN .t/ D S2k .t/ for some k 2 N satisfying 2k  2  N:

(21)

We have M j .k;t /j (9)  Q . .k;t // 

.k;t /

.k;t /   M 1 (6)  .k;t /  ˆ D

.k;t /

.k;t / (17)

jSN .t/j D jS2k .t/j D

(22)

  (5); (21) (4) D M  ˆ 2k  2 M  ˆ .N/ ; and the formula (20) follows from (22). The lemma is proved.

t u

6 The Q-Measures of Sets in G A covering of a set E  G is a family  of sets in G such that E

[

J:

J2

Let Q W .0; 1 ! .0; C1/ be a non-decreasing function. Then the Q-measure of a set E  G is defined as X Q . .J// ; (23) mesQ E WD lim mesıQ E; mesıQ E WD inf ı!0C0

where the infimum runs over all at most countable coverings fJg of E by dyadic intervals J with .J/  ı. The set function Q is a Borel measure on G. If Q.x/  xq (q-const), then the xq -measure is identical to the Hausdorff q-measure. We need the next definition. A non-trivial B-function  is said to be supported on a set E  G, if ./ D 0 for all  2 B such that  \ E D ;. Theorem 6.1 Suppose we are given a non-decreasing function Q W .0; 1 ! .0; 1, and a closed set F  G with mesQ F > 0. Then there is a non-trivial quasi-measure 2 HQ supported on the set F. Proof We introduce a B-function  by ./ WD mesıQ . \ F/;

 2 B:

(24)

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Clearly,  2 AB . Further, mesQ F > 0 implies that .G/ > 0, and that the B-function  is non-trivial. Let  be an arbitrary dyadic interval. If ./  ı, then the family containing only one set  is a covering of the set  \ F. Therefore 0  ./

(23); (24)



Q. .//:

(25)

Note that there are finitely many dyadic intervals whose diameter exceed ı. This fact along with (25) implies 0  ./  M  Q. .//

for each  2 B;

(26)

with some constant M > 0 independent of . Using Lemma 4.1, we find a non-trivial quasi-measure satisfying the formula (12). It follows from (12) and (26) that 2 HQ . Also we see from (12), (24), and (26), that the quasi-measure supported on the set F. The theorem is proved. t u

7 The Monotonicity Theorem for B-Functions Here we prove a monotonicity theorem B-functions. Below we apply this theorem to solve the uniqueness sets problem. Theorem 7.1 Suppose we are given a non-decreasing function Q W .0; 1 ! .0; 1 satisfying Q.x/ > 0; x!0C x lim

(27)

and a set E  G, and a B-subadditive B-function  2 HQ;  such that DB .t/  0

(28)

at every point t 2 G n E. If .J/ > 0 for some J 2 B, then E contains a closed subset F with mesQ F > 0. Proof This proof repeat in general the proof of the Theorem 2 in [7]. Divide the interval J into finitely many dyadic intervals. Since  2 AB , we have .J1 / > 0 for one of these intervals J1 . We introduce a B-function 1 W  2 B 7! ./"1  ./, where "1 > 0 is so small that 1 .J1 / > 0. Evidently, "1  ./ 2 AB . It follows from (10) and (27) that "1  ./ 2 HQ;  . Therefore 1 2 AB \ HQ;  . Let k0 be the rank of the J1 and consider the set FD

1 [ \ kDk0

;

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where the union runs over all dyadic intervals  J1 of rank k such that 1 ./ > 0. Being an intersection of closed sets, F is closed. The Lemmas 7.2 and 7.3 have proved in [7] (see the proof of the Theorem 2 in mentioned work). Lemma 7.2 F ¤ ;, F  E. Lemma 7.3 If t 2 F, then 1 ./ > 0 for every dyadic interval  of rank k  k0 containing t. We return to the proof of the theorem. For every M 2 P let FM be the set of points t 2 F such that the inequality 1 ./  M  Q . .//

(29)

S holds for all  D k;t . Since  2 HQ;  , we see that F D 1 MD1 FM . Since F is closed, it follows from the Baire Category Theorem [2] that there is M 2 P such that FM is dense on a non-empty portion of F. Hence, there is a dyadic interval J2  J1 with FM \ J2 D e F, where e F WD F \ J2 ¤ ;. By Lemma 7.3 we have 1 .J2 / D C > 0. Consider an arbitrary covering fXi g of the closed set e F by dyadic intervals. It was shown in [7] (see the proof of the Theorem 2 in the mentioned work) that there is a finite subcovering fYj g  fXi g of the set e F by disjoint dyadic intervals, and a finite family fZl g of disjoint dyadic intervals, such that 0 1 ! G G G J2 D @ Yj A Zl I j

X j

1 .Yj / C

l

X

l

1 .Zl /  1 .J2 /I

(30)

l

1 .Zl /  0

for all l:

(31)

We have X i

X

1 X .Yj / M j j   X (30) 1 (31) 1 .J2 / C 1 .J2 /  D :  1 .Zl /  M M M l

Q. .Xi // 

(29)

Q. .Yj // 

(32)

Since fXi g is an arbitrary covering of e F by dyadic intervals, the formula (32) yield that mesQe t u F > 0. Since F e F, we get mesQ F > 0. The theorem is proved. Theorem 7.4 Suppose we are given a non-decreasing function Q W .0; 1 ! .0; 1 satisfying the condition (27), and a set E  G, and a quasi-measure  2 HQ;  such

Q-Measures on the Dyadic Group

81

that DB .t/ D 0

(33)

at all points t 2 G n E. If .J/ ¤ 0 for some J 2 B, then E contains a closed subset F with mesQ F > 0. Proof Replacing, if necessary, the B-function  by the B-function , we obtain that .J/ > 0. All hypotheses of Theorem 7.1 are hold and, therefore, there is a t u closed set F  E with mesQ F > 0. The theorem is proved.

8 Uniqueness Sets for Haar Series Here we present new results for Haar series, concerning the uniqueness sets problem. Having a non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/, we denote by Uˆ the family of uniqueness sets for the series (8) whose Nth partial sums satisfy (20). Also we write Uˆ for the class of uniqueness sets for the series (8) whose partial sums SN .t/ satisfy (18) everywhere on G. Clearly, Uˆ Uˆ

(34)

for all admissible ˆ. Theorem 8.1 Consider an arbitrary non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/ satisfying (5), and the function Q W .0; 1 ! .0; 1 defined by (6). If E … Uˆ , then E contains a closed subset F with mesQ F > 0. Proof Since E … Uˆ , there is a non-trivial series .S/ of the form (8) such that .S/ satisfies the condition (18) everywhere on G, and .S/ converges to zero on G n E. Let be the quasi-measure isomorphic to this series. Since the series .S/ satisfies the condition (18) everywhere on G, Lemma 5.2 yields that the quasi-measure satisfies the condition (10) everywhere on G. Since .S/ converges to zero for all t 2 G n E, we have DB .t/ D 0 on G n E by Lemma 5.1. Finally,   Q.x/ (6) 1 1 Dˆ x x for every x > 0. Consequently, the function Q satisfies (27). Thus all hypotheses of Theorem 7.4 hold and, therefore, there is a closed subset F with mesQ F > 0. The theorem is proved. t u Theorem 8.2 Consider an arbitrary non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/ satisfying (5), and the function Q W .0; 1 ! .0; 1 defined by (6). If E contains a closed subset F with mesQ F > 0, then E … Uˆ .

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Proof By Theorem 6.1 there exists a non-trivial quasi-measure 2 HQ supported on the set F. We consider the series .S/ of the form (8) which is isomorphic to . The quasi-measure is non-trivial, hence the series .S/ is non-trivial, too. Further, 2 HQ and, therefore, .S/ satisfies the condition (20) by Lemma 5.3. Choose an arbitrary point t 2 G n E. Then t 2 G n F. Since the set G n F is open, there exists 0 2 B such that 0 3 t and 0  G n F. Since the quasi-measure 2 HQ is supported on the set F, ./ D 0 for each dyadic interval   0 . Hence, by Lemma 5.1 the series .S/ converges to zero at the point t. Thus there exists a non-trivial series .S/ of the form (8) satisfying (20) and converging to zero on G n E. Therefore, E … Uˆ . The theorem is proved. t u Along with the formula (34), Theorems 8.1 and 8.2 immediately imply the next result. Theorem 8.3 (Main Theorem) Consider an arbitrary non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/ satisfying (5), and the function Q W .0; 1 ! .0; 1 defined by (6). Then E 2 Uˆ \ Uˆ if and only if E contains no closed subsets F with mesQ F > 0. Corollary 8.4 Uˆ D Uˆ for every non-decreasing function ˆ W Œ1; C1/ ! Œ1; C1/ satisfying (5). Acknowledgements This work was completed with the support of the Russian Foundation for Basic Research (grant no. 14-01-00417), and of the program “Leading science schools” (grant no. NSh-3682.2014.1), and of the grant VGMHA-2014.

References 1. N.K. Bary (Bari), A Treatise on Trigonometric Series (Fizmatgiz, Moscow, 1961); Engl. transl., Pergamon Press, 1964 2. N. Bourbaki, General Topology: Chapters 5–10. Elements of Mathematics (Springer, Berlin, 1998) 3. B.I. Golubov, Series with respect to the Haar system. Itogi Nauki Ser. Mat. Anal. 109–146 (1971); Engl. transl., J. Soviet Math. 1(6), 704–726 (1973) 4. G.M. Mushegyan, Uniqueness sets for the Haar system. Izv. Akad. Nauk Armen. SSR Ser. Mat. 2(6), 350–361 (1967) (in Russian) 5. K.M. Ostaszewski, Henstock integration in the plane. Mem. Am. Math. Soc. 63 (1986), 1–106 6. M.G. Plotnikov, Uniqueness questions for some classes of Haar series. Mat. Zametki 75(3), 392–404 (2004); Engl. transl., Math. Notes 75(3), 360–371 7. M.G. Plotnikov, Quasi-measures, Hausdorff p-measures and Walsh and Haar series. Izv. RAN Ser. Mat. 74(4), 157–188 (2010); Engl. transl., Izvestia Math. 74(4), 819–848 (2010) 8. F. Schipp, W.R. Wade, P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis (Adam Hilger, Bristol/New York, 1990) 9. A.N. Shiryaev, Probability (Springer Science+Business Media New York, 1996) 10. V.A. Skvortsov, Calculation of the coefficients of an everywhere convergent Haar series. Mat. Sb. 75(117):3, 349–360 (1968); Engl. transl., Math. USSR Sb. 4(3), 317–327 (1968) 11. V.A. Skvortsov, Henstock-Kurzweil type integrals in P -adic harmonic analysis. Acta Math. Acad. Paedagog. Nyházi (N.S.) 20(2), 207–224 (2004)

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12. W.R. Wade, Sets of uniqueness for Haar series. Acta Math. Acad. Sci. Hungaricae 30(3–4), 265–281 (1977) 13. W.R. Wade, K. Yoneda, Uniqueness and quasi-measures on the group of integers of p-series field. Proc. Am. Math. Soc. 84(2), 202–206 (1982) 14. D. Williams, Probability with Martingales (Cambridge University Press, Cambridge, 1991) 15. A. Zygmund, Trigonometric Series, vol. I, II (Cambridge University Press, Cambridge, 1959)

Off-Diagonal and Pointwise Estimates for Compact Calderón-Zygmund Operators Paco Villarroya

Abstract We prove several off-diagonal and pointwise estimates for singular integral operators that extend compactly on Lp .Rn /. Keywords Calderón-Zygmund operator • Compact operator • Off-diagonal estimates, Singular integral

Mathematics Subject Classification (2010). Primary 42B20, 42C40; Secondary 47B07, 47G10

1 Introduction An operator is said to satisfy an off-diagonal estimate from Lp .Rn / into Lq .Rn / for p; q > 0 if there exists a function G W Œ0; 1/ ! Œ0; 1/ vanishing at infinity such that kT. f E / F kLq .Rn / . G.dist.E; F//k f kLp .Rn / for all Borel sets E; F  Rn and all f 2 Lp .Rn /, with implicit constant depending on the operator T and the exponents p; q. Some authors distinguish between properly off-diagonal estimates, when E \ F D ;, and the so-called on-diagonal estimates, when E \ F ¤ ;. However, we will not follow such convention and instead we will always call them off-diagonal estimates. In the specific case of singular integral operators, the study focuses on the exponents 1  p D q < 1. Very often, off-diagonal bounds are considered in

P. Villarroya () Centre for Mathematical Sciences, University of Lund, Lund, Sweden e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_7

85

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P. Villarroya

one of the two following dual forms: kT. I / .I/c kL1 .Rn / . G.`.I//jIj;

1 jIj

Z jT. .I/c /.x/jdx . G.`.I// I

(see [1, 11]), for any cube I  Rn and  > 1, where jIj denotes the volume of the cube and I is a concentric dilation of I. While their use in Analysis is very classical, the interest for this type of inequalities in modern Harmonic Analysis renewed in the 90s after the publication of new proofs of the T(1) Theorem that used the wavelet decomposition approach (see [2, 8] for example). These proofs were based on the development of estimates of the form jhT.

I /;

J ij

.

 jJj  12 C nı  jIj

1C

dist.I; J/ .nCı/ 1

jIj n

(1)

for all cubes I; J  Rn with jJj  jIj, under the appropriate hypothesis on the operator T, the parameter ı > 0 and the functions I ; J involved. The importance of off-diagonal inequalities lays mainly on two facts. On the one side, they are a satisfactory replacement for pointwise estimates of the operator kernel when these are not available or even when the operator kernel is unknown. On the other side, they completely enclose the almost orthogonality properties of the operator. For these reasons, they played a crucial role in the solution of the famous Kato’s conjecture [3] about boundedness of square root of elliptic operators and are nowadays extensively used in the study of second order elliptic operators. In the field, these estimates are typically established for one-parameter collections of operators .Tt /t>0 and the function G also depends on the parameter t in an appropriate manner (see [4–6, 9, 10, 12]). Finally, it is also worth mentioning that off-diagonal bounds provide very valuable information for the development of efficient algorithms to compress and rapidly evaluate discrete singular operators (see [7, 14]). One of the goals of the current paper is to establish similar type of estimates for singular integral operators that can be extended compactly on Lp .Rn / with 1 < p < 1. In [13], the author proved a characterization of these operators based on a new type of off-diagonal estimates for Calderón-Zygmund operators. Now, in the current paper, we aim to improve these bounds in several ways and also obtain some new estimates. More explicitly, we show in Sect. 3 that, in a broad sense and under the right hypotheses, these operators satisfy similar inequalities to (1) but with a new factor F that encodes the extra decay obtained as a consequence of their compactness properties: jhT.

I /;

J ij

.

 jJj  12 C ın  jIj

1C

dist.I; J/ .nCı/ 1

jIj n

F.I; J/:

(2)

Estimates for Compact Calderón-Zygmund Operators

87

The focus of this work is actually placed on obtaining a sharp and as detailed as possible description of the function F in the different cases under study. Furthermore, in Sect. 4 we establish pointwise estimates of the action of the operator over compactly supported functions. This allows to claim, in a broad sense as well, that the image of a bump function adapted and supported in a cube behaves as a bump function adapted, although not supported, to the same cube. As before, these estimates explicitly state an extra decay not present in the classical bounds that is again due to the compactness of the operator.

2 Notation and Definitions Q We say that the set I D niD1 Œai ; bi  is a cube in Rn if the quantity jbi ai j is constant when varying the index i. We denote by Qn the family of all cubes in Rn . For every cube I  Rn , we denote its centre by c.I/ D ..ai C bi /=2/niD1 , its side length by `.I/ D jbi  ai j and its volume by jIj D `.I/n . For any  > 0, we denote by I, the cube such that c.I/ D c.I/ and jIj D n jIj. We write j  jp for the lp -norm in Rn with 1  p  1 and j  j for the modulus of a complex number. Hopefully, the latter notation will not cause any confusion with the one used for the volume of a cube. We denote by B D Œ1=2; 1=2n and B D B D Œ=2; =2n . Given two cubes I; J  Rn , we define hI; Ji as the unique cube such that it contains I [ J with the smallest possible side length and whose center has the smallest possible first coordinate. In the last section, this notation will be applied also to points, namely hx; yi, as if they were considered to be degenerate cubes. We denote the side length of hI; Ji by diam.I [ J/. Notice that diam.I [ J/  `.I/=2 C jc.I/  c.J/j1 C `.J/=2  `.I/ C dist1 .I; J/ C `.J/ where dist1 .I; J/ denotes the set distance between I and J calculated using the norm j  j1 . Actually, 1 `.I/ `.J/ diam.I [ J/  C jc.I/  c.J/j1 C  diam.I [ J/: 2 2 2 We define the relative distance between I and J by rdist.I; J/ D

diam.I [ J/ ; max.`.I/; `.J//

which is comparable to max.1; n/ where n is the smallest number of times the larger cube needs to be shifted a distance equal to its side length so that the translated cube

88

P. Villarroya

contains the smaller one. The following equivalences hold: rdist.I; J/  1 C max.`.I/; `.J//1 jc.I/  c.J/j1  1 C max.`.I/; `.J//1 dist1 .I; J/: Finally, we define the eccentricity of I and J as ecc.I; J/ D

min.jIj; jJj/ : max.jIj; jJj/

Definition 2.1 In order to characterize compactness of singular integral operators, we use two sets of auxiliary bounded functions L; S; D W Œ0; 1/ ! Œ0; 1/ and F W Qn ! Œ0; 1/ satisfying the following limits lim L.x/ D lim S.x/ D lim D.x/ D 0;

x!1

lim F.I/ D lim F.I/ D

`.I/!1

(3)

x!1

x!0

`.I/!0

lim

jc.I/j1 !1

F.I/ D 0:

(4)

Remark 2.2 Since any dilation D L.x/ D L.1 x/, D F.I/ D F.1 I/ with L and F satisfying (3), (4) respectively still satisfies the same limits, we will often omit all universal constants appearing in the arguments. Definition 2.3 Let K W .Rn  Rn / n f.t; x/ 2 Rn  Rn W t D xg ! C. We say that K is a compact Calderón-Zygmund kernel if there exist constants 0 < ı  1, C > 0 and functions L, S and D satisfying the limits in (3), such that jK.t; x/  K.t0 ; x0 /j  C

.jt  t0 j1 C jx  x0 j1 /ı jt  xjnCı 1

FK .t; x/

(5)

whenever 2.jt  t0 j1 C jx  x0 j1 / < jt  xj1 , with FK .t; x/ D L.jt  xj1 /S.jt  xj1 /D.jt C xj1 /: We say that K is is a standard Calderón-Zygmund kernel if (5) is satisfied with FK  1. We first note that, without loss of generality, L and D can be assumed to be noncreasing while S can be assumed to be non-decreasing. This is possible because, otherwise, we can always define L1 .x/ D sup L.y/ y2Œx;1/

S1 .x/ D sup S.y/ y2Œ0;x

D1 .x/ D sup D.y/ y2Œx;1/

which bound above L, S and D respectively, satisfy the limits in (3) and are noncreasing or non-decreasing as requested.

Estimates for Compact Calderón-Zygmund Operators

89

On the other hand, we also denote  FK .t; t0 ; x; x0 / D L2 .jt  xj1 /S2 .jt  t0 j1 C jx  x0 j1 /D2 1 C

jt C xj1  1 C jt  xj1

and assume, in a similar way as before, that L2 and D2 are non-creasing while S2 is non-decreasing. Then, as explained in [13], (5) is equivalent to the following smoothness condition jK.t; x/  K.t0 ; x0 /j  C

.jt  t0 j1 C jx  x0 j1 /ı jt 

0 xjnCı 1

0

FK .t; t0 ; x; x0 /

(6)

whenever 2.jt  t0 j1 C jx  x0 j1 / < jx  tj1 , with possibly smaller ı 0 < ı. This resulting parameter ı 0 necessarily satisfies ı 0 < 1 since otherwise the kernel K would be a constant function. The proof of the equivalence between both formulations appears in [13]. However, to increase readability of the current paper, we sketch the proof that (5) implies (6). For any 0 <  < ı, let ı 0 D ı  . Then, from (5) we can write 0

jK.t; x/  K.t0 ; x0 /j  C

.jt  t0 j1 C jx  x0 j1 /ı Q FK .t; t0 ; x; x0 / 0 jt  xjnCı 1

with .jt  t0 j1 C jx  x0 j1 / FQ K .t; t0 ; x; x0 / D FK .t; x/: jt  xj1 Then, the functions L2 .y/ D S2 .y/ D D2 .y/ D

sup jxtj1 y

FQ K .t; t0 ; x; x0 /1=3

sup

jxx0 j1 Cjtt0 j1 y

sup jxCtj1 1C 1Cjxtj y 1

FQ K .t; t0 ; x; x0 /1=3

FQ K .t; t0 ; x; x0 /1=3

satisfy all the required limits in (3) and  FQ K .t; t0 ; x; x0 /  L2 .jt  xj1 /S2 .jt  t0 j1 C jx  x0 j1 /D2 1 C

jt C xj1  : 1 C jt  xj1

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As also proved in [13], the smoothness condition (5) and the hypothesis lim K.t; x/ D 0 imply the classical decay condition

jtxj1 !1

jK.t; x/j .

1 1 FK .t; x/ . n jt  xj1 jt  xjn1

(7)

for all t; x 2 Rn such that t ¤ x. Moreover, it is easy to see that we also get the decay jK.t; x/j .

 1 jt C xj1  ; L.jt  xj /S.jt  xj /D 1 C 1 1 jt  xjn1 1 C jt  xj1

(8)

which we will use later. Notice the change in the argument of S, which is now equal to the argument of L. Finally, we define two more sets of auxiliary functions which we will use in the next section. First, FK .I1 ; I2 ; I3 / D L.`.I1 //S.`.I2 //D. rdist.I3 ; B//

(9)

and FK .I/ D FK .I; I; I/. Second, Q rdist.I3 ; B// FQ K .I1 ; I2 ; I3 / D L.`.I1 //S.`.I2 //D.

(10)

and FQ K .I/ D FQ K .I; I; I/, where Q rdist.I; B// D D.

X

2jı D. rdist.2j I; B//:

(11)

j0

Q We note that for fixed `.I/, the Lebesgue Dominated Theorem guarantees that D Q rdist.I; B// D 0. satisfies lim D. jc.I/j1 !1

Definition 2.4 Let T W C0 .Rn / ! C0 .Rn /0 be a continuous linear operator. We say that T is associated with a compact Calderón-Zygmund kernel if there exists a function K fulfilling Definition 2.3 such that the dual pairing satisfies the following integral representation Z hT. f /; gi D

Rn

Z Rn

f .t/g.x/K.t; x/ dt dx

for all functions f ; g 2 C0 .Rn / with disjoint compact supports. Clearly, the integral converges absolutely since, by (7), we have for d D dist.supp f ; supp g/ > 0, ˇZ ˇ ˇ

Z Rn

ˇ 1 ˇ f .t/g.x/K.t; x/ dtdxˇ . k f kL1 .Rn / kgkL1 .Rn / n : n d R

Estimates for Compact Calderón-Zygmund Operators

91

Definition 2.5 Let 0 < p  1. We say that a bounded function  is an Lp .Rn /normalized bump function adapted to I with constant C > 0, decay N 2 N and order 0, if for all x 2 Rn  1 jr  c.I/j1 N j.x/j  CjIj p 1 C : `.I/

(12)

We say that a continuous bounded function  is an Lp .Rn /-normalized bump function adapted to I with constant C > 0, decay N 2 N, order 1 and parameter 0 < ˛  1, if (12) holds and for all t; x 2 Rn j.t/  .x/j  C

 jt  xj ˛ 1

`.I/

1

jIj p sup



1C

r2ht;xi

jr  c.I/j1 N ; `.I/

(13)

where ht; xi denotes the cube containing the points t and x with the smallest possible side length and whose centre has the smallest possible first coordinate. Unless otherwise stated, we will assume the bump functions to be L2 .Rn /normalized. Definition 2.6 We say that a linear operator T W C0 .Rn / ! C0 .Rn /0 satisfies the weak compactness condition, if there exists a bounded function FW satisfying (4) and such that for any cube I  Rn and any bump functions I , 'I adapted to I with constant C > 0, decay N and order 0, we have jhT.I /; 'I ij . CFW .I/ where the implicit constant only depends on the operator T. As explained in [13], this definition admits several other reformulations, but they all essentially imply that the dual pairing hT.I /; 'I i tends to zero when the cube involved is large, small or far away from the origin. Definition 2.7 We define CMO.Rn / as the closure in BMO.Rn / of the space of continuous functions vanishing at infinity. The following theorem, which is the main result in [13], characterizes compactness of Calderón-Zygmund operators. This is the reason why we say that the new off-diagonal bounds appearing in the current paper apply to operators that can be extended compactly on Lp .Rn /. Theorem 2.8 Let T be a linear operator associated with a standard CalderónZygmund kernel. Then, T extends to a compact operator on Lp .R/ for all 1 < p < 1 if and only if T is associated with a compact Calderón-Zygmund kernel and it satisfies the weak compactness condition and the cancellation conditions T.1/; T  .1/ 2 CMO.R/.

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3 Off-Diagonal Estimates for Bump Functions In the proof of Theorem 2.8, some off-diagonal estimates were developed. Now, we improve these inequalities in several directions: by extending the result to Rn , by weakening the smoothness requirements of the bumps, by shortening the proof and by obtaining a sharper bound for functions with compact support. This is the purpose of the three propositions of this section, which describe the action of a compact singular integral operator over bump functions with or without zero mean properties respectively. Later, in Sect. 4, we will use these bounds to obtain several pointwise bounds and other off-estimates of a more general type. We first set up some notation that appears in the statements of the three results. We consider K to be a compact Calderón-Zygmund kernel with parameter 0 < ı < 1 and T to be a linear operator with associated kernel K satisfying the weak compactness condition. We denote by I ^J and I _J the smallest and the largest of two given cubes I; J respectively. That is, I ^J D J, I _J D I if `.J/  `.I/, while I ^J D I, I _J D J, otherwise. We also remind the notation of FK , FW and FQ K provided in the previous section. Proposition 3.1 If the special cancellation conditions T.1/ D T  .1/ D 0 hold then, for all bump functions I , J adapted and supported on I, J respectively, with constant C > 0, order one, parameter ˛ > ı and such that I ^ J has mean zero, 1

ı

ecc.I; J/ 2 C n jhT. I /; J ij . C F.I; J/ rdist.I; J/nCı 2

(14)

where F is such that: (i) F.I; J/ D FK .hI; Ji; I ^J; hI; Ji/ when rdist.I; J/ > 3, (ii) F.I; J/ D FQ K .I _J; I ^J; I _J/ C FW .I ^J/ C FK .I ^J; I ^J; I _J/, otherwise. Proposition 3.2 For all bump functions I , J adapted and supported on I, J respectively, with constant C > 0, order one, parameter ˛ > ı and such that I ^ J has mean zero, we have 1

ecc.I; J/ 2 jhT. I /; J ij . C F.I; J/ rdist.I; J/nCı 2

(15)

with F.I; J/ D FQ K .I ^J/ C FW .I ^J/ C FK .I ^J; I ^J; I _J/ when rdist.I; J/  3. On the other hand, when rdist.I; J/ > 3, inequality (14) still holds with the same F.I; J/ D FK .hI; Ji; I ^J; hI; Ji/. Proposition 3.3 For all bump functions I , J adapted and supported on I, J respectively, with constant C > 0 and order zero, we have 1

jhT.

I /;

J ij

. C2

ˇ  ˇ ecc.I; J/ 2  1 C ˇ log ecc.I; J/ˇ F.I; J/ n rdist.I; J/

(16)

Estimates for Compact Calderón-Zygmund Operators

93

where (i) F.I; J/ D FK .hI; Ji/ and  D 0 when rdist.I; J/ > 3, (ii) F.I; J/ D FW .I ^J/ C FK .I ^J; I _J; I _J/ and  D 1 when rdist.I; J/  3. In all cases, the implicit constants depend on the operator T and the parameters ı and ˛ but they are universal otherwise. Needless to say that the actual value appearing in the condition rdist.I; J/ > 3 plays no special role and it could be easily changed by any other value strictly larger than one. As mentioned before, Proposition 3.1 is an improvement of the analog result in [13]. The result has been extended to non-smooth bump functions of several dimensions. At the same time, the proof has been largely simplified by using the extra hypothesis that the bump functions are compactly supported. Moreover, with this hypothesis, the last factor on the right hand side of the inequality turned out to be strictly smaller than the one appearing in [13]. In fact, when the bump functions are not longer compactly supported, as it happens in [13], the inequality (14) holds with a larger factor depending on six different cubes rather than only three cubes. Nevertheless, in both cases, the factors enjoy essentially the same properties and so, each of the two estimates suffices to prove compactness of the operator. We also note that in Proposition 3.2, the hypotheses that T.1/; T  .1/ 2 BMO.Rn / or T.1/; T  .1/ 2 CMO.Rn / are not needed. Moreover, in Proposition 3.3, the assumption of T satisfying the special cancellation conditions T.1/ D T  .1/ D 0 does not lead to any further improvement. Notation 3.4 For the following three proofs, we provide some common notation. For every cube I  Rn , we denote by ˆI 2 S.Rn / an L1 -normalized function adapted to I with arbitrary large order and decay such that 0  ˆI  1, ˆI D 1 in 2I and ˆI D 0 in .4I/c . This implies that ˆI .x/ D 1 for all jx  c.I/j1  `.I/ while ˆI .x/ D 0 for all jx  c.I/j1 > 2`.I/. As customary, we define the translation and dilation operators by Ta f .x/ D f .x  a/ and D f .x/ D f .1 x/ respectively with x; a 2 Rn and  > 0. We also define wI .x/ D 1 C `.I/1 jx  c.I/j1 and for any function D 1 ˝ 2 of tensor product type, we write ƒ. / D hT. 1 /; 2 i. Finally, by symmetry we can assume that `.J/  `.I/ and so, I ^J D J while I _J D I. Proof of Proposition 3.1 Let .t; x/ D I .t/ J .x/ which, by hypothesis, is supported and adapted to I  J with constant C2 , decay N, order 1, parameter ˛ > ı and, most importantly, it has mean zero in the variable x. (a) We first assume that 3`.I/ < diam.I [ J/ which implies .5I/ \ J D ; and so, diam.I [ J/ D `.I/=2 C jc.I/  c.J/j1 C `.J/=2  `.I/ C jc.I/  c.J/j1 . Then, since jt  c.I/j1  `.I/=2, we have jt  c.J/j1  `.I/=2 C jc.I/  c.J/j1  diam.I [ J/

(17)

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and jt  c.J/j1  jc.I/  c.J/j1  jt  c.I/j1  `.I/ C jc.I/  c.J/j1  3`.I/=2  diam.I [ J/  diam.I [ J/=2 D diam.I [ J/=2: On the other hand, 3`.I/ < diam.I [ J/  `.I/ C jc.I/  c.J/j1 also implies 2`.I/ < jc.I/  c.J/j1 and since jx  c.J/j1  `.J/=2, we get jt  c.J/j1  jc.I/  c.J/j1  jt  c.I/j1  2`.I/  `.I/=2  3`.J/=2  3jx  c.J/j1 : The last inequality implies that the support of is disjoint with the diagonal and so, we can use the Calderón-Zygmund kernel representation to write Z Z

Z Z ƒ. / D

.t; x/.K.t; x/  K.t; c.J/// dtdx

.t; x/K.t; x/ dtdx D J

I

where the second equality is due to the zero mean of in the variable x. Now, we denote QI;J D ft 2 Rn W diam.I [ J/=2 < jt  c.J/j1  diam.I [ J/g. Then, by the smoothness condition (6) of a compact Calderón-Zygmund kernel and the monotonicity properties of L, S and D, we bound as follows: Z Z jƒ. /j .

j .t; x/j J

I\QI;J

jx  c.J/jı1 jt  c.J/jnCı 1

jt C c.J/j1  dxdt 1 C jt  c.J/j1   `.J/ı jc.J/j1 . k kL1 .R2n / L.diam.I [ J//S.`.J//D 1 C diam.I [ J/nCı 1 C diam.I [ J/  L.jt  c.J/j1 /S.jx  c.J/j1 /D 1 C

`.J/ı L.`.hI; Ji//S.`.J//D. rdist.hI; Ji; B// diam.I [ J/nCı  jJj  12 C ın  diam.I [ J/ .nCı/ FK .hI; Ji; J; hI; Ji/ D C2 jIj `.I/ 1

1

. C2 jIj 2 jJj 2

as stated. To completely finish this case, we explain in more detail the reasoning to obtain the bounds for D used in the second and third inequalities above. Since jxj1  .jx  tj1 C jx C tj1 /=2, we have 1C

jt C xj1 jt C xj1  jxj1 1 3 : 1C 1C C  1 C jt  xj1 2 1 C jt  xj1 2 1 C jt  xj1

Estimates for Compact Calderón-Zygmund Operators

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Then, in the domain of integration, 1C

jc.J/j1 jt C c.J/j1  jc.J/j1 3 1C 1C  1 C diam.I [ J/ 1 C jt  c.J/j1 2 1 C jt  c.J/j1

and, since D is non-creasing, we have  D 1C

  jc.J/j1 jt C c.J/j1  D 1C 1 C jt  c.J/j1 1 C diam.I [ J/

omitting constants. On the other hand, since jc.I/j1  jc.J/j1  jc.I/  c.J/j1  diam.I [ J/, we can bound below the numerator of the argument of D in the last expression by 1 1 1 C diam.I [ J/ C jc.J/j1  1 C diam.I [ J/ C .jc.I/j1  jc.J/j1 / C jc.J/j1 2 2 1 1 D 1 C diam.I [ J/ C .jc.I/j1 C jc.J/j1 / 2 2  1 1  1 C diam.I [ J/ C jc.I/ C c.J/j1 : 2 2

Then, 1C

1 1 C diam.I [ J/ C jc.I/ C c.J/j1 =2 jc.J/j1 / 1 C diam.I [ J/ 2 1 C diam.I [ J/  jc.I/ C c.J/j1 =2  1 3 C :  3 2 diam.I [ J/

Finally, since j.c.I/ C c.J//=2  c.hI; Ji/j1  `.hI; Ji/=2 and `.hI; Ji/ D diam.I [ J/, we bound below previous expression by jc.hI; Ji/j1 1  1  13 jc.hI; Ji/j1  1 C   1C D rdist.hI; Ji; B/: 3 2 diam.I [ J/ 2 3 max.`.hI; Ji/; 1/ 3 (b) We now assume that diam.I [ J/  3`.I/ which implies 1  rdist.I; J/  3. In this case, we first show that we can assume .c.J/; x/ D 0 for any x 2 Rn . This assumption comes from the substitution of .t; x/ by .t; x/  .Tc.J/ D`.I/ ˆ/.t/ .c.J/; x/

(18)

where ˆ D ˆB as described in Notation 3.4. Then, we only need to prove that the subtracted term satisfies the desired bound.

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We denote Q .x/ D .c.J/; x/. Since I and J are adapted to I and J respectively with constant C > 0 and decay N for any N 2 N, we have 1

1

j Q .x/j  C2 jIj 2 jJj 2 wJ .x/N : 1 1 Then, k Q kL1 .Rn /  C2 jIj 2 jJj 2 . We also recall that Q is supported on J and has mean zero. Now, we write  D `.I/=`.J/  1 and take k 2 N so that 2k   < 2kC1 . Then,

Tc.J/ D`.I/ ˆ D Tc.J/ D`.J/ ˆ: To simplify notation, we write ˆ0 D Tc.J/ D`.I/ ˆ 2 S.Rn / and ˆ1 D 1  ˆ0 . We note that ˆ1 is a smooth bounded function supported on jt  c.J/j1 > `.J/. By the classical theory, we know that T.1/ can be defined as a distribution acting on the space of compactly supported functions with mean zero in the following way hT.1/; Q i D hT.ˆ0 /; Q i C D hT.ˆ0 /; Q i C

Z Z Z Z

ˆ1 .t/ Q .x/K.t; x/dtdx ˆ1 .t/ Q .x/.K.t; x/  K.t; c.J///dtdx

where the second equality is due to the mean zero of Q . Notice that, since jx  c.J/j1  `.J/=2  `.J/2k1  21 jt  c.J/j1 , the supports of ˆ1 and Q are disjoint and so the integral in the second line converges absolutely. Then, the hypothesis that T.1/ D 0 implies hT.ˆ0 /; Q i D 

Z Z

ˆ1 .t/ Q .x/.K.t; x/  K.t; c.J///dtdx:

Moreover, since 2jx  c.J/j1 < jt  c.J/j1 , we can use the the smoothness condition (6) of a compact Calderón-Zygmund kernel to write jhT.ˆ0 /; Q iij 

Z Z J

jˆ1 .t/jj Q .x/j

jx  c.J/jı1

jt  c.J/jnCı 1  L.jt  c.J/j1 /S.jx  c.J/j1 /D 1 C

`.J/ 0, T.I / does not have in general compact support and its decay is only comparable to jxjn . Both facts are typical of bounded singular integral operators. However, if the operator is associated with a compact CalderónZygmund kernel, the decay of T.I / improves depending on the rate of decay of the factor L.`.hI; xi// when x tends to infinity. On the other hand, note the gain in smoothness with respect bounded singular integrals provided by the factor jx  x0 jı1 S.jx  x0 j1 /. We show now an off-diagonal estimate for general functions deduced directly from the previous pointwise bound. Proposition 4.7 Let f an integrable function supported on a cube I. Then, for all 1 < p < 1 and all  > 1, we have kT. f / .I/c kL p .Rn / .

1 n

.1 C / p0

sup FK .hI; xi/k f kL p .Rn / :

x2.I/c

Proof From Proposition 4.5, we have Z p

kT. f / .I/c kL p .Rn / .

.I/c

FK .hI; xi/p p dxjIjp k f kL1 .I/ .1 C `.I/1 jx  c.I/j1 /np

.

`.I/n p sup FK .hI; xi/p jIjp k f kL1 .I/ .1 C /n.p1/ x2.I/c

D

1 p sup FK .hI; xi/p jIj.p1/ k f kL1 .I/ .1 C /n.p1/ x2.I/c

which, by Hölder, is smaller than the right hand side of the stated inequality.

t u

We end the paper by adding few remarks to the previous proposition. We first note that, in the particular case of f being a bump function, we obtain 1

1

kT.I / .I/c kL p .Rn / . jIj p  2

1 n

.1 C / p0

sup FK .hI; xi/:

x2.I/c

Estimates for Compact Calderón-Zygmund Operators

111

We also remind that, for bounded but not compact singular integral operators, the analog of Proposition 4.7 implies that for a fixed cube I and k f kLp .I/  1, we have lim kT. f / .I/c kLp .Rn / D 0

!1

n

with a rate of decay at most of order  p0 . However, for compact singular integral operators, the extra factor stated in Proposition 4.7 ensures that there is always an extra gain in decay. To see this, we note that for all x 2 .I/c we have I  3hI; xi and so, `.I/  3`.hI; xi/. Hence, FK .hI; xi/ . L.`.hI; xi// . L.`.I// and since n lim!1 L.`.I// D 0, the rate of decay is now at worst as fast as  p0 L.`.I//. Finally, since we also have the bound  jc.I/j1  FK .hI; xi/ . L.`.I//D 1 C 1 C `.I/ we deduce that for fixed  and k f kLp .I/  1 we have that lim kT. f / .I/c kLp .Rn / D 0

`.I/!1

while, for fixed , k f kLp .I/  1 and fixed `.I/, we also get lim

jc.I/j1 !1

kT. f / .I/c kLp .Rn / D 0:

The last two properties do not hold in general for bounded singular integral operators. Acknowledgements This work was completed with the support of Spanish project MTM201123164.

References 1. P. Auscher, Jose-Maria Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: off-diagonal estimates on spaces of homogeneous type. J. Evol. Equ. 7(2), 265–316 (2007) 2. P. Auscher, Ph. Tchamitchian, Bases d’Ondelettes sur les Courbes Corde-Arc, Noyau de Cauchy et Espaces de Hardy Associés. Rev. Mat. Ib. 5(3, 4), 139–170 (1989) 3. P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn . Ann. Math. 2, 633–654 (2002) 4. P. Auscher, C. Kriegler, S. Monniaux, P. Portal, Singular integral operators on tent spaces. J. Evol. Equ. 12(4), 741–765 (2012) 5. A. Axelson, S. Keith, A. McIntosh, Quadratic estimates and functional calculi of perturbated dirac operators. Invent. Math. 163(3), 455–497 (2006)

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6. A. Axelson, S. Keith, A. McIntosh, The Kato square problem for mixed boundary problems. J. Lond. Math. Soc. 74(1), 113–130 (2006) 7. G. Beylkin, R.R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44, 141–183 (1991) 8. R.R. Coifman, Y. Meyer, Wavelets, Calderon-Zygmund and Multilinear Operators (Cambridge University Press, Cambridge, 1997) 9. S. Hofmann, J.-M. Martell, Lp bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ. Math. 47, 497–515 (2003) 10. S. Hofmann, S. Mayboroda, A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in Lp , Sobolev and Hardy spaces. Ann. Sci. École Norm. Sup. 4(44), 723–800 (2011) 11. T. Hytönen, F. Nazarov, The local T(b) theorem with rough test functions. arxiv:1201.0648 (to appear) 12. A. Rosen, Square function and maximal function estimates for operators beyond divergence form equations. J. Evol. Equ. 13, 651–674 (2013) 13. P. Villarroya, A characterization of compactness for singular integrals. J. Math. Pures Appl. 104(3), 485–532 (2015) 14. Q.X. Yang, Fast algorithms for Calderón-Zygmund singular integral operators. Appl. Comput. Harmon. Anal. 3(2), 120–126 (1996)

Part II

Function Spaces of Radial Functions

Elementary Proofs of Embedding Theorems for Potential Spaces of Radial Functions Pablo L. De Nápoli and Irene Drelichman

Abstract We present elementary proofs of weighted embedding theorems for radial potential spaces and some generalizations of Ni’s and Strauss’ inequalities in this setting. Keywords Embedding theorems • Potential spaces • Power weights • Radial functions • Sobolev spaces

Mathematics Subject Classification (2000). Primary 46E35; Secondary 35A23

1 Introduction: Sobolev Spaces and Embedding Theorems The aim of this note is twofold: to review some known results from the theory of radial functions in Sobolev (potential) spaces, and to extend some of this results to the setting of weighted radial spaces. In both cases, we provide new elementary proofs that avoid the use of interpolation theory and sophisticated tools such as atomic or wavelet decompositions. This will have, at times, the limitation of not giving the most general possible result, in which case we will do our best to provide suitable references. However, we believe that the proofs presented here have the merit of being closer in spirit to some classical theorems (such as the Sobolev embedding theorem) and that the results obtained are good enough for common applications to the theory of partial differential equations. Before we state the results we are interested in, let us quickly recall that for Sobolev spaces of integer order, the most classical definition is in terms of

P.L. De Nápoli • I. Drelichman () Facultad de Ciencias Exactas y Naturales, IMAS (UBA-CONICET) and Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_8

115

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derivatives: given a domain   Rn , for integer k 2 N, W k;p ./ D fu 2 Lp ./ W D˛ u 2 Lp ./ for any ˛ 2 Nn0 with j˛j  kg where the derivatives D˛ u are understood in weak (or distributional) sense. It is also usual to denote W k;2 ./ by H k ./. When it comes to Sobolev spaces of fractional order, there exist several definitions in the literature, among others: a) Sobolev spaces of fractional order based on L2 .Rn / can be defined in terms of the Fourier transform: for real s  0, let Z H s .Rn / D fu 2 L2 .Rn / W jOu.!/j2 .1 C j!j2 /s d! < 1g: Rn

For integer s, we have that H s .Rn / D W s;2 .Rn /. b) One way to define spaces of fractional order based on Lp .Rn / when p ¤ 2 is given by the classical potential spaces H s;p .Rn / for s > 0, defined by: H s;p .Rn / D fu W u D .I  /s=2 f with f 2 Lp .Rn /g: Here, 1 < p < 1 and the fractional power .I  /s=2 can be defined by means of the Fourier transform (for functions in the Schwartz class): .I  /s=2 f D F 1 ..1 C j!j2 /s=2 F .f // D Gs f where Gs .x/ D F

1

2 s=2

..1 C jxj /

p Z .2 /n 1 t jxj2 sn dt /D e e 4t t 2 .s=2/ 0 t

(1)

is called the Bessel potential (see, e.g. [43, Chap. V]). Some classical references on potential spaces are [3, 43] and [5]. Notice that this family includes all the previous spaces (when  D Rn ) since we have that H 1;p .Rn / D W 1;p .Rn / for 1 < p < 1 by [5, Theorem 7]. Also H s;2 .Rn / D H s .Rn / for any s  0 by Plancherel’s theorem. c) Another (non-equivalent) definition of fractional Sobolev spaces is given by the Aronszajn-Gagliardo-Slobodeckij spaces, for 0 < s < 1; 1  p < 1: ( W ./ D u 2 L ./ W s;p

p

ju.x/  u.y/j n

jx  yj p Cs

) 2 L .  / : p

See [17] for a full exposition. We have that H s;2 .Rn / D W s;2 .Rn / D H s .Rn / for any 0 < s < 1, but for p ¤ 2, H s;p .Rn / and W s;p .Rn / are different.

Embedding Theorems for Potential Spaces of Radial Functions

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d) Even more general families of functional spaces are those of Besov-Lipschitz and Triebel-Lizorkin spaces. Indeed, the Littlewood-Paley theory says that H s;p .Rn / s coincides with the Triebel-Lizorkin space Fp;2 .Rn /, whereas W s;p .Rn / (as defined in c)) coincides with the Besov-Lipschitz space Bsp;p .Rn /. See for instance [2] for a full exposition of the theory of Sobolev spaces, and [50] for a discussion of the relations of the different functional spaces with the scales of Besov-Lipschitz and Triebel-Lizorkin spaces. In this note, we shall focus on potential spaces. A key result in the theory is the Sobolev embedding theorem, that reads as follows: Theorem 1.1 (Classical Sobolev Embedding) [3, Sect. 10],[5, Theorem 6] Assume that sp < n and define the critical Sobolev exponent p as p WD

np : n  sp

Then, there is a continuous embedding H s;p .Rn /  Lq .Rn /

for p  q  p :

(2)

Embedding theorems like this one play a central role when one wants to apply the direct methods of calculus of variations to obtain solutions of nonlinear partial differential equations. Consider, for simplicity, the following elliptic nonlinear boundary-value model problem in a smooth domain   Rn :

u D uq1 in  uD0 on @

(3)

It is well known that solutions of this problem can be obtained as critical points of the energy functional 1 J.u/ D 2

Z

1 jruj  q  2

Z jujq 

in the natural energy space, which is the Sobolev space H01 ./, i.e., the closure of C01 ./ in H 1 ./ (observe that we need to restrict u to this subspace in order to reflect the Dirichlet boundary condition; this implies in turn that the functional is coercive). The Sobolev embedding gives that H 1 ./  Lq ./

2  q  2 D

2n n2

which implies that J is well defined (and actually of class C1 ) for q within that range. Then, one can use minimax theorems like the mountain pass theorem of Ambrosetti and Rabinowitz (see [35] or [51]), in order to obtain critical points

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of J that correspond to non-trivial solutions of our elliptic problem. However, this kind of theorems usually need some compactness assumption like the Palais-Smale condition: For any sequence un in the space H01 ./ such that J.un / ! 0 and J 0 .un / ! 0, there exist a convergent subsequence unk In the subcritical case 2  q < 2 , this condition is easily verified by using the Rellich–Kondrachov theorem (see, e.g., [2, Chap. 6]): Theorem 1.2 If   Rn is a bounded domain, and 2  q < 2 then the Sobolev embedding H01 ./  Lq ./ is compact. See [35] or [51] for more information on variational methods for nonlinear elliptic problems. When  is unbounded, for instance  D Rn , the situation is different, since the embedding (2) is continuous but not compact, due to the invariance of the H s;p and Lq norms under translations. Indeed, if we choose u 2 C01 ./nf0g the sequence un .x/ D u.x C nv/ where v ¤ 0 is a fixed vector in Rn , is a bounded sequence in H s;p .Rn / without any convergent subsequence. Assume now that we want to find solutions of the analogous subcritical problem  u C u D uq1

2 < q < 2 D

2n n1

(4)

in Rn . Due to the lack of compactness of the Sobolev embedding, the previous approach does not work. However, we can still get compactness and hence nontrivial weak solutions, by restricting the energy functional J.u/ D

1 2

Z

jruj2 C Rn

1 2

Z

juj2  Rn

1 q

Z jujq Rn

1 to the subspace Hrad .Rn / of functions in H 1 .Rn / with radial symmetry, i.e. such that u.x/ D u0 .jxj/. Indeed we can make use of the following result, due to Lions [27]:

Theorem 1.3 Let n  2. For 2 < q < 2 , the embedding 1 .Rn /  Lq .Rn / Hrad

is compact.

Embedding Theorems for Potential Spaces of Radial Functions

119

Remark 1.1 When q D 2 we don’t have compactness even for radial functions, due to the rescaling invariance. The same happens for q D 2, since otherwise the Laplacian would have a strictly positive principal eigenvalue in Rn . Moreover, the orthogonal group O.n/ acts naturally on H 1 .Rn / by .g  u/.x/ D u.g  x/

g 2 O.n/

1 is precisely the and the functional J is invariant by this action. Furthermore, Hrad subspace of invariant functions under this action. Hence, the principle of symmetric 1 criticality [34] implies that the critical points of J on Hrad .Rn / are also critical points 1 n of J on H .R /, and hence weak solutions of the elliptic problem (4). This phenomenon of having better embeddings for spaces of radially symmetric functions (or more generally, for functions invariant under some subgroup of the orthogonal group) is well known, and goes back to the pioneering work by Ni [31] and Strauss [47], who proved the following theorems:

Theorem 1.4 ([31]) Let B be the unit ball in Rn .n  3/, Then every function in 1 H0;rad .B/ is almost everywhere equal to a function in C.B  f0g/ such that ju.x/j  Cjxj.n2/=2 krukL2 with C D .!n .n  2//1=2 , !n being the surface area of @B. 1 .Rn / is almost Theorem 1.5 ([47]) Let n  2. Then every function in Hrad n everywhere equal to a function in C.R  f0g/ such that

ju.x/j  C jxj.n1/=2 kukH 1 : In both cases, by density, it is enough to establish these inequalities for smooth radial functions. Once the inequality is known for smooth functions, if un is a sequence of smooth functions convergent to u in H 1 , the inequality implies that un converges uniformly on compact sets of Rn  f0g and hence, u is equal almost everywhere to a continuous function outside the origin that satisfies the same inequality. In the sequel we shall use this observation without further comments. Even though both inequalities look quite similar at first glance, they have different features: 1. Observe that the exponent of jxj in each inequality is different. As a consequence, 1 for a function in Hrad .Rn / the estimate provided by Ni’s inequality is sharper near the origin, whereas Strauss’ inequality is better at infinity. 2. Ni’s inequality is invariant under scaling. As a consequence, one has a.e. ju.x/j  Cjxj.n2/=2 krukL2

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1 for u 2 Hrad .Rn /. On the other hand, Strauss’ inequality is not invariant by rescaling. However, replacing u by u.x/ and minimizing over , one can get the following rescaling-invariant version: 1=2

1=2

ju.x/j  C jxj.n1/=2 kukL2 krukL2 : For elliptic problems other than (3) that involve other operators, we need different functional spaces, for instance, for the case of the p-Laplacian p u WD 1;p div.jrujp2 ru/ the natural space is W0 ./ (see, e.g., [16]), and for the fractional s=2 Laplacian ./ the natural space is H s .Rn / (see, e.g., [41]). The advantage of working with potential spaces is, as noted above, that they include all these cases in 1 a unified framework. As in the case of H 1 and Hrad , the corresponding subspaces of radially symmetric functions of the above spaces will be denoted by the subscript rad. This paper is organized as follows. In Sects 2 and 3 we recall some results on fractional integrals and derivatives, respectively. Then we give two easy proofs of Strauss’ inequality for potential spaces, one for p D 2 (in Sect. 4) and another one for general p (in Sect. 5). Section 6 is devoted to embedding theorems with power weights for radial functions, which include a generalization of Ni’s inequality in Rn . In Sect. 7 we analyze the compactness of the embeddings, both in the unweighted case (giving an alternative proof of Lions’ theorem that avoids the use of complex interpolation) and in the weighted case. Finally, in Sect. 8 we discuss a generalization of Ni’s inequality and embedding theorems for potential spaces in a ball.

2 Fractional Integral Estimates and Embedding Theorems Several ways of proving the Sobolev embedding are known. One of the most classical (which goes back to the original work by Sobolev), involves the operator Z I s f .x/ D c.n; s/

Rn

f .y/ dy 0 < s < n jx  yjns

(where c.n; s/ is a normalization constant), which is known as fractional integral or Riesz potential. This operator provides an integral representation of the negative fractional powers of the Laplacian, i.e: I s . f / D ./s=2 f for smooth functions f 2 S.Rn /. See also [28] for an extension to a space of distributions.

Embedding Theorems for Potential Spaces of Radial Functions

121

A basic result on this operator is the following theorem concerning its behaviour in the classical Lebesgue spaces: Theorem 2.1 (Hardy-Littlewood-Sobolev) If p > 1 and strong type .p; q/, i.e: there exist C > 0 such that

1 q

D

1 p



s n

then I s is of

kI s f kLq .Rn /  C kf kLp .Rn / : We recall that, using this result, the Sobolev embedding (Theorem 2) follows easily. Indeed, it is well know that the Bessel potential satisfies the following asymptotics for s < n (see, e.g., [5, Theorem 4]), Gs .x/ ' C

jxjsn if jxj  2 ejxj=2 if jxj  2

(5)

Since, in particular, Gs 2 L1 .Rn /, one immediately gets the embedding H s;p .Rn /  Lp .Rn /:

(6)

On the other hand, (5) gives the pointwise bound jGs f .x/j  C I s .jf j/.x/

(7)

whence, using the Hardy-Littlewood-Sobolev theorem, (6) and Hölder’s inequality one obtains the Sobolev embedding (2). It is therefore natural to ask ourselves which is the corresponding extension of Theorem 2.1 in the weighted case. In this article, we are mainly interested in power weights, for which the following theorem was proved in [45] by E. Stein and G. Weiss: Theorem 2.2 ([45, Theorem B*]) Let n  1, 0 < s < n; 1 < p < 1; ˛ < n ;˛ q

C ˇ  0, and

1 q

D

1 p

C

˛Cˇs . n

n ;ˇ p0


.n  1/. 1p  1q /.

3 Riesz Fractional Derivatives It is well known that the classical potential spaces can be characterized in terms of certain hypersingular integrals, which provide a left inverse for the Riesz potentials. We consider for that purpose, the hypersingular integral 1 D u.x/ WD dn;l .s/

Z

.ly u/.x/

s

Rn

jyjnCs

dy;

l>s>0

(8)

where .ly u/.x/

! l X j l D .1/ u.x C .l  j/y/ j jD0

denotes the finite difference of order l 2 N of the function u, and dn;l .˛/ is a certain normalization constant, which is chosen so that the construction (8) does not depend on l (see [39, Chapter 3] for details). We denote by Ds" u.x/ D

1 dn;l .s/

Z jxj>"

.ly u/.x/ jyjnCs

dy:

Then, we have the following result: Theorem 3.1 ([40, Theorem 26.3]) If u 2 Lp .Rn /, and 1  p < n=s then lim D" I s u.x/ D u.x/

"!0

in Lp .Rn /. Another way of inverting the Riesz potential (which holds for every s) is given by the Marchaud method (see [38]). Then, we have the following characterization theorem due to E. M. Stein (for 0 < s < 2, [44]) and S. Samko [40, Theorem 27.3] (see also [26, Proposition 2.1]).

Embedding Theorems for Potential Spaces of Radial Functions

123

Theorem 3.2 Let 1 < p < 1, 0 < s < 1, l > Œ.s C 1/=2 if s is not an odd integer, and l D s if s is an odd integer. Then u 2 H s;p .Rn / if and only if u 2 Lp .Rn / and the limit Ds u D lim Ds" u "!0

exists in the norm of Lp .Rn /. Moreover, kukH s;p ' kukLp C kDs ukLp : Remark 3.1 For 0 < s < 2, Ds coincides with the fractional Laplacian (as defined, e.g., in [17]) for smooth functions, i.e. we have that Ds u D ./s=2 u: Remark 3.2 For p D 2, we can express the L2 -norm of Ds u in terms of the Fourier transform (as a consequence of Plancherel’s theorem, see, e.g., [17, Proposition 3.4]) Z kD ukL2 '

2

s

Rn

2s

jOu.u/j j!j d!

1=2

:

Other related characterizations of potential spaces are given in [48] and [18].

4 A Version of Strauss’ Lemma for p D 2 Using the Fourier Transform In this section, we give an elementary proof of Strauss’ inequality for p D 2, namely, s Theorem 4.1 Let u 2 Hrad .Rn / and s > 1=2. Then u is almost everywhere equal to n a continuous function in R  f0g that satisfies that 1 1 2s

ju.x/j  Cjxj.n1/=2 kukL2

1

kDs ukL2s2 :

Remark 4.1 For an application of this result to non-linear equations involving the fractional Laplacian operator see [41]. We will make use of the following well-known lemmas (see, e.g., [46, Chapter 3]): Lemma 4.1 (Fourier Transform of Radial Functions) Let u 2 L1rad .Rn / be a radial function, u.x/ D urad .jxj/. Then its Fourier transform uO is also radial, and it

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is given by: uO .!/ D .2/n=2 j!j where  D

Z

1

urad .r/ J .rj!j/ rn=2 dr

0

 1 and J denotes the Bessel function of order .

n 2

Lemma 4.2 (Asymptotics of Bessel Functions) If  > 1=2, then there exists a constant C D C./ such that jJ .r/j  C r1=2 8 r  0: We observe that u.x/ can be reconstructed from uO using the Fourier inversion formula, and that for radial functions (that are even) the Fourier transform and the inverse Fourier transform coincide up to a constant factor. Hence, from Lemma 4.1, we have that: Z 1 j.Ou/rad .r/j jJ .rj!j/j rn=2 dr: ju.x/j  Cj!jn=2C1 0

Moreover, using Lemma 4.2, we have that: ju.x/j  C j!j

Z

.n1/=2

1

j.Ou/rad .r/j r.n1/=2 dr:

0

In order to bound this expression, we split the integral into parts: the low frequency part from 0 to r0 , and the high frequency part from r0 to 1, where r0 will be chosen later, and we bound each part using the Cauchy–Schwarz inequality. With respect to the low frequency part, we have that: Z

r0

0

j.Ou/rad .r/j r

.n1/=2

Z dr  0

r0

2

jOu.x/j r

1=2 Z n1

1=2

r0

dr

dr 0

1=2

 kukL2 r0 :

With respect to the high frequency part, we have that: Z

1

j.Ou/rad .r/j r

.n1/=2

Z

r0

Z

1

j.Ou/rad .r/j r.n1/=2 rs rs dr

r0

2 n1 2s

j.Ou/rad .r/j r



1

dr D

1=2 Z

1

r dr

r

r0

2s

1=2 dr

  kD ukL2 s

r0

r02s1 2s  1

Summing up, we obtain the following estimate: ju.x/j  C jxj

.n1/=2

1=2 r0

 kukL2 C

r02s1 2s  1

!

1=2 kD ukL2 s

1=2 :

Embedding Theorems for Potential Spaces of Radial Functions

125

which implies    2s1  r kDs uk2L2 : ju.x/j2  C2 jxj.n1/ r0 kuk2L2 C 0 2s  1

(9)

Minimizing over r0 we obtain the required inequality 1 1 2s

ju.x/j  Cjxj.n1/=2 kukL2

1

kDs ukL2s2 :

Remark 4.2 A similar technique was used in [6] to derive a whole family of related inequalities, including a generalization of Ni’s inequality. Remark 4.3 We observe that Lemma 4.1 actually expresses the Fourier transform of a radial function in terms of the (modified) Hankel transform. As a consequence, many results on embeddings for spaces of radial functions can also be derived by using results for the Hankel transform. For instance in [12], we have used this technique to prove Theorem 6.4 for p D 2, using results of De Carli [8]. Also some of the results of this paper can be alternatively derived from results obtained by Nowak and Stempak in [32].

5 Proof of Strauss’ Inequality In this section we prove a version of Strauss’ inequality for potential spaces for any p. We begin by the following lemma: Lemma 5.1 Let f 2 Lp .Rn / be a radial function, then jf B.0;R/ .x/j  CRn1=p jxj.n1/=p kf kLp .Rn / :

Proof Observe first that, for y 2 B.x; R/ we have that jxj  R  jyj  R C jxj. Now, taking polar coordinates y D ry0

r D jyj

y0 2 Sn1

x D x0

D jxj x0 2 Sn1

we have Z f B.0;R/ .x/ D

CR R

Z f0 .r/

Sn1

 B.x;R/ .ry0 / dy0 rn1 dr:

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To bound the inner integral, observe that B.x;R/ .ry0 / D Œt0 ;1 .x0  y0 / with t0 D r2 C 2 R2 , and that t0  1 because  R  r  R C . 2r Consider first the case > 2R. It follows that t0 > 1 and integrating over the sphere (see [13, Lemma 4.1] for details), Z Sn1

B.x;R/ .ry0 / dy0 D

Z Sn1

Œt0 ;1 .x0  y0 / dy0 D !n2

Z

1 1

 n3  Œt0 ;1 .t/ 1  t2 2 dt:

where !n2 denotes the area of Sn2 . Therefore, ˇZ ˇ jf B.0;R/ .x/j D !n2 ˇˇ

R

Z  !n2 Z

CR

Z

R

1 t0

R

CR



Z

CR

1

ˇ ˇ  n3  f0 .r/ 1  t2 2 rn1 dt drˇˇ 1=p

jf0 .r/jp rn1 dr



1t

 n3 2 2

!1=p0

p0 rn1 dr

dt

t0

D !n2 kf kLp .Rn / A. /: Notice that Z

1



1  t2

 n3 2

Z

1

dt D

t0

.1  t/

n3 2

.1 C t/

n3 2

dt  C .1  t0 /

n1 2

t0

which implies 0 A. / C @

Z

CR R

0 C @

Z

CR R

0 C @ n 0 DC @ n

Z

11=p0 .n1/p0  2 2 2 2 r C R 1 rn1 drA 2r 11=p0 0  .n1/p  2 1 C .r= /2  .R= /2 1 rn1 drA 2.r= /

1CR= 1R=

Z

1CR= 1R=

11=p0 .n1/p0  2 2 2 1 C .u/  .R= / 1 un1 duA 2u 

2

.R= /  .1  u/ 2u

2

.n1/p0 2

11=p0 un1 duA

Embedding Theorems for Potential Spaces of Radial Functions

0 C @ n

Z

1CR=



1R=

CRn1

0  .n1/p 2

1 .R= /2 u

n.n1/p0

Z

1CR=

u

127

11=p0 un1 duA

p0 .n1/.1 2

!1=p0 /

du

1R=

Since we are assuming > 2R, we have that 1  A. /  CR

n1

R

>

1 2

and

1=p0  n.n1/p0 R  CRn1=p .n1/=p

whence, in this case, jf B.0;R/ .x/j  CRn1=p .n1/=p kf kLp .Rn / : It remains to prove the case < 2R, where 0

jf B.0;R/ .x/j  CRn=p kf kLp .Rn /  CRn1=p .n1/=p kf kLp .Rn / : t u Theorem 5.1 (Strauss’ Inequality for Potential Spaces) Assume that 1 < p < 1 s;p and 1=p < s < n. Let u 2 Hrad .Rn /. Then u is almost everywhere equal to a continuous function in Rn  f0g that satisfies ju.x/j  Cjxj.n1/=p kukH s;p .Rn /

(10)

and, moreover, 11=.sp/

ju.y/j  Cjyj.n1/=p kukLp

1=.sp/

kDs ukLp

:

(11)

Proof Let u 2 H s;p .Rn / be a radial function, and f 2 Lp .Rn / radial, such that u D Gs f . Then, for a > 0 we have that, by Lemma 5.1

Gs f .x/ D

XZ k2Z



X k2Z

f2k1 ajyj2k ag

f .x  y/Gs .y/ dy

Z Gs .2k1 a/

fjyj2k ag

f .x  y/ dy

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D

X

Gs .2k1 a/ f B.0;2k a/ .x/

k2Z



X

Gs .2k1 a/ jf B.0;2k a/ .x/j

k2Z

 Ckf kLp .Rn / jxj.n1/=p

X

Gs .2k1 a/.2k a/n1=p

k2Z

so, letting rk D 2k1 a and rk D rkC1  rk D 2k1 a, we can write the above sum as C

X

Z Gs .rk /.rk /n1=p rk !

k2Z

1

Gs .r/rn1=p dr

0

when a ! 0, so we obtain jGs f .x/j Ckf kLp .Rn / jxj.n1/=p DCkf kLp .Rn / jxj.n1/=p

Z Z

1 0

Rn

Gs .r/rn1=p dr Gs .x/jxj1=p dx

Ckf kLp .Rn / jxj.n1/=p where the last inequality holds since we are assuming s > 1=p. Therefore, ju.x/j D jGs f .x/j  Cjxj.n1/=p kf kLp .Rn / D Cjxj.n1/=p kukH s;p .Rn / which proves (10). We proceed now to the proof of (11). Let u .x/ D u.x/. Then, using Theorem 3.2, it is easy to see that u 2 H s;p .Rn / and, moreover, we have that Ds u .x/ D s Ds u.x/: Hence, applying (10), ju .x/j  Cjxj.n1/=p ku kH s;p p p 1=p  Cjxj.n1/=p ku kLp C kDs u kLp p p 1=p  Cjxj.n1/=p n kukLp C spn kDs ukLp :

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129

Setting y D x, p p

ju.y/jp  Cjyj.n1/ n1 n kukLp C spn kDs ukLp p p

 Cjyj.n1/ 1 kukLp C sp1 kDs ukLp : Now, we choose  > 0 such that 1 kukLp D sp1 kDs ukLp p

p

i.e.,  D

kukLp kDs ukLp

1=s

:

Hence: ju.y/jp  2Cjyj.n1/ kukLp

1=s

p1=s

kDs ukLp

11=.sp/

kDs ukLp

ju.y/j  2Cjyj.n1/=p kukLp

1=.sp/

: t u

Remark 5.1 Inequality (10) was proved in [42] in the more general framework of Triebel-Lizorkin spaces, using an atomic decomposition adapted to the radial situation, which is much more technically involved. A version of Strauss’ inequality for a class of Orlicz-Sobolev spaces is given in [1].

6 Embedding Theorems with Power Weights for Radial Functions We begin this section by proving a generalization of Ni’s inequality. s Theorem 6.1 (Generalization of Ni’s inequality) Let 1 < p < 1, u 2 Hrad .Rn / and 1=p < s < n=p. Then u is almost everywhere equal to a continuous function in Rn  f0g that satisfies n

ju.x/j  Cjxj p Cs kukH s;p .Rn / :

s;p

p

Proof Let u 2 Hrad .Rn /, then there exists f 2 Lrad .Rn / such that u D Gs f , where Gs is as in (1). Using (7) we obtain kjxjn=ps ukL1 .Rn / D kjxjn=ps .Gs f /kL1 .Rn /  Ckjxjn=ps I s .jf j/kL1 .Rn / :

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t u

The above inequality combined with Remark 2.1 gives the proof.

For s D 1 and p D 2 this result coincides with Theorem 1.4 in R , and for arbitrary p and s D 1 gives the result in [49, Lemma 1]. Now we can proceed to an immediate extension of the result above, that gives an embedding theorem in the critical case (cf. [49, Lemma 2] when s D 1). n

Theorem 6.2 Let 1 < p < 1, 0 < s < n=p, c > n be such that .1  sp/c  .n  1/ps, and let pc D p.nCc/ nsp . Then 

kjxjc=pc ukLpc .Rn /  C kukH s;p .Rn / s;p

for any u 2 Hrad .Rn /. Proof Using the same argument as in the proof of Theorem 6.1 we obtain 



kjxjc=pc ukLpc .Rn /  Ckjxjc=pc I s .jf j/kLpc .Rn / : We apply Theorem 2.3 with q D pc , ˛ D 0 and ˇ D c . Clearly, since ˛ D 0, it p c c n n 0 holds that ˛ < n=p , and ˇ D p < p < q since c > n and q < pc . u t c

c

To prove the continuity of the embeddings in the subcritical case we will make used of the following weighted convolution theorem for radial functions, proved by the authors in [14, Theorem 5], which shows that the analogous result of Kerman [25, Theorem 3.1] for arbitrary functions can be improved in the radial case. For other results of weighted convolution inequalities see [4, 33] and references therein. Theorem 6.3 kjxj .f g/.x/kLr .Rn /  Ckjxj˛ f kLp .Rn / kjxjˇ gkLq .Rn / for f and g radially symmetric, provided 1. 1r D 1p C 1q C ˛CˇC  1; 1 < p; q; r < 1; 1r  1p C 1q ; n 2. ˛ < pn0 ; ˇ < qn0 ; < nr ; 3. ˛ C ˇ  .n  1/.1  1p  1q /; ˇ C  .n  1/. 1r  1q /; C ˛  .n  1/. 1r  1p / 4. maxf˛; ˇ; g > 0 or ˛ D ˇ D D 0. Theorem 6.4 Let 1 < p < 1, 0 < s < np , p  r  pc D continuous embedding s;p

Hrad .Rn /  Lr .Rn ; jxjc dx/

p.nCc/ nsp .

Then we have a

(12)

provided that  sp < c
0 or ˇ D  cr D 0. p.nCc/ Since we are assuming r < p.nCc/ , there exists " > 0 such that r D .nspC"p/ . For nsp n this value of " and sufficiently large q < 1 to be chosen later set ˇ D q0  s C ". This choice of ˇ clearly makes the scaling condition hold, so it suffices to check the remaining conditions: • 1r  1p C 1q follows from the fact that p  r. • ˇ < qn0 is equivalent to " < s. Let us check that this is the case: since ."  s/rp D pn C pc  rn, it suffices to check that the RHS is negative, but this is equivalent to c < n.rp/ , which is true because r  p and c < .n1/.rp/ by hypothesis. p p c n •  r < r is immediate from c > ps > n. • ˇ  .n  1/.1  1q  1p / is equivalent to c > pr C qr  r  n. Since we already know that c > n, it suffices to check that pr C qr  r < 0 which holds for sufficiently large q because p > 1. We pick any q that satisfies this condition. • ˇ  cr  .n  1/. 1r  1q / is equivalent to 1q  n C 1r which trivially holds. which holds by hypothesis. •  cr  .n  1/. 1r  1p / is equivalent to c < .n1/.rp/ p • maxfˇ;  cr g > 0 obviously holds for c < 0, so let us assume that c > 0. In this case, using the scaling condition we have that ˇn > 1r  1p  1q C 1 which is clearly positive since we chose q to satisfy  1p  1q C 1 > 0.

To complete the proof, we have to check that kjxjˇ Gs kLq < C which, using (5), holds provided Z

2 0

that is, ˇ >

n q0

rˇq r.sn/q rn1 dr < C1

 s, which is true by our choice of ˇ.

t u

Remark 6.1 A more general form of Theorem 6.4 can be proved in the context of weighted Triebel-Lizorkin spaces. Indeed, for power weights jxjc with n < c < s 0 n.p  1/ we have that H s;p .Rn / D Fp;2 .Rn / and Lp .Rn ; jxjc dx/ D Fp;2 .Rn ; jxjc dx/. One way of proving the theorem in this setting is by means of a weighted PlancherelPolya-Nikol’skij type inequality, which in turn makes use of a weighted convolution

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P.L. De Nápoli and I. Drelichman

theorem. In the general setting this approach was used by Meyries and Veraar in [29]. In the radial setting, the authors proved in [14] that a better Plancherel-PolyaNikol’skij equality holds in the case of radial functions, which in turn gives better weighted embeddings. More technically involved proofs use atomic decompositions or wavelet decompositions but have the advantage of allowing more general weights. See, e.g., [22–24, 30]. In the case of radial functions, this is the subject of a paper by the authors and Saintier [15].

7 Compactness of the Embeddings As announced in the introduction, we begin this section by proving the following theorem due to Lions [27], that we will need later to prove the compactness in the weighted case. Our proof is different from the original one of Lions in [27] and avoids the use of the complex method of interpolation. Theorem 7.1 For 1 < p < q < p , we have a compact embedding s;p

Hrad .Rn /  Lq .Rn / for any s > 0. s;p

Proof Let .un / be a bounded sequence in Hrad .Rn /, kun kH s;p .Rn /  M

(14)

i.e. un D Gs fn with radial fn such that kfn kLp  M: By the Kolmogorov–Riesz theorem (see [21]) we need to check three conditions: i) .un / is bounded in Lq .Rn /. This is clear by (14) and the continuity of the Sobolev embedding (2). ii) (equicontinuity condition) Let h u.x/ D u.x C h/. Given " > 0, there exists ı D ı."/ > 0 such that kh un  un kLq < " if jhj < ı.

Embedding Theorems for Potential Spaces of Radial Functions

133

This is easy since: kh un  un kLq D kh .Gs fn /  Gs fn kLq D k.h Gs  Gs / fn kLq  kh Gs  Gs kLr kfn kLp if 1 1 1 C1D C q r p by Young’s inequality, but, since Gs 2 Lq , kh Gs  Gs kLr
0 such that Z jxj>R

jun jq dx < " for all n 2 N:

In order to check this condition, we observe that if s > 1p  1q , by Theorem 6.4, we have that Z Z q R q jun jq  jxj q jun jq  Ckun kH s;p .Rn / jxj>R

jxj>R

where 

1 1 

D .n  1/ q p

 < 0:

Hence, Z jxj>R

jun jq 

C R q

Mq < "

(15)

if we choose R D R."/ large enough. We observe that ı D  q is exactly the exponent in Lions’ paper [27] (second equation on page 321), but he proves (15) using an interpolation argument.

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The case s 

1 p

1 q



cannot happen under the theorem hypotheses since

q < p ) s > n



1 1  p q

 

1 1  p q

t u

Theorem 7.2 The embedding s;p

Hrad .Rn /  Lr .Rn ; jxjc dx/ of Theorem 6.4 is compact, provided that 1 < p < 1, 0 < s < np , p < r < pc D p.nCc/ nsp

and sp < c
0; j D 1; 8: Then 

!˛1 ;˛2 f ; ı1 ; ı2

 q



 Zı1 Zı2 h 0

Cı1˛1



Z1 Zı2 ı1

Cı2˛2

0

1

1

.t1 t2 / p C q ! 1 ; 2 . f ; t1 ; t2 /p

0

iq dt dt  1 1 2 q t1 t2

h ˛1 C 1  1   1 C 1 iq dt dt  1 1 2 q p q t1 t2 p q ! 3 ; 4 . f ; t1 ; t2 /p t1 t2

 Zı1 Z1 h  1 C 1 ˛2 C 1  1  iq dt dt  1 1 2 q p q t1 p q t2 ! 5 ; 6 . f ; t1 ; t2 /p t1 t2 0 ı2

Cı1˛1 ı2˛2

 Z1 Z1 h ˛1 C 1  1  ˛2 C 1  1  iq dt dt  1 1 2 q p q p q t2 ! 7 ; 8 . f ; t1 ; t2 /p t1 t1 t2 ı1 ı2

 B1 C B2 C B3 C B4 :

170

M.K. Potapov and B.V. Simonov

Proof For everyone ıi 2 .0; 1/ there exists an integer non-negative number ni such that 2ni1C1  ıi < 21ni ; i D 1; 2: Using Lemma 1.4, we have    1 1  I D !˛1 ;˛2 f ; ı1 ; ı2 q  !˛1 ;˛2 f ; n ; n : 21 22 q Using Theorem 2.1, and then further Lemma 1.4 (property f), we get   1 1 h  iq  1    X X 1 1 1 1 1  1 q 2.1 C2 / p  q ! 1 ; 2 f ;  1 I  !˛1 ;˛2 f ; n ; n C ;  1 21 22 q Œ2 1  C 1 Œ2 2  C 1 p  Dn  Dn 1

C

C

C

n1 1 1 X X h

2n1 ˛1

21

2

2

˛1 C 1p  1q



22



1 1 pq



1 D0 2 Dn2

n2 1 1  X Xh

2n2 ˛2

21



1 1 pq



22



˛2 C 1p  1q

1 Dn1 2 D0

n1 n2 1 X Xh

1

1



2n1 ˛1 2n2 ˛2

21



˛1 C 1p  1q



22





 ! 3 ; 4 f ;  ! 5 ; 6 f ;

˛2 C 1p  1q



1 Œ21 1  1 Œ21 1 

 ! 7 ; 8 f ;

1 D0 2 D0

C1

C1

;

;

 iq 

1 Œ22 1 

C1

p

 iq 

1 Œ22 1 

C1

1 q

1 q

p

C

C

 iq   1 1 q : ; Œ21 1  C 1 Œ22 1  C 1 p 1

Using properties of the mixed module of smoothness (Lemma 1.4) and a fact that 1 R2  dt 2

t t ; we get I  B1 C B2 C B3 C B4 : Thus, the proof of Theorem 2.2 1 2C1



is complete. Let us note a special case of Theorem 2.2.

Theorem 2.3 Let f 2 L0p ; 1  p < q  1; ˇi > ˛i > 0; ıi 2 .0; 1/; i D 1; 2: Then    I D !˛1 ;˛2 f ; ı1 ; ı2 q

Zı1 Zı2 h 0

Cı1˛1



Z1 Zı2 ı1

Cı2˛2

0

0

1

1

.t1 t2 / p C q !˛1 C 1p  1q ;˛2 C 1p  1q . f ; t1 ; t2 /p

iq dt dt  1 1 2 q C t1 t2

h ˛1 C 1  1   1 C 1 iq dt dt  1 1 2 q p q t1 t2 p q !ˇ1 C 1  1 ;˛2 C 1  1 . f ; t1 ; t2 /p p q p q t1 t2

 Zı1 Z1 h  1 C 1 ˛2 C 1  1  iq dt dt  1 1 2 q p q t1 p q t2 !˛1 C 1  1 ;ˇ2 C 1  1 . f ; t1 ; t2 /p p q p q t1 t2 0 ı2

C

ı1˛1 ı2˛2

 Z1 Z1 h ˛1 C 1  1  ˛2 C 1  1  iq dt dt  1 1 2 q p q p q t1 t2 !ˇ1 C 1  1 ;ˇ2 C 1  1 . f ; t1 ; t2 /p p q p q t1 t2 ı1 ı2

 A1 . f ; ı1 ; ı2 / C A2 . f ; ı1 ; ı2 / C A3 . f ; ı1 ; ı2 / C A4 . f ; ı1 ; ı2 /:

(1)

Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness

171

The Theorem 2.3 is sharp in the sense that there is a function f0 .x; y/ such that the symbol  in (1) can be replaced by for f D f0 . Let us prove this fact. We shall consider the function f0 .x; y/ D sin x sin y: For everyone ıi 2 .0; 1/ there exists integer non-negative number ni such, that 2ni1C1  ıi < 21ni ; i D 1; 2: Using lemmas 1.1 and 1.4, for any ri > 0; i D 1; 2; and any p 2 Œ1; 1 we have    1 1  1 !r1 ;r2 f0 ; ı1 ; ı2 p !r1 ;r2 f0 ; n ; n

n r Cn r ı1r1 ı2r2 : 1 2 1 1 2 2 2 2 p 2 But then   A0 . f0 ; ı1 ; ı2 /  !˛1 ;˛2 f0 ; ı1 ; ı2 q ı1˛1 ı2˛2 ; A1 . f0 ; ı1 ; ı2 / ı1˛1 ı2˛2 ; A2 . f0 ; ı1 ; ı2 / ı1˛1 ı2˛2 ; A3 . f0 ; ı1 ; ı2 / ı1˛1 ı2˛2 ; A4 . f0 ; ı1 ; ı2 / ı1˛1 ı2˛2 : From this estimates it follows that A0 . f0 ; ı1 ; ı2 / A1 . f0 ; ı1 ; ı2 / C A2 . f0 ; ı1 ; ı2 / C A3 . f0 ; ı1 ; ı2 / C A4 . f0 ; ı1 ; ı2 /: This means that for the function f0 .x; y/ symbol  can be replaced by in (1). Note that for some p and q the second, third and fourth terms in the right-hand side part of inequality (1) in the Theorem 2.3 can be omitted. It follows from the theorems stated below.

3 Analogues of Ul’ynov Inequalities for Mixed Moduli of Smoothness Theorem 3.1 Let f 2 L0p ; where ˛i > 0; ıi 2 .0; 1/; i D 1; 2, and 1 D p < q D 1: Then 

!˛1 ;˛2 f ; ı1 ; ı2

 q



 Zı1 Zı2 h

1

1

.t1 t2 / p C q !˛1 C 1  1 ;˛2 C 1  1 . f ; t1 ; t2 /p p

0

0

q

p

q

iq dt dt  1 1 2 q : t1 t2 (2)

The Theorem 3.1 is sharp in the sense that there is a function f0 .x; y/ such that symbol  in (2) can be replace by for this function. Sharp Ulyanov inequality (2) in the case 1 < p < q < 1 was proved in [9] and [7]. Here we study the case when 1 D p < q D 1: Similar theorem is proved in [11] for functions of one variable in the case of 1 D p < q D 1.

172

M.K. Potapov and B.V. Simonov

Proof For everyone ıi 2 .0; 1/ there exists an integer non-negative number ni such, that 2ni1C1  ıi < 21ni ; i D 1; 2: Therefore    1 1  I D !˛1 ;˛2 f ; ı1 ; ı2 1  !˛1 ;˛2 f0 ; n ; n : 21 22 1 Using Lemma 1.1, we have I  I1 C 2n1 ˛1 I2 C 2n2 ˛2 I3 C 2n1 ˛1 n2 ˛2 I4 : Using lemmas 1.2 a), 1.3 a), and 1.4 f), we get I1  Y2n1 ;2n2 . f /1 

1 X 1 X

21 C2 Y21 ;22 . f /1

1 Dn1 2 Dn2



1 X 1 X 1 Dn1 2 Dn2

 1 1  21 C2 !˛1 C1;˛2 C1 f ;  ;  : 21 22 1

 .˛ ;0/   Now we shall estimate I2 D V2n11;1 f  V1;2n2 . f / 1 : Let us denote  .˛ ;0/  .˛ ;0/ .˛ ;0/ A  V2n11;1 f  V1;2n2 . f / D V2n11;1 . f /  V2n11;2n2 . f /: For f 2 L01 and for almost all x and y and any N2 > n2 we have N2 X  .˛1 ;0/  .˛ ;0/ .˛ ;0/ .˛ ;0/ V2n1 ;22 C1 . f /  V2n11;22 . f / D V2n11;2N2 C1 . f /  V2n11;2n2 . f / D

2 Dn2

 .˛ ;0/   .˛ ;0/  .˛ ;0/ .˛ ;0/ D V2n11;1 . f /  V2n11;2n2 . f /  V2n11;1 . f /  V2n11;2N2 C1 . f /  A  B: This implies that jjAjj1 

N2 X   .˛1 ;0/ V n  C1 . f /  V .˛n11 ;0/2 . f / C kBk1 : 2 ;2 1 2 1 ;2 2

(3)

2 Dn2

Let us estimate jjBjj1 : Using Lemma 1.5, we have .˛ ;0/

jjBjj1  kV2n11;1 . f  V1;2N2 C1 . f //k1  2n1 ˛1 kV2n1 ;1 . f  V1;2N2 C1 . f //k1 : Using Lemma 1.2 b), we get jjBjj1  2n1 ˛1 k f  V1;2N2 C1 . f /k1 :

Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness

173

Since f 2 L01 ; then V01 . f / D V02N2 C1 . f / D 0: Therefore, using Lemma 1.2 a), we have jjBjj1  2n1 ˛1 k f V0;1 . f /V1;2N2 C1 . f /CV0;2N2 C1 . f /k1  2n1 ˛1 Y0;2N2 C1 . f /1 : Using Lemma 1.3, we have jjBjj1  2n1 ˛1

1 X

1 X

21 C2 Y21 1;22 1 . f /1 :

1 D0 2 DN2 C1

Using Lemma 1.4 f), we get jjBjj1  2

n1 ˛1

 1 1  21 C2 !˛1 C1;˛2 C1 f ;  ;   21 22 1 C1

1 X

1 X

1 D0 2 DN2 1

Z1 2ZN2 C1 dt1 dt2  2n1 ˛1 .t1 t2 /1 !˛1 C1;˛2 C1 . f ; t1 ; t2 /1 : t1 t2 0

0

This implies that jjBjj1 ! 0 while N2 ! 1: Since jjBjj1 ! 0 while N2 ! 1 then implies the following inequality jjAjj1 

1 X  .˛1 ;0/  V n  C1 . f /  V .˛n11 ;0/2 . f / : 2 ;2 1 2 1 ;2 2

2 Dn2

Using Lemma 1.8 a), we get jjAjj1 

1 X 2 Dn2



1 X 2 Dn2

C

1 X 2 Dn2



1 X 2 Dn2

  .˛ C1;0/ .˛ C1;0/ 22 V2n11;22 C1 . f /  V2n11;22 . f /1 

  .˛ C1;0/ .˛ C1;0/ 22 V2n11;1 . f /  V2n11;1 .V1;22 . f //1 C

  .˛ C1;0/ .˛ C1;0/ 22 V2n11;1 . f /  V2n11;1 .V1;22 C1 . f //1 

1 X    .˛ C1;0/  .˛ C1;0/ 22 V2n11;1 . f  V1;22 . f //1 C 22 V2n11;1 . f  V1;22 C1 . f //1 : 2 Dn2

174

M.K. Potapov and B.V. Simonov

Using Lemma 1.1, we have 1 X

jjAjj1 

2 Dn2

 1 1  22 2n1 .˛1 C1/ !˛1 C1;˛2 C1 f ; n ;  ; 21 22 1

i.e. 1 X

I2 

2 Dn2

 1 1  22 2n1 !˛1 C1;˛2 C1 f ; n ;  2n1 ˛1 : 21 22 1

Similarly, we get an estimate 1 X

I3 

1 Dn1

 1 1  21 2n2 !˛1 C1;˛2 C1 f ;  ; n 2n2 ˛2 : 21 22 1

Let us estimate I4 : Using Lemma 1.8 c), we have  .˛ C1;˛ C1/  I4  V2n11;2n2 2 . f /1 : Using Lemma 1.1, we get  1 1  I4  2n1 .˛1 C1/Cn2 .˛2 C1/ !˛1 C1;˛2 C1 f ; n ; n : 21 22 1 Finally, combining estimates for I1 ; I2 ; I3 and I4 ; we have I

1 X 1 X 1 Dn1 2 Dn2

Zı1 Zı2  0

 1 1  21 C2 !˛1 C1;˛2 C1 f ;  ;  21 22 1

.t1 t2 /1 !˛1 C1;˛2 C1 . f ; t1 ; t2 /1

0

dt1 dt2 ; t1 t2

i.e., the proof of Theorem 3.1 is complete for 1 D p < q D 1:

t u

Theorem 3.2 Let f 2 L0p ; where ˛i > i > 0; ıi 2 .0; 1/; i D 1; 2; and 1 D p < q < 1 or 1 < p < q D 1: Then 

!˛1 ;˛2 f ; ı1 ; ı2

 q



 Zı1 Zı2 h

1

1

.t1 t2 / p C q ! 1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p p

0

0

q

p

q

iq dt dt  1 1 2 q : t1 t2 (4)

Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness

175

The Theorem 3.2 cannot be improved in the sense that if we replace even one of i by ˛i in the right-hand side part of inequality (4), then the received inequality will be false. Remark that similar results for functions of one variable where proved in [13] for 1  p < q < 1 and in [6] for 1 < p < q D 1. Proof Let us take 1 D 3 D 5 D 7 D 1 C

2 C 1p  1q in Theorem 2.2. Then we have 

I D !˛1 ;˛2 f ; ı1 ; ı2

 q



 Zı1 Zı2 h

1

q

p

q

0

iq dt dt  1 1 2 q t1 t2

 Z1 Zı2 h ˛1 C 1  1   1 C 1 iq dt dt  1 1 2 q p q t2 p q ! 1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p t1 p q p q t1 t2 0

ı1

Cı2˛2

 1q ; 2 D 4 D 6 D 8 D

.t1 t2 / p C q ! 1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p p

0

Cı1˛1

1

1 p

 Zı1 Z1 h  1 C 1 ˛2 C 1  1  iq dt dt  1 1 2 q p q ! 1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p t1 p q t2 p q p q t1 t2 0 ı2

Cı1˛1 ı2˛2

 Z1 Z1 h ˛1 C 1  1  ˛2 C 1  1  iq dt dt  1 1 2 q p q p q t1 t2 ! 1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p p q p q t1 t2 ı1 ı2

 D1 C D2 C D3 C D4 :

Using Lemma 1.4 (item d) and taking into account, that i < ˛i ; we get Z1 Zı2 " !

D2 D ı1˛1

ı1



ı1˛1

Zı2 "

0

1 C 1p  1q ; 2 C 1p  1q . f ; t1 ; t2 /p ˛    t11 1 t22

1 C 1p  1q

2 C 1p  1q t1 t2

! 1 C 1  1 ; 2 C 1  1 . f ; ı1 ; t2 /p p  q  p q

1 C 1  1

2 C 1  1

#q

q t22

Z1

#q

dt1 dt2 t1 t2

. ˛ /q t1 1 1

!

dt1 dt2 t1 t2

p q p q ı1 t2 ı1 #q ! 1 Zı2 " ! 1 1 q

1 C p  q ; 2 C 1p  1q . f ; ı1 ; t2 /p

2 q 1 q dt2     t ı 2 1 t2

1 C 1p  1q

2 C 1p  1q ı1 t2 0 #q ! 1 Zı2 " ! 1 1 Zı1 q

1 C p  q ; 2 C 1p  1q . f ; ı1 ; t2 /p

2 q

1 q dt1 dt2     t t 2 1 t1 t2

1 C 1p  1q

2 C 1p  1q ı1 t2 0 0

0

1 q

!

1 q

176

M.K. Potapov and B.V. Simonov



Zı1 Zı2 " !

1 C 1p  1q ; 2 C 1p  1q . f ; t1 ; t2 /p

0

D



t1

0

 Zı1 Zı2 h

1

1 C 1p  1q

 t2

q q t22 t11



1

.t1 t2 / p C q ! 1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p p

0

2 C 1p  1q

#q

q

p

q

0

dt1 dt2 t1 t2

!

1 q

iq dt dt  1 1 2 q  D1 : t1 t2

Similarly we can show, that D3  D1 and D4  D1 : Thus, we have proved, that I  D1 : The proof of Theorem 3.2 is complete. t u Let us show now that if we replace even one of i by ˛i ; the received inequality will be false. For example we show that for any function f 2 L0p and any ı1 2 .0; 1/; ı2 2 .0; 1/ for 1 D p < q < 1; or 1 < p < q D 1 it is impossible to find a constant C; independent of f ; ı1 and ı2 , such that inequality 

!˛1 ;˛2 f ; ı1 ; ı2

 q

C

 Zı1 Zı2 h

1

1

.t1 t2 / p C q !˛1 C 1  1 ; 2 C 1  1 . f ; t1 ; t2 /p p

0

q

p

q

0

iq dt dt  1 1 2 q t1 t2

(5) holds true. First of all, let us consider a case of 1 < p < q D 1: Let us define the function f1 .x; y/ D '1 .x/ 1 .y/; where '1 .x/ D

1 X kD1

'1 .x/ D

cos kx   1 ; for ˛1 ¤ 2l  1; l 2 N; k˛1 C1 ln.k C 1/ p

1 X kD1



sin kx

k˛1 C1 ln.k C 1/ 1 .y/

 1p ; for ˛1 D 2l  1; l 2 N;

D sin y:

For the function '1 .x/ it was proved in [10] that !˛1 .'1 ; ı1 /1  ı1˛1 Zı1

1

t1 p !˛1 C 1 .'1 ; t1 /p p

0

!1 1p ;

 dt1 3  1p  ı1˛1 ln ln : t1 ı1

It is proved in [12] that for the function following relation holds !ˇ .

1 ln ı1

1 ; ı 2 /p

1 .y/

and any ˇ > 0; p 2 Œ1; 1/ the ˇ

ı2 :

Analogues of Ulyanov Inequalities for Mixed Moduli of Smoothness

177

This implies that !˛2 . Zı2

1 ; ı 2 /1

1

t2 p ! 2 C 1 . p

 ı2˛2 ;

1 ; t2 /p

0

dt1

 ı2 2 : t1

Since j1 D !˛1 ;˛2 .'1 ; ı1 ; ı2 /1 D !˛1 .'1 ; ı1 /1  !˛1 .  1 1p ı2˛2 : Since ı1˛1 ln ı11 Zı1 Zı2 j2 D

p

0

Zı1 D 0

1

.t1 t2 / p !˛1 C 1 2 C 1 . f1 ; t1 ; t2 /p p

0

1 dt1 t1 p !˛1 C 1 .'1 ; t1 /p p t1

Zı2 

1

t2 p ! 2 C 1 . p

1 ; ı 2 /1 ;

then j1 

dt1 dt2 D t1 t2 1 ; t2 /p

0

dt2 ; t2

  1p

then j2  ı1˛1 ln ln ı31 ı2 2 :

.0/

For any fixed ı2 2 .0; 1/ and sufficiently small ı1 ; i.e. for ı1 2 .0; ı1 /; there exists a constant C1 > 0 dependent only on ı2 such that the inequality  ln ln

 1 1 1p 3  1p ˛   C1 ı2 2 2 ln ı1 ı1

holds. This yields that for any fixed ı2 2 .0; 1/ there is no constant C2 > 0 independent of ı1 and ı2 such that j1  C2  j2 for any ı1 2 .0; 1/: Hence inequality (5) does not hold for the function f1 .x; y/. Now we consider a case of 1 D p < q < 1: Let us define the function f2 .x; y/ D 1 P .˛2 C1 1q / '2 .x/ 2 .y/; where '2 .x/ is the function such that '2 D ak cos kx; the sequence fak g is such that ak  2akC1 C akC2  0 and ak   k 2 NI

2 .y/

kD1 1

ln.kC1/

 1q for any

D sin y: The following inequalities were proved in [10] and [12]:  3  1q ; !˛1 .'2 ; ı1 /q  ı1˛1 ln ln ı1 !1 Zı1 h iq dt q 1C 1q 1 !˛1 C1 1 .'2 ; t1 /1  ı1˛1 ; t1 q t1 0

178

M.K. Potapov and B.V. Simonov

!˛2 . Zı2 h 0

j4 D

0

D

Zı1 h 0

 ı2˛2 ;

iq dt 1C 1 2 t2 q ! 2 C1 1 . 2 ; t2 /1 q t2

Since j3 D !˛1 ˛2 . f2 ; ı1 ; ı2 /q   1q ı1˛1 ln ln ı31 ı2˛2 : Since Zı1 Zı2 h

2 ; ı 2 /q

.t1 t2 /

1C 1q

0

D

! 1q

 ı2 2 :

!˛1 .'2 ; ı1 /q  !˛2 .

2 ; ı 2 /q ;

iq dt dt 1 2 !˛1 C1 1 ; 2 C1 1 . f2 ; t1 ; t2 /p q q t1 t2

iq dt 1C 1 1 t1 q !˛1 C1 1 .'2 ; t1 /p q t1

! 1q 

Zı2 h 0

then j3



! 1q D

iq dt 1C 1 2 t2 q ! 2 C1 1 . 2 ; t2 /p q t2

! 1q ;

then j4  ı1˛1 ı2 2 : .0/ For any fixed ı2 2 .0; 1/ and sufficiently small ı1 ; i.e. for ı1 2 .0; ı1 /; there exists the constant C3 > 0; dependent only on ı2 ; such that inequality ˛  2

ı2 2

 3  1q  C3 ln ln ı1

holds. This yields that for any fixed ı2 2 .0; 1/ it is impossible to find constant C4 > 0; independent of ı1 and ı2 ; such that j3  C4  j4 for any ı1 2 .0; 1/: Hence inequality (5) does not hold for the function f2 .x; y/. Acknowledgements This research was supported by the RFFI (grant N 16-01-00350) and program of support of leading Scientific schools (grant NSH-3862-2014-1).

References 1. N.K. Bary, A Treatise on Trigonometric Series (MacMillan, New York, 1964) 2. O.V. Besov, V.P. Il’in, S.M. Nikol’skii, Integral Representation of Function and Imbedding Theorems (Wiley, New York, 1978/1979; translated from Russian: Nauka, Moscow, 1975) 3. M.K. Potapov, On “angular” Approximation, Proc. Conf. Constructive Functions Theory (Budapest, 1969) (Acad. Kiado, Budapest, 1971), pp. 371–399 4. M.K. Potapov, About one imbedding theorem. Mathematica (Clui) 14(37), 123–146 (1972) 5. M.K. Potapov, Imbedding of classes of function with a dominating mixed modulus of smoothness. Trudy Mat. Inst. Steklov. 131, 199–210 (1974)

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6. M.K. Potapov, B.V. Simonov, Interrelations between the moduli of smoothness in metrics Lp and C. Vestn. Mosk. Univ. (to be printed) 7. M.K. Potapov, B.V. Simonov, S. Yu. Tikhonov, Relations between the mixed moduli of smoothness and embedding theorems for Nikol’skii classes. Proc. Steklov Inst. Math. 269, 197–207 (2010) (translation from Russian: Trudy Matem. Inst. V.A. Steklova 269, 204–214 (2010)) 8. M.K. Potapov, B.V. Simonov, S. Yu. Tikhonov, Constructive characteristics of mixed moduli of smoothness of positive orders, in Proceedings of the 8th Congress of the International Society for Analysis, its Applications, and Computation ( 22–27 August 2011 ), vol. 2, 314–325 (2012) 9. M.K. Potapov, B.V. Simonov, S. Yu. Tikhonov, Mixed moduli of smoothness in Lp ; 1 < p < 1: a survey. Surv. Approx. Theor. 8, 1–57 (2013) 10. B. Simonov, S. Tikhonov, Sharp Ul’yanov-type inequalities using fractional smoothness. J. Approx. Theor. 162, 1654–1684 (2010) 11. S. Tikhonov, Weak type inequalities for moduli of smoothness: the case of limit value parameters. J. Fourier Anal. Appl. 16(4), 590–608 (2010) 12. S. Tikhonov, On moduli of smoothness of fractional order. Real Anal. Exch. 30(2), 507–518 (2005) 13. S. Tikhonov, W. Trebels, Ulyanov inequalities and generalized Liouville derivatives. Proc. Roy. Soc. Edinburgh Sec. A 141(1), 205–224 (2011)

Reconstruction Operator of Functions from the Sobolev Space N.T. Tleukhanova

Abstract For function classes with dominant mixed derivative we study the reconstruction operator of functions by their values at a given number of nodes. We prove that the error by the order coincides with the corresponding orthogonal width. Keywords Fourier series • Hyperbolic cross • Pitt inequality • Quadrature formulas • Reconstruction operator

Mathematics Subject Classification (2000). Primary 31C15, Secondary 44A35, 46E30

1 Introduction In this work we study the reconstruction problem of periodic functions from the spaces Wp˛ with dominant mixed derivative by values of a function at a given number of nodes. Let .X; Y/ be a couple of function spaces of 1-periodic functions and X  n Y. We study the following problem: to find nodes ftk gM kD1  Œ0; 1 and functions M fk .x/gkD1 such that the errors ıM .X; Y/ D sup kf ./  kf kX D1

M X

f .tk /k ./kY

kD1

are minimal with respect to the decay order with increasing M. In what follows we consider X D Wp˛ Œ0; 1n and Y D Lp Œ0; 1n . This problem was considered earlier in [6, 7, 15–17, 20, 23–25]. In particular, in these papers there were considered the reconstruction operators such that the decay orders of ıM .X; Y/ differ from the decay N.T. Tleukhanova () L.N. Gumilyov Eurasian National University, Munatpasova str. 7, 010010 Astana, Kazakhstan e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_12

181

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N.T. Tleukhanova

order of orthogonal diameter given by ? .X; Y/ D inf dM

sup kf 

fgj gM jD1 kf kX D1

M X . f ; gj /gj kY ; jD1

n where the infimum is taken over all orthogonal systems fgj gM jD1 from L1 Œ0; 1 . The notion of the orthogonal diameter was introduced by Temlyakov in [19], see also [3]. The main goal of this paper is to find the reconstruction operator such that the ? decay orders of ıM .Wp˛ Œ0; 1n ; Lq Œ0; 1n / and dM .Wp˛ Œ0; 1n ; Lq Œ0; 1n / coincides, i.e., M to find ftk gM kD1 and fk .x/gkD1 such that ? ıM .Wp˛ Œ0; 1n ; Lq Œ0; 1n /  dM .Wp˛ Œ0; 1n ; Lq Œ0; 1n /;

in other words ? .Wp˛ Œ0; 1n ; Lq Œ0; 1n /  C2 ıM .Wp˛ Œ0; 1n ; Lq Œ0; 1n /: C1 ıM .Wp˛ Œ0; 1n ; Lq Œ0; 1n /  dM

Let 1 < p < 1, f be an 1-periodic function from Lp Œ0; 1n , and the series 2ikx O be the Fourier series of f . We say that f 2 Wp˛ Œ0; 1n , ˛ > 0, k2Zn f .k/e if there exists f .˛/ 2 Lp Œ0; 1n , the Fourier series of which coincides with the series P

X

kN ˛ fO .k/e2ikx ;

k2Zn

where kN D

n Y

maxf2jkj j; 2g. Moreover,

jD1

kf kWp˛ Œ0;1n WD kf .˛/ kLp Œ0;1n : Furthermore, x D

Pn jD1

j xj , jkj D k1 C : : : C kn ,

./ D f D . 1 ; : : : ; n / 2 Nn W 2j 2  j j j < 2j 1 g; where Œx is integer part of number x. Let us define the following transform for a function f 2 CŒ0; 1n : Fm . f I x/ D

X 1 X r1 rn r1 rn f . k ; : : : ; k /k;r .x1 C k ; : : : ; xn C k /; m 1 n 1 2 2 2 2 2n k jkjDm k2Nn

0r 1p ; then ? sup kf  Fm . f /kLq  dM .Wp˛ ; Lq /:

(3)

kf kWp˛ D1

2 Reconstruction of the Fourier Coefficients Let us denote AŒ0; 1n the space of absolutely convergent trigonometric P through 2i x series f D . Let  2 Nn ; 2 ./; m  jj: Let us define

2Zn a e the functional Pm . f I / D D

X jkCjDm kj 0

1 2m

X

.1/

0r jBj j, j D 1; :::; n, I D r2Zn ..B1 C r1 d1 /  : : :  .Bn C rn dn //: If f 2 CŒ0; 1n ; f  P P fO . /e2i x , then the series 2I fO . /e2i x is the Fourier series of the function

2Zn

dX dX 1 1 n 1  1 r1 rn   r1 rn  DB ; ::: f x1 C ; : : : ; xn C ;:::; d1 : : : dn r D0 r D0 d1 dn d1 dn 1

n

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N.T. Tleukhanova

where DB .x/ D

P

e2i x is the Dirichlet kernel corresponding to the paral-

2B n

lelepiped B from Z .

Lemma 2 Let b D fbs gs2Z 2 l1 ; m 2 N; then for any 0    m the following representation holds m X

.1/sgn.k/

kD

X

b2k1 .2rCsgn.k// D

X

r2Z

b2m r :

r2Z

Proof m m X X X X X sgnk  .1/ b2k1 .2rCsgn.k// D b2 r  b2k1 .2rC1/ kD

r2Z

D b0 C

1 X

X

1 X X

m X

b2k1 .2rC1/ 

kDC1 r2Z

D b0 C

kDC1 r2Z

r2Z

X

b2k1 .2rC1/

kDC1 r2Z

b2k1 .2rC1/ D b0 C

kDmC1 r2Z

X

b2m r D

r2Z

X

b2m r :

r2Z

r¤0

t u Theorem 2 Let  2 Nn ; jj  m; 2 ./, then for f 2 AŒ0; 1n the following equality holds Pm . f I / D fO . / C



X

n X

X

lD1

k1 C:::Ckl Dmjj kj 0

.1/

Pl1

jD1 sgnkj



fO .2k1 C1 1 .2r1 C sgnk1 / C 1 ; : : : ; rl 2kl Cl C l ; lC1 ; : : : ; n /:

(4)

r2Zl rl ¤0

In the case of D .0; : : : ; 0/, formula (4) is the quadrature formula that was considered in the papers [12] and [13]. Proof The proof of Theorem 2 follows the same lines as the proof of Theorem 1 in [13] using Lemmas 1 and 2. t u Theorem 3 Let  2 Nn ; jj  m; have

˛ >

1 pQ

D max. 1p ; 12 /. Then for 2 ./ we

.m C 1  jj/ sup jfO . /  Pm . f ; /j  C 2m˛ kf kW ˛ D1 p

n1 pQ

:

Reconstruction Operator of Functions from the Sobolev Space

185

Proof Let f 2 Wp˛ Œ0; 1n and 2 ./ D f 2 Nn W Œ2i 2   i < 2i 1 g. Using Theorem 2 and Hölder’s inequality, we get 11=Qp0 0 11=Qp pQ0 X  1 pQ X @ A jfO . /  Pm . f ; /j  @ sN˛ jfO .s/j A ˛ s N s2M s2M 0





11=Qp0 0 11=Qp pQ0 X  1 pQ X @ A jf .˛/ .s/j A @ ˛ s N s2M s2M 0

b





0

11=Qp 0 11=Qp X  1 pQ X  1 pQ A D Ckf kW ˛ @ A ;  Ckf .˛/ kLpQ @ p ˛ ˛ s N s N s2M s2M



S S where M D nlD1 k1 C:::Ckl Dmjj f.2k1 C1 1 .2r1 C sgnk1 / C 1 ; : : : ; 2kl Cl rl C

l ; lC1 ; : : : ; n / W r 2 Z l ; rl ¤ 0g: Taking into account that for ˛ > 1=p 0

11=Qp n1 X  1 pQ .m C 1  jj/ pQ @ A C sN˛ 2m˛ s2M

(5)



we obtain the statement of the theorem.

t u

3 The Pitt Inequality In this section we present some inequalities for anisotropic Lorentz spaces Lp;q ; where p D .p1 ; : : : ; pn /; q D .q1 ; : : : ; qn /I see [8] and [11] for the definitions of these spaces. We will use these estimates in the proof of Theorem 1. Let D .j1 ; : : : ; jn / be some rearrangement of sequence of numbers f1; 2; : : : ; ng: By f  .t/ D f j1 :::jn .t1 ; : : : ; tn / let us denote the function obtained by applying non-increasing rearrangements with respect to as follows: first with respect to the variable xj1 of xj1 ; : : : ; xjn in Œ0; 1n with fixed other variables, then sequentially with respect to the second variable with fixed others and etc. The space Lp;q .Œ0; 1n / is defined as the space of functions for which 0 @

Z

1 0

Z :::

1 0

1 q1 ˇ 1 n ˇq1  qq2 1 1 ˇ p1 ˇ dt dt 1 n ˇt : : : tnpn f j1 :::jn .t1 ; : : : ; tn /ˇ A ::: < 1; ˇ1 ˇ t tn 1

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N.T. Tleukhanova

where the expression

R 1 0

.F.t//q dtt

1=q

for q D 1 we understand as sup F.t/. t>0

If D .1; 2; : : : ; n/, then the space Lp;q .Œ0; 1n / is denoted by Lp;q .Œ0; 1n /: Remark 1 In contrast to the scalar case, when p D q space Lp;q does not coincide with the Lebesgue space with mixed norm Lp D Lp1 ;:::;pn . Similarly we define the discrete spaces lp;q D fa D fas gs2Zn W 9 1q2 =q1 11=qn > > = 1 j :::jn q 1 C B 1 1 D@ kn :::@ k1 .ak1 :::kn / A

> kn D1 k1 D1 ; 0

kaklp;q

1 X

0

qn pn

1 X

q1 p1

Let us formulate the Paley type inequalities for trigonometric series. P Lemma 3 ([11, Theorem 4]) Let f  k2Zn fO .k/e2ikx , D .j1 ; : : : ; jn / be some rearrangement of the sequence .1; 2; : : : ; n/, and 0 D .jn ; jn1 : : : ; j1 /. Then for 2 < p D .p1 ; : : : ; pn / < 1, 0 < q D .q1 ; : : : ; qn /  1, p0 D p=.p  1/ kf kLp;q Œ0;1n  ckfO kl0 ; p0 ;q

for 1 < p D .p1 ; : : : ; pn / < 2, 0 < q D .q1 ; : : : ; qn /  1, p0 D p=.p  1/ kfO kl0  ckf kLp;q Œ0;1n : p0 ;q

Lemma 4 (Pitt Type Inequality) 0

a) If 1 < p < 2; 0 < q  minj pj ; q D .q; : : : ; q/, then X

k

q p0

1

jfO .k/j

! 1q  ckf kLp;q :

q

(6)

k2Zn

b) If 2 < p < 1; max1jn p0j  q  1; q D .q; : : : ; q/, then kf kLp;q  c

X

k

q p0

1

jfO .k/j

! 1q q

:

(7)

k2Zn 0

Proof Let f 2 Lp;q .Œ0; 1n /; where 1 < p < 2; 0 < q  minj pj ; q D .q; : : : ; q/. P Without loss of generality we can assume that f D k2Nn fO .k/e2ikx . By Lemma 3 we have 1 X kn D1

:::

1 X k1 D1

q 0

p

k1 i

1

q 0

p

   kn i

1



fO n :::1 .k1 ; : : : ; kn /

q

q

 ckf kLp;q :

Reconstruction Operator of Functions from the Sobolev Space

187

Since q  p0 , then 1 X

q 0 pn

1

kn

kn D1



1 X

q 0

1

 q fO n :::1 .k1 ; : : : ; kn /

q 0 p1

1



k p1

k1 D1 q 0 1 pn

kn



kn D1



1 X

:::

1 X m1 D1

X

q p01

1

m1

m1

q 0

: : : mnpn

fO n :::2 .m1 ; k2 ; : : : ; kn /

1

q

ˇ ˇq ˇO ˇ ˇf .m1 ; : : : ; mn /ˇ :

m2Nn

Thus, part a) of Lemma is proved. Part b) may be shown similarly using the second statement of Lemma 3. t u Different variants of Pitt’s inequalities of type (6)–(7) are contained in [4]; see also the papers [2, 5, 8, 11, 14, 18, 22] for the previous results related to Pitt’s inequalities for periodic functions.

4 Proof of Theorem 1 Let f 2 L1 Œ0; 1n and f .˛/ 

P k2Zn

˛ k fO .k/e2ikx . Let us first introduce the anisotropic

Sobolev and Besov spaces as follows: ˚  ˛ Wp;q Œ0; 1n D f W kf .˛/ kLp;q < 1 and 8 9 0 0   1q 11=q ˆ > ˆ >   < = X B Pn X   C C B ˛ n ˛ jD1 kj  2i x  O : < 1 Bp;q Œ0; 1 D f 2 Lp W @ f . /e @2 A A   ˆ > ˆ >  2 .k/  : ; k2ZnC Lp Second, we note that the functionals Pm . f ; / and the operators Fm . f I x/ are connected by Fm . f I x/ D

X

2Gm

Pm . f ; /e2i x D

X X jjm 2Nn

2 ./

Pm . f ; /e2i x :

(8)

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N.T. Tleukhanova

From [21] we get ? sup kf  SGm . f /kLq  dM .Wp˛ ; Lq /:

kf kWp˛ D1

Therefore, it is enough to prove that ? .Wp˛ ; Lq /: sup kSGm . f /  Fm . f /kLq  cdM

kf kWp˛ D1

Let ˛ > 1=2 and q > 2: From Pitt’s inequality (see Lemma 4), we have 0 kSGm . f /  Fm . f /kLq  @

X

11=2

N

2 q0

1

jfO . /  Pm . f ; /j

2A

:

(9)

2Gm

By Theorem 2, we have jfO . /  Pm . f I /j 

n X

X

X

lD1

k1 C:::Ckl Dmjj ki 0

r2Dkl

jfO .r/j D

X

jfO .r/j:

r2M

Then, using Hölder inequality and taking into account that ˛ > 1=2 and inequality (5), we obtain that 0 jfO . /  Pm . f I /j  @

X

11=2 0 jNr˛ fO .r/j2 A

@

r2M

X

11=2 rN2˛ A

r2M

0 11=2 .m C 1  jj/.n1/=2 @ X ˛ O 2 A C jNr f .r/j : 2m˛ r2M

(10)



Hence, from (9) we have kSGm . f /  Fm . f /k2Lq 

C

m C X

22˛m

2.˛C q10  12 /m

X X

mu N q2

jkjDm 2 .k/

sDn

1 2

.m C 1  s/n1

kf k2W ˛ D C 2

1 2

2.˛ 12 C 1q /m

ˇ2 X ˇˇ ˇ ˇr˛ fO .r/ˇ r2M

kf k2W ˛ : 2

Reconstruction Operator of Functions from the Sobolev Space

189

Thus, for q > 2 sup kSGm . f /  Fm . f /kLq  C

kf kW ˛ D1 2

1 2

?  dM .W2˛ ; Lq /:

.˛ 12 C 1q /m

Let 1 < p < 2 < q < 1: Applying Pitt’s inequality and (5), we get 0 kSGm . f /  Fm . f /kLq  @

X

11=q ˇq ˇ ˇ ˇ

q2 ˇfO . /  Pm . f ; /ˇ A

2Gm

20 11=q 0 X q 1 X X 6 B @

q2 4@ rN p0 jNr˛ fO .r/jq A @ 0

2Gm

r2M

r2M

!q0 11=q0

1 r

A

˛C p10  1q

3q 11=q 7 C 5 A

0 C

1 2

.˛C p10  1q /m

11=q 1 ˇq X X q 1 ˇˇ .n1/q X X ˇ @ .m  s C 1/ q0

q2 rN p0 ˇNr˛ fO .r/ˇ A jkjDs 2 .k/

sD0

C

X

1 1

1

2.˛ p C q /m

rN

q p0

r2M

!1=q 1

˛O

jNr f .r/j

:

q

r2Zn

Therefore, kSGm . f /  Fm . f /kLq  C

X

1 1

1

2.˛ p C q /m

r

q p0

!1=q 1

˛O

jNr f .r/j

q

:

(11)

r2Zn

By Pitt’s inequality, we get kSGm . f /  Fm . f /kLq  C

1 2

.˛ 1p C 1q /m

˛ : kf kWp;q

(12)

˛ ˛ Taking into account that q > p and using the embedding Wp˛ D Wp;p ,! Wp;q , we have

sup kSGm . f /  Fm . f /kLq 

kf kWp˛ D1

C 2

.˛ 1p C 1q /m

?  dM .Wp˛ ; Lq /

The proof is now complete. Finally, we state the result similar the one given by Theorem 1 but for the Besov spaces.

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N.T. Tleukhanova

Theorem 4 If 1 < p  2  q  1; ˛ > 1p ; then sup kf kB˛p;q D1

? ˛ kf  Fm . f /kLq  dM .Bp;q ; Lq /:

Proof The statement follows from inequality (12) and the following Lemma 5 ([10]) Let p D .p; : : : ; p/; q D .q; : : : ; q/; r D .r; : : : ; r/; ˛ 

 1r ; ˇ > ˛, then ˛ B r;q Œ0; 1n ,! Wp;q Œ0; 1n :

1 p

D

t u

Acknowledgements This research was partially supported by the Ministry of Education and Science of the Republic of Kazakhstan (3311/GF4, 4080/GF4).

References 1. K.I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Sov. Math. Dokl. 1, 513–516 (1960) 2. J.J. Benedetto, H.P. Heinig, Weighted Fourier inequalities: new proofs and generalizations. J. Fourier Anal. Appl. 9, 1–37 (2003) 3. E.M. Galeev, Orders of the orthoprojection widths of classes of periodic functions of one and of several variables. Mat. Zametki 43(2), 197–211 (1988) 4. D. Gorbachev, S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates. J. Approx. Theor. 164(9), 1283–1312 (2012) 5. H.P. Heinig, Weighted norm inequalities for classes of operators. Indiana Univ. Math. J. 33, 573–582 (1984) 6. L.K. Hua, Y. Wang, Applications of Number Theory to Numerical Analysis (Springer, Berlin, Heidelberg, New York, 1981) 7. N.M. Korobov, Number-Theoretic Methods in Approximate Analysis. 2nd edn., p. 285 (Tsentr Nepreryvnogo Matematicheskogo Obrazovaniya, Moscow, 2004) 8. E.D. Nursultanov, Interpolation properties of some anisotropic spaces and Hardy-Littlewood type inequalities. East J. Approx. 4(2), 243–275 (1998) 9. E.D. Nursultanov, On multipliers of Fourier series in a trigonometric system. Math. Notes 63(1–2), 205–214 (1998) 10. E.D. Nursultanov, Interpolation theorems for anisotropic function spaces and their applications. Dokl. Akad. Nauk 394(1), 22–25 (2004) [Russian] 11. E.D. Nursultanov, On the application of interpolation methods in the study of the properties of functions of several variables. Mat. Zametki 75(3), 372–383 (2004) [Russian]; translation in Math. Notes 75(3–4), 341–351 (2004) 12. E.D. Nursultanov, N.T. Tleukhanova, On the approximate computation of integrals for functions in the spaces Wp Œ0; 1n. Math. Surv. 55(6), 1165–1167 (2000) 13. E.D. Nursultanov, N.T. Tleukhanova, Quadrature formulae for classes of functions of low smoothness. Sb. Math. 194(9–10), 1559–1584 (2003) 14. H.R. Pitt, Theorems on Fourier series and dower series. Duke Math. J. 3, 747–755 (1937) 15. V.S. Rjabenki, Tables and interpolation of a certain class of functions. Sov. Math. Dokl. 1, 382–384 (1960)

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16. S.A. Smolyak, Interpolation and quadrature formulas for classes Ws˛ and Es˛ . Dokl. Akad. Nauk SSSR 131(5), 1028–1031 (1960) 17. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 148(9), 1042–1045 (1963) [Russian] 18. E.M. Stein, Interpolation of linear operators, Trans. Am. Math. Soc. 83, 482–492 (1956) 19. V.N. Temlyakov, Widths of some classes of functions of several variables. Sov. Math. Dokl. 26, 619–622 (1982) 20. V.N. Temlyakov, On the approximate reconstruction of periodic functions of several variables. Sov. Math. Dokl. 31, 246–249 (1985) 21. V.N. Temlyakov, Approximations of functions with bounded mixed derivative. Proc. Steklov Inst. Math. 178, 1–121 (1989) 22. S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326 (1), 721–735 (2007) 23. N.T. Tleukhanova, On the approximate computation of multiplicative transformations of functions from the Korobov and Sobolev classes. Mat. Zh. 2(3), 79–88 (2002) [Russian] 24. N.T. Tleukhanova, An interpolation formula for functions of several variables. Math. Notes 74(1), 151–153 (2003) 25. N.T. Tleukhanova, An interpolation formula for multiplicative transformations of functions of several variables. Dokl. Akad. Nauk 390(2), 169–171 (2003) [Russian] 26. R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions (Kluwer Acad. Publ. Dordrecht, 2004)

Part IV

Optimization Theory and Related Topics

Laplace–Borel Transformation of Functions Holomorphic in the Torus and Equivalent to Entire Functions L.S. Maergoiz Abstract The object under study of this paper is the class A.Tn / of functions holomorphic in the torus Tn and equivalent to entire functions. We present an approach to constructing some growth theory of this class with the use of the growth theory of entire functions of several variables. This approach is illustrated by investigations of Laplace–Borel transformation of functions of A.Tn /. Keywords Entire function • Growth characteristics (order, type, indicator) • Laplace–Borel transformation • Monomial mapping • Multiple Laurent series

Mathematics Subject Classification (2010). Primary 32A22, 32A15

1 Introduction Let Tn D Cn ; where C D Cnf0g; n > 1; be the multidimensional torus, and H.Tn / be the class of functions, holomorphic in Tn . This class is the natural extension of the class H.Cn / of entire functions of n complex variables. The first problem is to find in H.Tn / a subclass of functions “equivalent” to H.Cn /. The natural analog of entire functions of one variable is given by the analytic functions on the Riemann sphere without a point. We cannot expect such an analog for several variables: the analytic functions of several variables have no isolated singularities. Some possible approach to finding a multidimensional analog of this result is suggested by the following version of the Hadamard–Valiron theorem:

L.S. Maergoiz () Siberian Federal University, pr. Svobodny 79, 660041 Krasnoyarsk, Russia e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_13

195

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L.S. Maergoiz

Theorem A Let f .z/ D f .z1 ; : : : ; zn / be a function, holomorphic in Tn ; while Mf .r/ D maxfj f .z/j W jzk j D rk ; k D 1; : : : ; ng;

r 2 Rn0 ;

where R0 D fr > 0 W r 2 Rg; be the maximum modulus of f . Then Mf .r/ and ln Mf .r/ are convex functions of ln r1 ; : : : ; ln rn . In the case that f is the trace of an entire function on Tn , note that Mf .r/ is an increasing function in each variable in contrast, for example, to the Zhukovski˘ı function f0 .z/ D 12 .zC1=z/ whose maximum modulus attains the (global) minimum at some point r0 > 0. Therefore, Vf .u/ D ln Mf .eu1 ; : : : ; eun /;

u 2 Rn

(1)

is a convex function whose directions of decrease comprise the convex decrease direction cone KV which includes Rn , where R D fu 2 R W u  0g: On the other hand, if for a given convex function W.v/; v 2 Rn , different from a constant, KW satisfies the condition dim KW D n then there exists a nondegenerate linear mapping FW W Rn ! Rn with a property V.u/ WD ŒW ı FW .u/; u 2 Rn is a convex function increasing in each variable. This fact generates the conjecture of the existence of a proper subclass A.Tn / in the class H.Tn / of the holomorphic functions in Tn that are equivalent to entire functions in the following sense: a function g 2 H.Tn /, n > 1, belongs to A.Tn / if there exists a monomial mapping F W Tn ! Tn I

z D F .w/; zj D

n Y

s

wi ij ; j D 1; : : : ; n;

(2)

iD1

where B D ksij k is an integer nondegenerate square n  n-matrix,1 such that the function f .w/ D Œg ı F .w/ admits an analytical continuation F.w/ to Cn (i.e. F is an entire function). Main motivation of this paper is to present an approach to constructing some growth theory for the extension A.Tn /; with the use of the growth theory of entire functions of several variables [1, 2]. This approach is illustrated by investigations of Laplace–Borel transformation of functions of A.Tn /:

1

For fixed i 2 f1; : : : ; ng ituple .si1 ; : : : ; sin / of B consists of exponents of wi in (2).

Laplace–Borel Transformation of Functions Holomorphic in the Torus and. . .

197

2 Description of a Multiple Laurent Series Equivalent to a Power Series Main results of this section were announced in [3]. Consider the multiple Laurent series g.z/ D

X

ak zk ;

zk D zk11 : : : zknn ;

(3)

k2Zn

with nonempty domain D; of absolute convergence which we conventionally assume to be situated in Tn (in contrast to multiple power series). The set Sg D fk 2 Zn W ak ¤ 0g is called the support of g: Definition 1 Call a multiple Laurent series g.z/ of the form (3) equivalent to a power series if there exists a monomial mapping of the form (2) where B D ksij knn is an integer nongenerate square n  n matrix such that f .w/ D Œg ı F .w/ D

X

cm wm

(4)

m2BŒSg 

is a convergent power series. In this case we write g  f . In particular, if g 2 H.Tn / then, under the condition (2), the function g.z/ is said to be equivalent to an entire function. Here the notion of equivalence means that, after a change of variables of the form (2), the function f .w/ extends analytically to a neighborhood of the origin (in particular, in Cn ). Remark The inverse mapping F1 is a monomial mapping with fractional exponents, consequently, it is multi-valued one. In spite of it, g.z/ D Œ f ı F 1 .z/ is the same Laurent series as Sg D B1 ŒSf  (see (3), (4)). Therefore, let us agree to say that the power series f is equivalent to the Laurent series g; i.e. f  g: 1 2 Example Let g.z/ D az1 z22 Cbz3 1 z2 Ccz2 =z1 be a Laurent polynomial in T ; where a; b; and c are nonzero complex numbers. Under the mapping

z D F w; w 2 T2 ;

2 3 z1 D w1 1 w2 ; z2 D w1 w2

(5)

it goes to the polynomial f .w1 ; w2 / D aw51 C bw52 C cw41 w32 : The determinant of the exponents of w1 ; w2 (see (5)) is nonzero. Therefore, the polynomials g and f are equivalent by Definition 1. Let K be the closed convex cone in R2 presenting an obtuse angle with the sides 1 D fu 2 R2 W u1 D s; u2 D 2s; s  0g; 2 D fu 2 R2 W u1 D 3t; u2 D t; t  0g:

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Since .1; 2/ 2 1 ; .3; 1/ 2 2 ; .1; 1/ D 0:8.1; 2/ C 0:6.3; 1/ 2 K; therefore, Sg  K:2 For description of a multiple Laurent series equivalent to a power series we use of the concept of a simplicial cone. This is the cone generated by the simplex, where the apex of the cone coincides with one of the vertices of the simplex. A simplicial cone with the apex at 0 2 Rn is called a simplicial rational cone if there is a simplex generating this cone such that all its vertices have only rational coordinates. Note that this conception together with monomial mappings of the form (2) are used in describing the structure of toric varieties [4, 5]. A key property of a simplicial cone is as follows. Theorem 1 ([3, Theorem 7]) Let g.z/ be a n-dimensional Laurent series of the form (3). In the notations of formulas (4) the series g is equivalent to the power series f D g ı F .see Definition 1/ if and only if there exists a simplicial rational cone K .with the apex at 0 2 Rn / that contains the support Sg of g: Moreover, the support Sf of f lies in the sublattice LK D fm 2 ZnC W m D Bk; k 2 K \ Zn g of ZnC ; where B is the matrix associated with F : Proof NECESSITY. Let g  f . In the notations of Definition 1 the matrix B WD ksij knn defines some bijective linear mapping B W Rn ! Rn : Then (see (4)) Sf D BŒSg   RnC ; Sg  K; where Sg and Sf are the supports of the series g and f , respectively, and K D B1 ŒRnC : Here B1 is the inverse matrix of B: Let ej ; j D 1; : : : ; n be the standard basis in RnC : Combining this with a structure of B, we deduce that vectors B1 .ej /; j D 1; : : : ; n have only rational coordinates and make up the basis of K. Therefore, K is the simplicial rational cone, containing Sg . SUFFICIENCY. Suppose that K is a simplicial rational cone such that Sg  K. We may assume without loss of generality that dim K D n: Denote by cj D .c1j ; : : : ; cnj /; j D 1; : : : ; n

(6)

direction vectors of the cone generators with properties: they have only integer coordinates, and, moreover, their sequence defines the positive orientation of Rn . This means that  WD det A > 0; where A D kcij knn (the vectors fcj gn1 are the columns of A (see (6)). Look for a basis in K in the form bj D tcj ; j D 1; : : : ; n; for some t > 0: Each element k D .k1 ; : : : ; kn / 2 Sg  K is representable as k D m1 b1 C    C mn bn D tAm; m D .m1 ; : : : ; mn / 2 RnC :

2

There are more complicated examples in [3].

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Hence, using the inverse matrix A1 of A; we obtain, tm D A1 k. Choosing t D 1=; we have: m D A k; mj D A1j k1 C    C Anj kn ;

j D 1; : : : ; n;

(7)

where A is the adjoint matrix of A, whose elements are the cofactors to the elements of A. Consequently, m 2 ZnC : In the notations of (7) we consider the mapping F W Tn ! Tn I

z D F .w/; zi D

n Y

A

wj ij ; i D 1; : : : ; n:

jD1

Putting gk .z/ WD zk ; k 2 Sg ; we find mn 1 fk .w/ WD Œgk ı F .w/ D wm D wm 1 : : : wn ;

with the exponent of the monomial wm (see (7)). Therefore, f .w/ D Œg ı F .w/ is a power series, and, moreover, in the notations of Definition 1 we have B D A ; i.e. g  f. t u Remark Let in the notations of Theorem 1 Xg be the closed convex hull of the support Sg of g with property dim Xg D m < n D dim K. Then it is possible to choose the cone K and the mapping F of the form (2) such that f D g ı F is a power series of m variables (see Example 1 and Theorem 7 in [3]). Corollary 1 In the notations of Theorem 1 and its proof it is possible to choose the cone K and the matrix B; associated with the mapping of the form (2), such that B.bj / D ej ; j D 1; : : : ; n;

B.K/ D RnC ;

K D B1 ŒRnC :

(8)

It is known that the domain D of absolute convergence of a multiple Laurent series g (see (2)) has the following property: ln D D fu D ln jzj WD .ln jz1 j; : : : ; ln jzn j/ 2 Rn W z 2 D  Tn g is a convex set in Rn : Moreover, it is valid Multidimensional Analogue of the Abel’s Lemma (see, e.g., [6]) Let g be a Laurent series with nonempty domain D; of absolute convergence, while K is an arbitrary cone containing the closed convex hull of the support Sg of g, and a is its apex. Then every point z0 2 D satisfies the condition ln jz0 j C Kaı  ln D; Kaı D fv 2 Rn W hu; vi  0 8 u 2 Ka g where Kaı is the polar of the cone Ka D K  a with the apex at 0 2 Rn .

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If the series g equivalent to a power series, then its domain of absolute convergence has the following similar geometric property. Corollary 2 (cf. Theorem 2 in [3]) Let g be a multiple Laurent series g with a nonempty domain D  Tn of absolute convergence. Suppose that g is equivalent to a power series. Denote by K any fixed simplicial rational cone .with the apex at 0 2 Rn /, containing the support Sg of g. Then every point z0 2 D satisfies the condition ln jz0 j C K ı  ln D; where K ı is the polar of the cone K:

3 Growth Characteristics of Functions of H.Tn /: Remind the notions of growth characteristics of functions of H.Tn / which were defined in [3] by analogy with the asymptotic characteristics of entire functions of several variables (see [2], Chaps. 6–8). The growth of each function g 2 H.Tn / was compared to its maximum modulus Mg .r/; r 2 Rn0 (see Theorem A). This function is in general not increasing in each variable. Definition 2 (cf. [2], Definition 6.2.4; [3]) The order function of a function g 2 H.Tn / is defined in Rn as g .u/ D lim .ln t/1 lnC lnC Mg .tu /; u 2 Rn I t!1

tu D .tu1 ; : : : ; tun /:

Call g a function of finite order if its order function g is a finite function in Rn . For fixed x 2 Rn n f0g the value g .x/ is called the x-order of g: If g 2 H.Tn / is a function of finite order (see Definition 2), then D D fu 2 Rn W g .u/ > 0g is the cone of all parabolic directions of its normal growth order. This set is defined by its section Tg D fu 2 Rn0 W g .u/ D 1g

(9)

which we, by analogy with the case of entire functions(see [2], Definition 6.2.16), call the order hypersurface of g: Definition 3 (cf. [2], Definition 6.3.1; [3]) Let g 2 H.Tn /; and let g be the order function of g. Assume that g .x/ 2 .0; 1/ for fixed x 2 Rn n f0g: Call g .rI x/ D lim t g .x/ lnC Mg .rtx /; t!1

r 2 Rn0 ;

the x-type function for g and refer to g .x/ WD g .II x/ as its type in the direction x. Therefore, without loss of generality, under the assumptions of Definition 3, it suffices to require the fulfillment of the condition x 2 Tg (see (9)).

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Since g .˛u/ D ˛ g .u/

8 ˛ > 0;

u 2 Rn ;

(10)

where g is the order function of g (see Definition 2), we have by Definition 3 that g .rx I x/ D g .rI x/ g .x/

8 r 2 Rn0 ;  > 0I

g .I x/ D g .I x/

(11)

8  > 0:

Therefore, without loss of generality, under the assumptions of Definition 3, it is suffices to require the fulfillment of the condition x 2 Tg (see (9)). Such a property can be used in defining the x-indicator of a function g: Definition 4 (cf. [2], Chap. 8; [3]) Let g 2 H.Tn / be a function of finite order. Assume that Tg ¤ ;, and x 2 Tg : The x-indicator of g is defined in Tn as hg .zI x/ D lim lim t1  ln jg.tx /j; !z r!1

z 2 Tn I tx D .1 tx1 ; : : : ; n txn /:

Study some properties of these asymptotic characteristics of functions of the class A.Tn /: Theorem 2 Let g 2 A.Tn /, and K be a simplicial rational cone, containing the support Sg of g such that dim K D n. Assume that f D g ı F is an entire function of n variables, where F is a mapping of the form (2), satisfying condition (8). Put L Then the following KL D K ı ,3 where K ı is the polar of K. Suppose x 2 intK: assertions are true (see Definitions 2, 3, 4): 1) the order function g is finite, if g .x/ < 1I 2) the x-type function g .I x/ of g is finite, if 0 < g .x/ < 1; and g .x/ < 1; where g .x/ is the x-type of gI 3) under the conditions of 2/, the xO -indicator hg .zI xO / of g is a finite plurisubharmonic function in Tn ; moreover, hg .zxO I xO / D hg .zI xO /;

z 2 Tn ;  > 0; xO D x= g .x/I

sup hg Œ.r1 e'1 ; : : : ; rn e'n I xO  D g .rI xO /

j'j j

To prove Theorem 2, we need an auxiliary assertion.

3

This set is called the dual cone.

8 r 2 Rn0 :

(12) (13)

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Lemma 1 Let, under the conditions of Theorem 2, KV .g/ be the cone of decrease directions of Vg .u/ D ln Mg .eu /; where Mg is the maximum modulus of g (see (1)). In the notations of Theorem 2 we have L D RnC ; ŒB0 1 ŒK

K ı  KV .g/;

(14)

where B WD ksij knn is associated with F the matrix satisfying equalities (8), and B0 is the transpose of B. Proof By Corollary 14.6.1 in [7], we have dim K ı D n; where the cone K ı does not contains lines. Using (8), we find KL D fx 2 Rn W hx; B1 ui  0 8 u 2 RnC g Hence, taking into consideration that B1 is a linear operator in Rn , we deduce: ŒB1 0 KL D Rn : Since ŒB1 0 D ŒB0 1 ; the equality in (14) follows. Therefore, C

ŒB0 1 ŒK ı  D Rn

(15)

On the other hand, consider the trace of F (see (2)) in Rn0 W r D F . p/; p 2 Rn0 I

rj D

n Y

s

pi ij ; j D 1; : : : ; n;

(16)

iD1

Denote by Mf the maximum modulus of f : Under the conditions of Theorem 3, the functions f .w/ and Mf .r/ are functions of n variables. Then Mf . p/ D ŒMg ı F . p/; p 2 Rn0 :

(17)

Taking the logarithm of both sides of Eq. (16), we find (see (1)) Vf .v/ D Vg ŒB0 .v/; v 2 Rn I Vg .u/ D ŒVf ı .B0 /1 .u/; u 2 Rn :

(18)

Here B0 is the transpose of B; and .B0 /1 is the inverse matrix of B0 : But Mf . p/ and Vf .v/ are increasing functions in each variable, therefore, Rn  KV . f /; where KV . f / is the cone of decrease directions of Vf .v/. Combining this with (15), (18), we obtain B0 ŒRn  D K ı  KV .g/: t u

Laplace–Borel Transformation of Functions Holomorphic in the Torus and. . .

203

Proof of Theorem 2 1. By Definition 2 and (18), we obtain that g .u/  Œ f ı .B0 /1 .u/; u 2 Rn ;

(19)

L we find where g ; f are the order functions of g and f , respectively, Since x 2 intK; by Lemma 1 (see (14)): y WD ŒB0 1 .x/ 2 Rn0 : Under the condition of Theorem 1 (see 1)), together with (19), (10), we receive: g .tx/ D f .ty/ D t g .x/ < 1 8 t > 0:

(20)

Combining this with (19), we deduce that the order function f .v/; increasing in each variable, is finite. By (19), g is a function of finite order, according to Definition 2. Therefore, the statement 1) of Theorem 1 is valid. 2. The conclusions of item 1 have a complete analog for the x-type function of g: In the above notations, together with (16) and (17), we find F . pty / D tx F . p/; Mg .rtx / D Mg Œtx F . p/ D Mf . pty /; p 2 Rn0 ;

(21)

where x D B0 .y/; y 2 Rn0 : Now by Definition 3, we have g .rI x/ D g ŒF . p/I B0 .y/  f . pI y/; p 2 Rn0 : Specifically, taking into consideration (11), we obtain g .x I x/ D g .x/ g .x/ D f .y/ f .y/ D f .y I y/ < 1

8  > 0:

Here, the quantities g .x/; f .y/ are the x-types of g and f ; respectively. Next, using the same arguments as in item 1, we conclude that the statement 2) of Theorem 2 is true. 3. Note that the element xO WD x= g .x/ 2 Tg (see (9)). At first, formula (12) follows from Definition 4. In the notations of item 1 put yO WD y= f .y/. Taking into account (10) and the equality g .x/ D f .y/ (see (19)), we have xO D B0 .Oy/; yO 2 Tf : Combining Definition 4 with (2) and (21), we obtain gŒtxO F .w/ D gŒF .wtyO / D f .wtyO /I hg .zI xO /  hf .wI yO /; w 2 Tn :

(22)

Growth characteristics of f , its yO -type function f . pI yO / D lim lim t1 lnC Mg .btyO / b!r r!1

(23)

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L.S. Maergoiz

(see [2], Definition 6.3.1, Proposition 6.3.2) and its yO -indicator hf .wI yO / are defined in RnC and Cn ; respectively. Moreover, by item 1 and 2 of the proof, f .Oy/ D g .Ox/ D 1;

f .Oy/ D g .Ox/ < 1:

(24)

Therefore, properties 8.1.1 and 8.1.3 in [2] (see also Chap. 3, §5 in [1]) imply that yO -indicator hf .wI yO / of f is a finite plurisubharmonic function in Cn and satisfies the formula of the form (13), where the function g and the vector xO are replaced by f and yO , respectively. Now the statement 3) of Theorem 2 follows from (22). t u Remark Theorem 2 is true also in the case if f D g ı F is an entire function of m variables, where m < n: Then there exist more complicated relations between growth characteristics of functions g and f : Corollary 3 If under the conditions of the statement 2/ of Theorem 2, specifically, g .x/ D 0, then hg .zI x/  0; z 2 Tn : Proof By analyze of item 2 of the proof (see (21), (23)), we verify, in the notations of Theorem 2, that g .rI x/  0; r 2 Rn0 I

f . pI y/  0; p 2 RnC :

Then hf .wI y/  0; w 2 Cn ; and hf .w0 I y/ D 0 for some w0 2 Cn : Together with Theorem 2.1.4 in [1], we deduce that hf .wI y/  0; w 2 Cn : Now Corollary 3 follows from (22). t u

4 Laplace–Borel Transformation of Functions Equivalent to Entire Functions with Supports in a Simplicial Cone The object under study at the first part of this section are geometrical growth characteristics of functions of the class H.Tn /: Introduce a natural generalization of the geometrical image in Cn ; n > 1 of the nonnegative x-indicator of an entire function in the case x 2 Rn0 (see [8]; Definition 8.1.5 in [2]). Definition 5 Let g 2 H.Tn /; and x 2 Rn n f0g. Put n h.z/ D hC g .zI x/ WD maxfhg .zI x/; 0g; z 2 T ;

where hg .I x/ is the x-indicator of g (see Definition 4). The set [ … j 0; … D fz 2 Tn W h. x z/ < Reg; h WD h .gI x/ D

(25)

(26)

where  xk D jjxk expfixk argg; k D 1; : : : ; n; is called the x-indicator diagram of g:

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205

The geometrical structure of h (see (26)) is similar to the above case (see Sect. 8.2.1 in [2]). Consider the disk K D f 2 C n f0g W Re  1  1g: For w 2 Tn ; we put Ox .w/ D f x w D . x1 ; : : : ;  xn / 2 Tn W  2 Kg: Definition 6 A set  in Tn is said to be x-circular if, together with every point z;  contains the set Ox .w/ with some w D w.z/ 2 Tn : The set ! D fw 2 Tn W Ox .w/  g is called the kernel of : The set f.z1 ; z2 / 2 T2 W jz1 z2 j > 1g is the example of the .1; 1/-circular domain. By Definition 6, in the notations of Definition 5 the x-indicator diagram h of g 2 H.Tn / is the x-circular set; moreover, the set …1 (see (26)) is the kernel of h : The set h and its kernel …1 are x-parabolically star-like sets, i.e., if, for example, z 2 h , then ztx 2 h 8 t 2 .0; 1/ (see (12)). Besides, the function h is the x-parabolic Minkowski functional of h , i. e. h.z/ D inffs > 0 W zsx 2 h g;

z 2 Tn

(see [9], Definition 3; [2], Chap. 8). Fix a simplicial rational cone K  Rn such that dim K D n > 1; and associated with K a mapping F of the form (2), satisfying the condition (8). We consider two classes of functions. The first class is the subclass AK .Tn / D fgg of A.Tn / with properties Sg  K; where Sg is the support of Laurent series expansion of g; and f D g ı F is the entire function of n variables (see Theorem 2). Next, using the notations of Theorem 2, fix L and suppose that x 2 intK; g .x/ D 1; 0 < g .x/ < 1

8 g 2 AK .Tn /:

(27)

n Denote by Pn .x/ D fp D hC g .I x/ W g 2 AK .T /g the class of nonnegative functions of the form (25). By Theorem 2, every element p 2 Pn .x/ is a plurisubharmonic function in Tn such that

0  p.z/ < 1;

p.ztx / D tp.z/;

z 2 Tn ; t > 0:

(28)

On the other hand, we consider the following class of entire functions of n variables. Let in the above notations y WD ŒB0 1 .x/ 2 Rn0 (see (14)), where B is the matrix, associated with F and satisfying the equalities (8), and ŒB1 0 is the

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transpose of B1 . Denote by DK .Cn / D ff g the subclass of H.Cn /, consisting of entire functions with the property (see Theorem 1) Sf  LK ;

LK D fm 2 ZnC W m D Bk; k 2 K \ Zn g:

(29)

Here Sf is the support of the power series which represents f . Suppose that f .y/ D 1; 0 < f .y/ < 1

8 f 2 DK .Cn /;

(30)

where f .y/; f .y/ are the y-order and the y-type of f , respectively. Let Qn .y/ be the class of the nonnegative truncations of the y-indicators for functions of the class DK .Cn /: This is the class of plurisubharmonic functions in Cn with the properties 0  q.w/ < 1;

q.wty / D tq.w/;

w 2 Cn ; t  0:

(31)

Besides, taking into account Remark to Definition 1 and (29), we conclude that every element q 2 Qn .y/ possesses the following property: q ı F1 is an univalent function in Tn : Combining this with the proof of Theorem 2 and Definition 5, we find relations between growth characteristics of the functions g 2 AK .Tn / and f D g ı F 2 DK .Cn /: Proposition 1 In the above notations consider the operator T W AK .Tn / ! DK .Cn /; T .g/ D g ı F D f :

(32)

This mapping is an isomorphism with the following properties (see (24), (26), (22), (32)): g .x/ D 1 D f .y/;

0 < g .x/ D f .y/ < 1I

hg ŒF wI x  hf .wI y/; w 2 Tn I hf ŒF 1 zI y/  hg .zI x/; z 2 Tn I p D F Œq \ Tn ; q \ Tn D F 1 Œp ; C where p D hC g .I x/; q D hf .I y/ are the nonnegative truncations of the x-indicator hg .I x/ and the y-indicator hf .I y/, respectively.

Now we study some properties of Laplace–Borel transformation of functions of the class AK .Tn /, based on investigations in the case K D RnC ; x 2 Rn0 (see [2], Chap. 8).

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Definition 7 Let K  Rn be a simplicial rational cone such that dim K D n > 1; and KL be its dual cone (see the footnote to Theorem 2). Put g 2 AK .Tn /: The Laplace–Borel transformation of g at fixed x 2 intKL is called the system of integrals Z 1.arg  D / G .z/ D g.z x /e d ;

jj < =2:

(33)

0

Fix any function p 2 Pn .x/ (see (28)) and introduce the subclass AK . pI x/ D fg 2 AK .Tn / W hg .zI x/  p.z/ 8 z 2 Tn g;

(34)

where hg .I x/ is the x-indicator of g (see Definition 4). Proposition 2 Let g 2 AK . pI x/; and Sg  K be the support of its expansion with the Laurent series (3). The system of integrals fG ;  2  WD .=2; =2/g (see (33)) is an analytic extension of the Laurent series G.z/ D

X

ak .hk; xi C 1/zk

(35)

k2Sg

to the x-circular domain (see (26), Definition 6) p D […./j 2 ;

…./ D fz 2 Tn W pŒz.ei /x  < cos g:

(36)

In particular, integral G converges in …./ for any  2 : Proof Take the mapping F of the form (2), fixed in this section. By (22), we obtain after elementary manipulations of the integral G for any  2  (see (33), (14)): Z F .w/ WD ŒG ı F .w/ D

1.arg  D /

f .w y /e d ; x D B0 .y/; y 2 Rn0 ;

(37)

0

where f D g ı F : Put q D p ı F ; and (see (29)) DK .qI y/ D fg 2 DK .Cn / W hf .wI y/  q.w/ 8 w 2 Cn g:

(38)

According to Proposition 1, the operator T W AK . pI x/ ! DK .qI y/ is an isomorphism (see (32)). Therefore, f 2 DK .qI y/: Consider the expansion of f with the Laurent series (4) in Tn : Together with Lemma 8.3.2 in [2], the power series F.w/ D

X m2BŒSg 

cm .hm; yi C 1/wm

(39)

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L.S. Maergoiz

is holomorphic in a neighborhood of the origin in Cn ; and this series extends analytically to the domain q D […./j 2 ;

…./ D fw 2 Cn W qŒw.ei /y  < cos g

(40)

by the system of integrals fF g (see (37) ). Then the series Œ F ı F 1 .z/ has a nonempty domain of absolute convergence in Tn : But hm; yi D hBk; yi D hk; B0 .y/i D hk; xi; m 2 Sf D BŒSg ; k 2 Sg : Consequently, in notations of (35) Œ F ı F 1 .z/  G.z/: Recalling that p D q ı F 1 ; p D F Œq \ Tn  (see Proposition 1), we deduce the statement of Proposition 2 by (37), (39), (40). t u Remark In the above notations the set p (36) is called the generalized Borel xpolygon of the series G: Put H.p / the class of holomorphic functions in p : It is possible to consider the Laplace–Borel transform as the operator Lx . p/ W AK . pI x/ ! H.p /; Lx .g/ D G

(41)

(see Definition 7, (34), (35)). In similar form it is used in [10] for description of the domain of absolute convergence of the integral G0 (see (33)) in the case K D RnC ; x 2 Rn0 . Now we formulate the following analog of the Polya–Martineau–Ehrenprice theorem (cp. [2], Sect. 8.3). Theorem 3 Let p 2 Pn .x/ (see (28)). In the notations of Definition 7, (34), (3) the operator Lx is an isomorphism between AK . pI x/ and H.p /: Moreover, the following statements are equivalent: 1) for a function g 2 AK . pI x/; its x- indicator diagram is equal to p , or the nonnegative trunkation of its x-indicator hC g .I x/ D pI 2) the largest x-circular domain (see Definition 6) to which the function G D Lx .g/ extends analytically coincides with p : Proof A scheme of the proof is the same as in the case of Proposition 2: the statement of Theorem 3 is true if (and only if) it is true the same assertion for the Laplace–Borel transform Ly W DK .qI y/ ! H.q /; Ly . f / D F; where q D p ı F 2 Qn .y/; H.q / is the class of holomorphic functions in q (see (31), (38), (40)). By Theorem 8.3.1 in [2], this statement is valid for more wide class of entire functions ff g: relation (29) is replaced by the condition Sf 2 RnC : The proof of Theorem 3 is the same in the present case. t u

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209

Remark As was noted by the author in [11], Kiselman [8] proved a multidimensional analog of the Polya theorem. In that analog, the geometrical image h of the radial indicator h of an entire function (i.e. x D I WD .1; : : : ; 1/) lies in the projective space Pn : !h D

[

…h ./j 2 CI …h ./ D f.z1 ; : : : ; zn ; t/ 2 Pn W h.z/ < 0 and "k ! 0 as k ! 1, a. y/ D 0 for y  0 and a. y/ D y3 if y < 0. By monotony method (see [6], Sect. 2, Theorem 2.1) for any v 2 L2 ./ the Eq. (2) has a unique solution y D yk Œv from the space H01 ./ \ H 2 ./, and the mapping yk Œ  W L2 ./ ! H01 ./

Optimization Control Problems for Systems Described by Elliptic Variational. . .

213

is weakly continuous. Besides yk Œv ! y weakly in H01 ./ after extracting a subsequence by Theorem 5.2 (see [6], Chap. 3), where y is a solution of the variational inequality (1) for this control. Note that the norm of the solution of Eq. (2) is estimated by the norm of the absolute term by this theorem. Then the mentioned convergence is uniform with respect to v from any bounded subset of L2 ./. Consider convex closed bounded subsets V of L2 ./ and Y of H01 ./. The pair .v; y/ from the set V  Y is called admissible if it satisfies the inequality (1) (see [22]). By U denote the set of all admissible pairs. Suppose this set is non-empty for nontriviality of the problem. Consider the functional I.v; y/ D

1 2

Z



. y  y@ /2 C v 2 dx;



where y@ is a given function from H01 ./, > 0. We have the following optimization control problem. Problem P1 Minimize the functional I on the set U. Prove the weak continuity of the solution of the variational inequality (1) with respect to the control. By yŒv denote its solution for the control v. Lemma 2.1 If fvs g  V and vs ! v weakly in L2 ./, then yŒvs  ! yŒv weakly in H01 ./ after extracting a subsequence. Proof We have yŒvs   yŒv D



     yŒvs   yk Œvs  C yk Œvs   yk Œv C yk Œv  yŒv :

Then yk Œw ! yŒw weakly in H01 ./ uniformly with respect to w 2 V after extracting a subsequence as k ! 1. So yk Œv ! yŒv and . yŒvs   yk Œvs  ! 0 weakly in H01 ./. Besides yk Œvs  ! yk Œv weakly in H01 ./ for all k as s ! 1. Hence the assertions of the lemma follow from the last equality. t u Theorem 2.2 Problem P1 is solvable.  ˚ Proof Let the sequence of pairs .vs ; ys / be minimizing. So we have the inclusions vs 2 V, ys 2 Y, the variational inequality Z .ys C vs /.z  ys /dx  0 8z 2 Z; 

and the convergence I.vs / ! inf I.U/. The sequence fvs g is bounded in L2 ./ by the boundedness of V. Then vs ! v weakly in L2 ./ after extracting a subsequence. Using Lemma 2.1, we get ys ! yŒv weakly in H01 ./ after extracting a subsequence. So we obtain the inclusions v 2 V and yŒv 2 Y by the convexity

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and the closeness of the sets V and Y. Then   v; yŒv 2 U: Using the lower semicontinuity of the square of the norm for Hilbert space, we have   I v; yŒv  lim I.vs ; ys /: s!1

Thus   I v; yŒv  I.U/:   Therefore the pair v; yŒv is a solution of our problem. This completes the proof of the Theorem 2.2. t u Hence the Problem P1 has a solution. Our aim is the development and the substantiation of the method of its resolution.

3 Approximation of the Optimization Control Problem The optimization control problems for systems described by equations are easier than for systems described by variational inequalities. So we will use the known approximation of the system (1) by the nonlinear elliptic equation (2) for the analysis of Problem P1. Consider the set ˚  Vk D v 2 Vj yk Œv 2 Y and the functional Ik .v/ D

1 2

Z h

i . yk Œv  y@ /2 C v 2 dx;



Problem P2 Minimize the functional Ik on the set Vk . Prove the non-triviality of the set Vk at first. We supposed that the set U is nonempty. Use now the more strong assumption. Suppose the existence of a point v 2 V such that the state yŒv belongs to the interior of the set Y with respect to the weak topology of H01 ./. Lemma 3.1 Under this supposition the set Vk is non-empty for large enough value k. Proof By our assumption the state yŒv belongs to the interior of the set Y for some control v 2 V. Then there exists a neighborhood O of yŒv such that it is the subset

Optimization Control Problems for Systems Described by Elliptic Variational. . .

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of this set. By convergence yk Œv ! yŒv weakly in H01 ./ the point yk Œv belongs to O for a large enough k. Then yk Œv 2 Y. So the set Vk is non-empty. t u Using the weakly continuity of the map yk Œ  W L2 ./ ! H01 ./; we obtain the following result. Lemma 3.2 Problem P2 is solvable. By vk denote a solution of Problem P2. Prove the convergence of the approximation method.   Theorem 3.3 We have the convergence I vk ; yŒvk  ! inf I.U/ as k ! 1 and vk ! v in L2 ./ after extracting a subsequence, where v is a solution of Problem P1. Proof We have Ik .vk / D min Ik .Vk /  Ik .v /: Using the definition of the approximate functional, we get 1 Ik .v / D 2

Z n

o . yk Œv   y@ /2 C v 2 dx



Z o 1 n . yk Œv   y@ /2  Œ. yŒv   y@ /2 dx: D I.v / C 2 

Then Ik .vk /  inf I.U/ C

  1    yk Œv  C yŒv   2y@  yk Œv   yŒv  ; 2 2 2

where k kp is the norm of the space Lp ./. The sequence fyk Œv g is bounded in the space H01 ./. Besides yk Œv  ! yŒv  weakly in H01 ./ and strongly in L2 ./ by Rellich–Kondrashov Theorem. Then we obtain lim Ik .vk /  inf I.U/:

k!1

(3)

The sequences fvk g and fyk g, where yk D yk Œvk , are bounded in the spaces L2 ./ and H01 ./ because of the boundedness of the set V and Y. Then we get vk ! v weakly in L2 ./ and yk ! y weakly in H01 ./ after extracting subsequences. Using convexity and closeness of the set V and Y, we get v 2 V and y 2 Y. We have the

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equality  yk C

1 a. yk / D vk : "k

(4)

Then a. yk / D "k .vk C yk /: By boundedness of the sequence fyk g in H01 ./ the sequence fyk g is bounded in H 1 ./. Using the convergence yk ! y weakly in H01 ./, we have yk ! y weakly in H 1 ./. After passing to the limit in the last equality, we get a. yk / ! 0 weakly in H 1 ./. By Sobolev Theorem we have the continuous embedding H01 ./  L4 ./ and L4=3 ./  H 1 ./. Then the sequence fyk g is bounded in the space L4 ./. Using the definition of the function , we obtain Z Z  ˇ ˇ ˇ  ˇ   a. yk /4=3 D ˇa. yk /ˇ4=3 dx D ˇyk ˇ4 dx  yk 4 ; 4=3 4 

k

where ˚  k D x 2 j yk .x/  0 : Then the sequence fa. yk /g is bounded in the space L4=3 ./. By Rellich– Kondrashov Theorem we have the convergence yk ! y strongly in L2 ./ and a.e. in  after extracting a subsequence. So a. yk / ! a. y/ a.e. in  . Using Lemma 1.3 [6, Chap. 1], we have a. yk / ! a. y/ weakly in L4=3 ./ and in H 1 ./ too. Then a. y/ D 0, so y  0 on . Hence the inclusion y 2 Z is true. Using the equality (1), we have Z 

  1 yk Cvk .zyk /dx D "k D 

1 "k

Z a. yk /.zyk /dx D  

Z k

1 "k

Z



a.z/a. yk / .zyk /dx



3

z  . yk /3 .z  yk /dx 8z 2 Z:

(5)

Optimization Control Problems for Systems Described by Elliptic Variational. . .

217

Besides we get Z

Z yk . yk y/dx D 



vk . yk y/dx C



Z D  

1 vk . yk  y/dx C "k

Z

1 "k

Z

Z a. yk /. yk y/dx D  

a. yk /. yk y/dx





a. yk /  a. y/ . yk  y/dx  



Z vk . yk  y/dx: 

Then Z

Z yk . yk  y/dx   lim

lim

k!1

vk . yk  y/dx D 0:

k!1



(6)



By inequalities (5) and (1) we have Z .y C v/.z  y/dx D lim

Z h

k!1



i yk .z  y/ C vk .z  y/ dx



Z D lim

k!1



yk .z  yk / C vk .z  yk / C yk . yk  y/ dx



Z

Z

 lim

.yk C vk /.z  yk /dx C lim

k!1

yk . yk  y/dx  0 8z 2 Z:

k!1





So y D yŒv, then .v; y/ 2 U. Using the convergence vk ! v weakly in L2 ./ and yk ! y weakly in H01 ./, we get kvk2  inf lim kvk k2 ; ky  y@ k2  inf lim kyk  y@ k2 : k!1

k!1

Then Ik .vk / ! inf I.U/. We have the inequality Z ˇ ˇ 1 ˇ ˇ ˇIk .vk /  I.vk ; yk /ˇ  2

ˇ 2  2 ˇˇ ˇ ˇ yk Œvk   y@  yŒvk   y@ ˇdx



 

   1 yk Œvk   yŒvk  yk Œvk  C yŒvk   2y@  2 2 2

    o 1 n yk   y C yŒvk   yŒv yk Œvk  C yŒvk   2y@  : 2 2 2 2

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By the convergence vk ! v weakly in L2 ./ we get yŒvk  ! yŒv weakly in H01 ./. Using the convergence yk ! y weakly in H01 ./, we obtain yŒvk  ! yŒv and yk ! y strongly in L2 ./. Using the last inequality, we have ˇ ˇ lim ˇIk .vk /  I.vk /ˇ! 0;

k!1

so I.vk ; yk / ! inf I.U/. We proved that a subsequence of solutions of Problem P2 is minimizing for the Problem P1. Suppose the existence of a subsequence of fI.vk ; yk /g such that it does not have inf I.U/ as a limit point. Using considered technique, extract its subsequence that convergences to inf I.U/. So the whole sequence fI.vk ; yk /g converges to inf I.U/. By the convergence vk ! v weakly in L2 ./ and yk ! y strongly in L2 ./ we have kvk2  lim kvk k2 ; kyŒv  y@ k2 D lim kyk Œv  y@ k2 : k!1

k!1

Then I.v; y/  lim I.vk ; yk / D inf I.U/: k!1

Using the inclusion .v; y/ 2 U, we prove that v is a solution of Problem P1. By fvk g denote the subsequence, which correspond the lower limit of last inequalities. Suppose the strong inequality kvk2 < inf lim kvk k2 : k!1

Then we obtain the strong inequality I.v; y/ < inf I.U/: This contradiction prove the convergence kvk k2 ! kvk2 . Using the convergence vk ! v weakly in L2 ./, we prove that vk ! v strongly in L2 ./. This completes the proof of Theorem 3.3. t u Remark 3.4 Problem P1 can have many solutions. In this case different subsequences of fvk g can converge to different solutions of this problem. However our conclusions are true for all its convergent subsequence. Therefore the set of limit points of fvk g consists of solutions of Problem P1. However it is possible that some solution does not belong to this set. The known results the optimization control problems for systems described by variational inequalities include as a rule the justification of the necessary conditions of optimality (see, for example, [7–14]). However we solve optimization control

Optimization Control Problems for Systems Described by Elliptic Variational. . .

219

problems only approximately. The known necessary conditions of optimality are difficult enough. So it is more naturally to find the approximate solution of the problem, rather than necessary conditions of optimality. This idea was used in [19– 21] in the case of insolvability of extremum problems. By Theorem 2 we can choose the optimal control for Problem P2 for large enough value of k as an approximate solution of Problem P1. So we will solve the solution of Problem P2. It is easier than Problem P1 because the system is described by equation, rather than variational inequality.

4 Second Approximation of the Problem The general difficulty of Problem P2 is the state constraint. We cannot to use the standard variational method for this case because we do not know how we can change the control for saving the state constraint. We could apply results of the general extremum theory (Lagrange principle and some other methods, see, for example, [23–25]). But it uses very difficult properties of the linearized operator and the state constraint. However some results for optimization control problems for nonlinear elliptic equations with state constraints are known (see, for example, [26–32]). Our aim is the search of minimizing sequences in contrast to these results. Then we transform our problem to an easier one. Using the penalty method [22], we change our optimization problem by the minimization problem for the penalty functional on the set of admissible “control-state” pairs. Note that this technique was used in [22] for the case of the absence of the state constraint. The unique solvability of the state equation was not guarantee there. However our boundary problem is well-posed, but we have the state constraint. Define the functional Z

2 o 1 n 1 Ikm .v; y/ D . y  y@ /2 C v 2 C dx; y C "1 k a. y/ C v 2 ım 

where ım > 0 and ım ! 0 as m ! 1. Define the space W D L2 ./  H01 ./ and the set U@ D V  Y. We have the following problem. Problem P3 Minimize the functional Ikm on the set U@ . Lemma 4.1 Problem P3 is solvable.  ˚ Proof Let fus g D vs ; ys be a minimizing sequence for the Problem P3, so us 2 U@ and Ikm ! inf Ikm .U@ / as s ! 1. Using the boundedness of the set U@ , we prove that the sequence fus g is bounded in the space W. By definition of the functional we

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have the equality ys D "1 k a. ys / C vs C fs ; where the sequence ffs g is bounded in the space L2 ./. Using the boundedness of the sequence fys g in H01 ./ and in L6 ./ too because of Sobolev Embedding Theorem, we prove the boundedness of the sequence fa. ys /g in the space L2 ./. Then the term in the right side of the last equality is bounded in the space L2 ./. So fysg is bounded in L2 ./. Hence we get vs ! v weakly in L2 ./, ys ! y weakly in H01 ./, a. ys / ! ' weakly in L2 ./, ys ! y weakly in L2 ./ after extracting subsequences. Using the convexity and the closeness of the sets V and Y, we have the inclusions v 2 V and y 2 Y, then u 2 U@ , where u D .v; y/. By Rellich–Kondrashov Theorem we get ys ! y strongly in L2 ./ and a.e. on , then a. ys/ ! a. y/ a.e. on . Using Lemma 1.3 (see [6], Chap. 1), we obtain a. ys / ! a. y/ weakly in L2 ./, so ' D a. y/. By the weak lower semicontinuous of the norm in Hilbert spaces we get Ikm .u/  inf Ikm .U@ /; so u is a solution of Problem P3. This completes the proof of Lemma 4.1.  m m Let um k D vk ; yk be a solution of Problem P3.

t u

Theorem 4.2 For any k Ik .vkm / ! inf Ik .Vk / as m ! 1, besides vkm ! vk in L2 ./ after extracting a subsequence. Proof We have the inequality   m m Ikm .um k / D min Ik .U@ /  Ik vk ; yk Œvk  D Ik .vk /:

(7)

By boundedness of the set U@ the sequence fum k g is bounded in the space W. Using the inequality (7) and the definition of the functional Ikm , we get 1 m m  ym k D "k a. yk / C vk C

p ım fkm ;

(8)

where the sequence ffkm g is bounded in L2 ./. Then (see the proof of Lemma 4.1), m m  the sequence fym k g is bounded in L2 ./. Then vk ! vk weakly in L2 ./ , yk ! m m  1  yk weakly in H0 ./, fk ! fk weakly in L2 ./, and yk ! yk weakly in L2 ./ as m ! 1 after extracting subsequences. Using the technique from the proof of  Lemma 4.1, we obtain vk 2 V, yk 2 Y, and a. ym k / ! a. yk / weakly in L2 ./. After   passing to the limit in the equality (8) we get yk D yk Œvk . By definition of the functional Ikm we have Ikm .um k/ 

1 2

Z h  

2  m 2 i ym dx:  y @ C vk k

Optimization Control Problems for Systems Described by Elliptic Variational. . .

221

Hence min Ik .Vk / D Ik .vk /  Ik .vk / D



1 lim 2 m!1

Z h 

1 2

Z n 

2  2 o yk Œvk   y@ C vk dx



2  m 2 i ym dx  lim Ikm .um  y @ C vk k k /: m!1



Using (7), we obtain Ikm .um k / ! min Ik .Vk /. By inequalities Ik .vk /  Ik .vkm /  Ikm .um k/ we have Ik .vkm / ! inf Ik .Vk /. The proof is ended with using the technique from Theorem 4.2 . t u Remark 4.3 All assertions of Remark 3.4 are true in this case. By proved theorem a sequence of solutions of Problem P1 minimizes the functional Ik on the set Vk . So the value vkm for large enough m can be chosen as an approximate solution of Problem P2. Then the control vkm for large enough value m and k can be chosen as an approximate solution of Problem P1. Our next step is finding of this control. We will prove that the obtained result is sufficient for the analysis of the given optimization problem without any constraints.

5 Necessary Conditions of Optimality We have the minimization problem for an integral functional on a convex set. The necessary condition of the minimum at the point u of Gateaux differentiable functional J on a convex set W is the variational inequality hJ 0 .u/; w  ui  0 8w 2 W;

(9)

where h'; i is the value of a linear continuous functional ' at a point . We prove the differentiability of the functional Ikm for using this result in our case. Lemma 5.1 The functional Ikm has the partial derivatives m m m 1 0 m Ikv .v; y/ D v C pm k .v; y/; Iky .v; y/ D y  y@ C pk .v; y/ C "k a . y/pk .v; y/; (10)

at the arbitrary point .v; y/, where pm k .v; y/ D

1 y C "1 k a. y/ C v : ım

(11)

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S. Serovajsky

Proof For any function h 2 L2 ./ and the value  we have the equality Z h 

2

Ikm .v C h; y/  Ikm .v; y/ D

i v C h/2  v 2 dx



Z n

2

2 o 1 1 y C "1 dx C k a. y/ C v C h  Œy C "k a. y/ C v 2ım 

Z D  



 v C pm k .v; y/ hdx C 2

Z n

2 o 2 C ım p m h dx: k .v; y/ 

So the first equality (10) is true. For any function h 2 H01 ./ and the value  we get Ikm .v; y C h/  Ikm .v; y/ D

1 2

Z



. y  y@ C h/2  . y  y@ /2 dx



Z n

2

2 o 1 1 . y C h/ C "1 dx C a. y C h/ C v  Œy C " a. y/ C v k k 2ım 

D 

Z n



o 1 0 . y  y@ /h C pm k .v; y/ h C "k a . y/h dx C ./



C

Z n



o 1 0 m . y  y@ / C pm .v; y/ C " a . y/p .v; y/ hdx C ./; k k k



where a0 . y/ D 0 for y  0, a0 . y/ D 3y2 for y < 0 and ./ ! 0 as  ! 0. So the first equality (10) is true. This completes the proof of Lemma 5.1. t u Thus by the inequality (9) we get a necessary condition of optimality.   Theorem 5.2 The solution vkm ; ym k of Problem 3 satisfies the following system Z



  vkm C pm v  vkm dx  0 8v 2 V; k

(12)



Z h

m 1 0 m m ym k  y@ C pk C "k a . yk /pk

i

 y  ym k dx  0 8y 2 Y;

(13)

 1 m m m ym k C "k a. yk / C vk D ım pk :

(14)

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We obtain the standard necessary condition of optimality. It can be solved with using an iterative method (see, for example, [33–35]). Then the control vkm can be chosen as an approximate solution of the initial optimization problem for large enough values of k and m. Remark 5.3 This system is simplified in the case of the absence of the state constraint. The variational inequality (13) can be transformed to the standard adjoint equation 1 0 m m m pm k C "k a . yk /pk D y@  yk

in this case. Hence necessary conditions of optimality include the state equation (14), this adjoint equation and classical variational inequality (12). If we do not have any constraints, then we can find the control vkm D  pm k from (12). Then we obtain two elliptic equations 1 0 m m m pm k C "k a . yk /pk D y@  yk ; 1 m m m ym k C "k a. yk / C vk D ım pk :

After solving this system we can find vkm by the obtained formula. Analogical results could be obtained for controls systems described by parabolic and hyperbolic variational inequalities. Laplace operator can be substituted by general linear elliptic operators and some nonlinear elliptic operators. We could consider also a general integral functional with corresponding assumptions.

References 1. G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali 834(2), 138–142 (1963) 2. G. Duvaut, J.-L. Lions, Les inKequations en mKecanique et en physique (Dunod, Paris, 1972) 3. D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (Academic, New York, 1980) 4. R. Glowinski, J.-L. Lions, R. TrKemolier, Numerical Analysis of Variational Inequalities (North Holland, Amsterdam, 1981) 5. C. Baiocchi, A. Capelo, Variational and Quasivariational Inequalities: Applications to FreeBoundary Problems (Wiley, New York, 1984) 6. J.-L. Lions, Quelques MKethods de resolution des problJemes aux limites Non linJeaires (Dunod, Gautier-Villars, Paris, 1969) 7. V. Barbu, Optimal Control of Variational Inequalities. Research Notes in Mathematics, vol. 100 (Pitman, Boston, 1984) 8. Z.X. He, State constrained control problems governed by variational inequalities. SIAM J. Control Optim. 25, 1119–1145 (1987) 9. D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems. Lecture Notes in Mathematics, vol. 1459 (Springer, Berlin, 1990)

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10. J. Bonnans, E. Casas, An extension of Pontryagin’s principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities. SIAM J. Control Optim. 33, 274–298 (1995) 11. G. Wang, Y. Zhao, W. Li, Some optimal control problems governed by elliptic variational inequalities with control and state constraint on the boundary. J. Optim. Theory Appl. 106, 627–655 (2000) 12. Y. Ye, Q. Chen, Optimal control of the obstacle in a quasilinear elliptic variational inequality. J. Math. Anal. Appl. 294, 258–272 (2004) 13. Q. Chen, D. Chu, R.C.E. Tan, Optimal control of obstacle for quasi-linear elliptic variational bilateral problems. SIAM J. Control Optim. 44, 1067–1080 (2005) 14. Y.Y. Zhou, X.Q. Yang, K.L. Teo, The existence results for optimal control problems governed by a variational inequality. J. Math. Anal. Appl. 321, 595–608 (2006) 15. S. Serovajsky, State-constrained optimal control of nonlinear elliptic variational inequalities. Optim. Lett. 8(7), 2041–2051 (2014) 16. P. Neittaanmaki, D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms, and Applications (Marcel Dekker, New York, 1994) 17. D.R. Adams, S. Lenhart, Optimal control of the obstacle for a parabolic variational inequality. J. Math. Anal. Appl. 268, 602–614 (2002) 18. Q. Chen, Optimal obstacle control problem for semilinear evolutionary bilateral variational inequalities. J. Math. Anal. Appl. 307, 677–690 (2005) 19. J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972) 20. V. F. Krotov, Global Methods in Optimal Control Theory (Marcel Dekker, New York, 1996) 21. M.I. Sumin, Suboptimal control of a semilinear elliptic equation with state constraints. Izv. Vuzov 6, 33–44 (2000) 22. J.-L. Lions, ContrOole Optimal de systJemes Distributes Singuliers (Gautier-Villars, Paris, 1983) 23. A.D. Ioffe, V.M. Tihomirov, Extremal Problems Theory (Nauka, Moscow, 1974) 24. A.S. Matveev, V.A. Yakubivich, Abstract Theory of Optimal Control (St. Petersburg University, St. Petersburg, 1994) 25. K. Madani, G. Mastroeni, A. Moldovan, Constrained extremum problems with infinitedimensional image: selection and necessary conditions. J. Optim. Theory Appl. 135, 37–53 (2007) 26. P. Michel, Necessary conditions for optimality of elliptic systems with positivity constraints on the state. SIAM J. Control Optim. 18, 91–97 (1980) 27. E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 33, 993–1006 (1993) 28. N. Arada, J.P. Raymond, State-constrained relaxed problems for semilinear elliptic equations J. Math. Anal. Appl. 223, 248–271 (1998) 29. J.F. Bonnans, A. Hermant, Conditions d’optimalitKe du second ordre nKecessaires ou suffisantes pour les problJemes de commande optimale avec une contrainte sur l’Ketat et une commande scalaires. C. R. Acad. Sci. Paris Ser. I. 343, 473–478 (2006) 30. A. RRosch, F. TrRoltzsch. Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17, 776–794 (2006) 31. A. RRosch, F. TrRoltzsch, On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46, 1098–1115 (2007) 32. E. Casas, J.C. Reyes, F. TrRoltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19, 616–643 (2008) 33. R. Fletcher, Practical Methods of Optimization (Wiley, New York, 2000) 34. J.F. Bonnans, J.C. Gilbert, C. LemarKechal, C. SagastizKabal, Numerical Optimization: Theoretical and Practical Aspects. Universitext (Springer, Berlin, 2006) 35. C. Floudas, P. Pardalos (eds.), Encyclopedia of Optimization (Springer, Berlin, 2008)

Two Approximation Methods of the Functional Gradient for a Distributed Optimization Control Problem Ilyas Shakenov

Abstract An optimization control problem for one-dimensional parabolic equation is considered. Given a value for Gateaux derivative of the target functional. Obtained problem is solved numerically by considering approximation of functional gradient and gradient of functional approximation. Some experiments are done to compare the effects of two methods on resolvability of the problem on final time. Keywords Approximation • Control • Gateaux derivative • Gradient • Optimization, Parabolic equation

Mathematics Subject Classification (2000). 49K20

1 Introduction Problem setting. We consider a problem for parabolic equation in one dimension. Mathematical statement looks as following: @t u.t; x/ D @2x u.t; x/ C f .x; t/; 0 < x < L; 0 < t < T;

(1)

u.0; x/ D '.x/; 0 < x < L;

(2)

@x u.t; 0/ D b.t/; 0 < t < T;

(3)

@x u.t; L/ D y.t/; 0 < t < T;

(4)

Function y.t/ is unknown and must be determined. For that purposes we use an additional information u.t; 0/ D a.t/. We transform this problem to optimization

I. Shakenov () al-Farabi Kazakh National University, 71 al-Farabi Avenue, 050040 Almaty, Kazakhstan e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_15

225

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I. Shakenov

problem which requires to minimize a functional ZT  2   u t; 0I y.t/  a.t/ dt: ID 0

If the functional gets its minimum value then u.t; 0/ is the most close to a.t/ and additional information is fulfilled. The most commonly used method for solving such problems is gradient method. For that, we construct a sequence   ynC1 .t/ D yn .t/  ˛n I 0 yn .t/ ; where ˛n > 0 with an appropriate initial value for y0 .t/.

2 Gradient in Continuous Form   Question arises, what is I 0 yn .t/ ? Expression for Gateaux derivative is given by the following Theorem 1 Gateaux derivative of the functional I at the point y is determined by the formula   I 0 y.t/ D where

.t; L/;

.t; x/ is the solution of the adjoint system: @t .t; x/ C @2x .t; x/ D 0; 0 < x < L; 0 < t < T; .T; x/ D 0; 0 < x < L; @x

@x .t; L/ D 0; 0 < t < T;   .t; 0/ D 2 u.t; 0I y.t//  a.t/ ; 0 < t < T;

Proof The proof of this theorem you can see in [1].

(5) (6) (7) (8) t u

According to this theorem we can write ynC1 .t/ D yn .t/  ˛n .t; L/:

(9)

We choose an initial approximation y0 .t/ and the process is ready to be started. But at this moment we encounter a huge problem.

Two Approximation Methods of the Functional Gradient for a Distributed. . .

227

3 Problem of Improvement on the Right Side of Time Interval One of the boundary conditions says that .T; x/ D 0 and also .T; L/ D 0. If we put t D T in (9) we obtain that ynC1 .T/ D yn .T/ and it means that on the right side of time interval we can’t get more precise values than the initial approximation has. One of the methods to avoid this problem is to cut time interval and consider a numerical solution only at a part of 0 < t < T interval. This approach is described in [1].

4 Numerical Solution Lets consider an approximation of direct problem (1)–(4) jC1

jC1

jC1

j u  2ui C uiC1  ui j D i1 C fi ;  h2

jC1

ui

i D 1; 2; : : : ; N  1; j D 0; 1; : : : ; M  1; u0i D 'i ; i D 0; 1; : : : ; N; j

(11)

j

u1  u0 D bj ; j D 0; 1; : : : ; M; h j

(12)

j

uN  uN1 D yj ; j D 0; 1; : : : ; M; h j

(10)

(13)

j

where ui D u.j; ih/, fi D f .j; ih/,  D MT , h D NL are steps by time and space variables. The following is an approximation of adjoint problem (5)–(8) [2, 3]. jC1 i

 

j i

C

j i1

j

j iC1

2 i C h2

D 0;

j D M  1; M  2; : : : ; 0; i D 1; 2; : : : ; N  1; M i j N j 1

 h

 h j 0

D 0; i D 0; 1; : : : ; N;

j N1

D 0; j D M; M  1; : : : ; 0;

 j  D 2 u0  aj ; j D M; M  1; : : : ; 0:

(14) (15) (16) (17)

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Solving (10)–(13) and (14)–(17) numerically and using iterative process to determine new improved values of y.t/ we can solve the stated problem. As we expected, there is no any movement of the endpoint where t D T. Numerical experiments show it very clearly. It also must be noted that we considered Q D I.y/

M1 X

j

u0  aj

2

(18)

jD0

as an approximation of functional (“left rectangles”). Here are three experiments that were done with different initial approximations in which the following common input data is used: 1. 2. 3. 4.

" D 0:052 (it is how close is I to zero). T D 1, L D 1. Both space and time intervals are divided  by 100 steps.  ˛n D 1 in iterative process ynC1 .t/ D yn .t/  ˛n I 0 yn .t/ . Exact solution is y.t/ D 2e2t .

In all figures ./ imply exact solution and .: : : / is numerical solution and i is a number of iterations. The results of experiments are shown on Figs. 1, 2, 3.

Fig. 1 y0 D 1, i D 2767, jjyn  yexact jjH D 2:299

Two Approximation Methods of the Functional Gradient for a Distributed. . .

229

Fig. 2 y0 D 10  10t, i D 2995, jjyn  yexact jjH D 1:453

Fig. 3 y0 D 16, i D 9708, jjyn  yexact jjH D 2:630

5 Gradient of Approximation Method One more method is to derive an adjoint problem from (10)–(13), that is, we do not approximate adjoint problem, we derive an adjoint problem from approximation. Usually it is much more tedious and requires a lot of rigorous computations [4]. j Let uQ i is a solution of (10)–(13) when yj is replaced with yj C Hj , where Hj are j j arbitrary values. Lets subtract two Eqs. (10)–(13) with solutions uQ i and ui from each

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I. Shakenov

j j j  uQ  ui ıu  j other and we take assuming zi D i D i   jC1

jC1

jC1

j z  2zi C ziC1  zi D i1 ; i D 1; 2; : : : ; N  1; j D 0; 1; : : : ; M  1;  h2 (19)

jC1

zi

z0i D 0; i D 0; 1; : : : ; N; j z1

 h

j

j z0

(20)

D 0; j D 0; 1; : : : ; M;

(21)

j

zN  zN1 D Hj ; j D 0; 1; : : : ; M: h j

We multiply both sides of (19) by unknown values 1; 2; : : : ; N  1 and j D 0; 1; : : : ; M  1: 0D

N1 X M1 X

jC1

jC1

j

 zi 

zi

iD1 jD0

j i

(22)

i

jC1

and sum by all i D !

jC1

z  2zi C ziC1  i1 h2

j i

:

Using summation by parts we obtain: N1 P

zM i

M i



iD1 M1 P

jC1

z0  i 

j N

zN

h2

jD0 M1 P N1 P jD0 iD1

jC1

0 i



jD0

jC1



M1 P

z1

j 1

h2

jC1

ziC1  zi h

 jC1 zi jC1

zN1 h2



j iC1

 h



j N

 

jC1

 ! j

i

!

j

jC1 i

z0

h2



i

! j

1

C

:

here we use statements (21) and (22) and convert the last expression to the following by extracting one addend for i D N  1 from the last expression: N1 P

M i



M1 P



jC1 i

!

j

M1 P   iD1 jD0 jD0 jC1  j jC1 j j M1 z  P zjC1  z  N N1 N1  N C N1 N 2 h h h jD0 !  j j j N2 P jC1 iC2  2 iC1 C i : ziC1 h2 iD1

zM i

jC1 zi

 

i

j N Hj

h j N1

C



jC1 



z1

j 2  h2

j 1



Two Approximation Methods of the Functional Gradient for a Distributed. . .

231

After further rearranging: 

M1 P N1 P

jD0 iD1 M N1 P zM i i



iD1



! j j  2 i C i1 C C h2  j  M1  j j M1 P jC1 P jC1 N Hj N  N1 C  zN z 1 h h2 jD0 jD0

jC1 i

M1 P jD0

j iC1

j

 

jC1 zi

i

j

Now we put some restrictions on j N

jC1 i

 

 h

j N1

j i

C

i

j 1

 h2

j 0

to simplify the last expression:

  D 0; j D 0; 1; : : : ; M  1; corresponds to @x .t; L/ D 0 ;

j iC1

2

j

i h2

M i

j i1

C

D 0; j D 0; 1; : : : ; M  1; i D 1; 2; : : : ; N  1;   @t .t; x/ C @2x .t; x/ D 0 ;   D 0; i D 1; 2; : : : ; N  1; .T; x/ D 0 ;

From these we conclude that: 

M1 X jD0

j N Hj

h



M1 X

jC1



z1

j 1

jD0

j 0

 h2

D 0:

jC1

jC1

One interesting fact follows from (21) and says that z1 D z0 . Our next step is to find variation of functional in discrete form: Q C H/  I.y/ Q 2  jC1 2  P  jC1 I.y  M1 uQ 0  ajC1  u0  ajC1 D D   jD0   M1 P P  jC1 2  M1 jC1  jC1 ıu0 2ıu0 u0  ajC1 C :  jD0  jD0 Pay attention that we get another method for approximation of functional (“right  M1 P jC1  jC1 rectangles”). As  ! 0 and using notation of z we get:  2z0 u0  ajC1 . jD0

This is a time to summarize what we obtained before. M1 X jD0

j N Hj

D

M1 X jD0

 jC1

z0

j 1

 h

j 0

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and we assume

j 1

 h 

  jC1 D 2 u0  ajC1 . It follows that

j 0

M1 X

jC1

2z0

M1   X jC1 u0  ajC1 D 

jD0

j N Hj :

jD0

Reasonings above allows us to formulate a Theorem 2 Gateaux derivative of the functional I at the point y is determined by the formula IQ0 .y/ D j N

where jC1 i

 

j N;

is the solution of the adjoint system: j i

C

j iC1

j

j i1

2 i C h2

D 0;

j D M  1; M  2; : : : ; 0; i D 1; 2; : : : ; N  1; M i j N j 1

 h

 h j 0

D 0; i D 0; 1; : : : ; N;

j N1

D 0; j D M; M  1; : : : ; 0;

 jC1  D 2 u0  ajC1 ; j D M  1; : : : ; 0:

(23) (24) (25) (26)

6 Comparing Two Methods There is a slight difference if you compare (17) and (26). In the right hand side index by time variable has changed to j C 1 instead of j. That was a hope to avoid a problem that we met in first part of the paper. Only numerical experiments can answer the question does the new method resolve the problem or not. The numerical experiments are done with the same parameters as were introduced in graphs of Figs. 1, 2, 3. If graphs of Figs. 1, 2 and 3 are compared with graphs of Figs. 4, 5 and 6 respectively, then there is no visible changes. It means that method of obtaining gradient of approximated problem doesn’t resolve problem of

Two Approximation Methods of the Functional Gradient for a Distributed. . .

233

Fig. 4 y0 D 1, i D 2472, jjyn  yexact jjH D 1:297

Fig. 5 y0 D 10  10t, i D 2669, jjyn  yexact jjH D 1:450

improvement on the right side of time interval. But some notable effect presents in a number of iterations. The second method allows to decrease a number of iterations by approximately 10 %. One more experiment was done to see what happens if time interval becomes larger and now varies as 0 < t < 5. All other parameters kept the same. See Figs. 7 and 8.

234

Fig. 6 y0 D 16, i D 8942, jjyn  yexact jjH D 2:613

Fig. 7 y0 D 10, i D 4620, jjyn  yexact jjH D 3:168, ˛n D 0:1; T D 5

I. Shakenov

Two Approximation Methods of the Functional Gradient for a Distributed. . .

235

Fig. 8 y0 D 10, i D 4134, jjyn  yexact jjH D 3:171, ˛n D 0:1; T D 5

7 Conclusions 1. Approximation of gradient of the functional I and gradient of approximation of the functional I are not the same. The only difference arises in boundary condition. Obtaining a gradient of approximation of the functional is more difficult. 2. The second method of obtaining the adjoint system (gradient of approximation) doesn’t avoid the problem that the error of numerical solution is large when t comes to T. 3. The number of iterations becomes less approximately by 10 % if we use the second method to obtain the adjoint system. That is the new method gives some advantages to reach solution more quickly. 4. If time interval increases (that is Œ0; T becomes larger) the situation doesn’t change cardinally.

References 1. I.K. Shakenov, Inverse problems for parabolic equations on unlimited horizon of time. Bull. Kazakh Natl. Univ. Ser. Math. Mech. Inform. 70(3), 36–48 (2011) 2. S.I. Kabanihin, Inverse and Ill-Conditioned Problem (Nauka, Novosibirsk, 2009) 3. A.A. Samarsky, Theory of Difference Schemes (Nauka, Moscow, 1983) 4. S.Ya. Serovajsky, Optimization and Differentiation, vol. 2 (Publisher of Kazakh University, Almaty, 2009)

Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods Kanat Shakenov

Abstract Four models of linear relaxational filtration is considered. Initial and boundary conditions are set for them (Dirichlet, Neumann and mixed). Obtained problem solved by Monte Carlo methods—“random walk on spheres“, “random walk on balls“ and “random walk on lattices“ of Monte Carlo methods and by probability difference methods. Keywords Approximation • Difference • Expectation • Filtration • Method • Model • Monte Carlo • Numerical modeling • Probability • Random walk • Relaxation

Mathematics Subject Classification (2000). 65C05

1 Introduction The linear relaxational filtration is described by the conservation law of pulse of resistance force, by the linearized conservation law of a fluid mass and determining relations for pulse of resistance forces and fluid mass. After exception of a pulse density of resistance forces (J) and (m ) this system with respect to pressure (p) and velocity of filtration (W) is F.0/ˆ.0/ @2 p.x; t/ p.x; t/ D C 0 @t2 1 0

t0

Z 0

Z1 0

ˆ.0/ dF.t0 / F.0/ dˆ.t0 / C C 0 dt0 0 dt0

! @2 p.x; t  t0 / 0 dF./ dˆ.t0  / d dt ; d d.t0  / @.t  t0 /2

(1)

K. Shakenov () al-Farabi Kazakh National University, 71 al-Farabi avenue, 050040 Almaty, Kazakhstan e-mail: [email protected] © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. Tikhonov (eds.), Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-27466-9_16

237

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K. Shakenov

@W.x; t/   F.0/ @t

Z1 0

dF.t0 / @W.x; t  t0 / 0 dt D gradx p.x; t/: dt0 @.t  t0 /

(2)

Here F.t/ and ˆ.t/ are relaxation kernels of the filtration law and fluid mass [1]. We consider four models of relaxational filtration. I. A model of classical elastic filtration. This model of filtration relates to the kernels of relaxation F.t/ D  t.t/, ˆ.t/ D 0 ˇ.t/, and Eqs. (1) and (2) take form p.x; t/ D W.x; t/ D 

@p.x; t/ ; @t

 gradx p.x; t/:

(3) (4)

II. The simplest model of filtration with a constant speed of disturbance spread. This model is defined with kernels of relaxation: F.t/ D  .t C /.t/, ˆ.t/ D 0 ˇ.t/. For the given model the system (1)–(2) has a form: p.x; t/ D 

@2 p.x; t/ @p.x; t/ C ; @t @t2

 @W.x; t/ C W.x; t/ D  gradx p.x; t/: @t

(5)

(6)

III. Filtration model in relaxationaly-compressed porous environment realized by the linear Darcy law. Corresponding kernels are F.t/ D  t.t/, ˆ.t/ D      0 ˇ  mm p ˇc exp  tm .t/. System (1)–(2) has a form:     @ @p.x; t/ @p.x; t/ D p.x; t/ C 0m ;  p.x; t/ C m @t @t @t  W.x; t/ D  gradx p.x; t/:

(7) (8)

where 0m D m ˇˇ . In a particular case of incompressible fluid, ˇf D 0 and p D 0, instead of (7)–(8) we have   @p.x; t/ @p.x; t/  p.x; t/ C m D ; @t @t  W.x; t/ D  gradx p.x; t/:

(9) (10)

Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods

239

Model (9)–(10) describes a filtration of incompressible fluid in relaxationalycompressed porous environment for p D 0, and also in fractured-porous environment with infinitesimal elasticity of fractures and conductivity of blocks. IV. Model of filtration by the simplest unbalanced law in elastic  porous environ 

ment. Here the kernels of relaxation have form: F.t/ D  t  tW  tp 1     .t/, ˆ.t/ D 0 ˇ.t/. For this model system (1)–(2) lead to exp  tp form:     @ @p.x; t/ @p.x; t/ D p.x; t/ C W ;  p.x; t/ C p @t @t @t    @W.x; t/ @p.x; t/ W : C W.x; t/ D  gradx p.x; t/ C W @t

@t

(11) (12)

We describe functions and parameters incoming in four filtration models.  is time of relaxation,  is penetrability coefficients, m is a porosity, t is time, ˇ is elasticity capacity coefficient of the layer, ˇ D ˇc C m0 ˇf , ˇc is compressibility coefficient of the porous environment, m0 is a fluid porosity in the unperturbed layer conditions, ˇf is compressibility coefficient of the fluid, is a fluid viscosity, is a  fluid density, D ˇ is piezoconductivity coefficient of the layer, .t/ is Heaviside function, .t/ D 1 for t > 0, .t/ D 1=2 for t D 0, .t/ D 0 for t < 0, W and p nonnegative constants relaxation times of filtration velocity and pressure, 0m D  ˇˇ , ˇ D m0 ˇf C ˇc p =m is dynamics coefficient of elasticity capacity, m is the relaxation time of porosity under the constant overfull of pressure, p is the relaxation time of pressure under the constant porosity, p0 is pressure in the unperturbed layer conditions, 0 is density in the unperturbed layer conditions. All parameters are nonnegative given numbers [1]. Mathematical problems for models I–IV. Initial conditions for all four models. First of all in all four models in bounded region of filtration  2 R3 with boundary @ and for t 2 Œ0; T and for pressure p.x; t/ we consider Eqs. (3), (5), (7), (9) and (11). Then we set initial conditions for them. For Eqs. (3) and (9): p.x; t/ D a.x/; while t D 0;

(13)

and for Eqs. (5), (7) and (11) besides condition (13) we give an additional condition @p.x; t/ D b.x/; while t D 0: @t Boundary conditions for all four models.

(14)

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K. Shakenov

Problem 1 (Dirichlet Problem) In bounded filtration region  2 R3 with boundary @ and for time t 2 Œ0; T, function p.x; t/ satisfies the boundary condition p.x; t/ D p1 .x; t/ for x 2 @  Œ0; T:

(15)

Problem 2 (Neumann Problem) In bounded filtration region  2 R3 with boundary @ and for time t 2 Œ0; T, function p.x; t/ satisfies the boundary condition @p.x; t/ D p2 .x; t/ for x 2 @  Œ0; T; @n

(16)

where n is an internal normal. Problem 3 (Mixed Problem) In bounded filtration region  2 R3 with boundary @ and for time t 2 Œ0; T, function p.x; t/ satisfies the boundary condition ˛1 p.x; t/ C ˇ1

@p.x; t/ D p3 .x; t/ for x 2 @  Œ0; T; @n

(17)

where n is an internal normal. An idea for solving by Monte Carlo methods. Initial-boundary problem with respect to pressure p.x; t/ is discretized only by variable t. For that, interval Œ0; T split into N equal steps of  D NT , tn D n, n D 1; 2; : : : ; N. As a result we get discrete boundary problem by time variable. Obtained, the problems for elliptic type PDEs (Helmholtz equation) solved by Monte Carlo methods.

2 Solution of the Initial Boundary Value Problems by Monte Carlo Methods We demonstrate a solution of the initial boundary value problems by Monte Carlo methods on the following model—Filtration in relaxationaly-compressed porous environment realized by the linear Darcy law, that is model III. For this model we have a mathematical problem:     @ @p.x; t/ 0 @p.x; t/  p.x; t/ C m D p.x; t/ C m ; @t @t @t

(18)

p.x; t/ D a.x/; while t D 0;

(19)

@p.x; t/ D b.x/; while t D 0; @t

(20)

p.x; t/ D p1 .x; t/ for x 2 @  Œ0; T:

(21)

Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods

241

2.1 Solution of the Dirichlet Problem (18)–(21) Let coefficients ; m ; 0m are while positive fixed values. Let us divide interval t 2 Œ0; T into N equal parts with length . So that tn D n  ; n D 0; 1; : : : ; N;  D NT ;  > 0; and we digitize only with respect to t using implicit scheme. In result taking into account 0m , we obtain Eq. (18) on time layer tnC1 pnC1 .x/  a1  pnC1 .x/ D f n .x/;

(22)

m , 2 C m 4.m0 ˇf m C ˇc p / , d1 D D  }

where f n .x/ D b1  pn .x/ C c1  pn1 .x/ C d1  pn1 .x/, c1 D

m0 ˇf . C 2m / C ˇc . C 2p / , b1 } m0 ˇf .2m  / C ˇc .2p  / , } D  .2 C m /  .ˇc C mˇf /. } The algorithm “Random walk on spheres“ of Monte Carlo methods. It is clear that a1 > 0, as parameters m0 ; ˇf ; ; m ; ˇc ; p ; are positive. Combining the initial condition with (22) we obtain a1 D

p0 .x/ D a.x/; x 2 ;

p1 .x/  p0 .x/ D b.x/; x 2 ; 

(23)

which are the difference analogues of the initial data (19) and (20) respectively. For this problem the boundary condition transformed to: pnC1 .x/ D pnC1 1 .x/; x 2 @:

(24)

We shall call the boundary @ (and @" ) satisfying the Dirichlet condition as absorbing boundary. It is known that the problem (22)–(24) (Dirichlet problem for the Helmholtz equation of a time layer tnC1 ), is solved with the help of “random walk on spheres“ algorithm of Monte Carlo methods. The constructed "–displaced estimation of the solution pnC1 .x/ with the help of “random walk on spheres“ algorithm has a uniformly bounded variance by " [2–7]. The algorithm “Random walk on balls“ of Monte Carlo methods. “Random walk on balls“ algorithm for solving a Dirichlet problem. This algorithm is similar to algorithm “random walk on spheres“. In algorithm “random walk on balls“ a “particle“ passes from the center of the ball to a random point inside the ball and including a bound of ball (sphere), that is the following state of Markov chain inside the ball and including a bound of ball. It can be proved that Markov’s chain converges in the same manner as for “random walk on spheres“ algorithm and for finite number of steps to the "-bound of @" . But it is obvious that convergence of Markov chain for “random walk on balls“ algorithm is slower than for “random walk on spheres“. For that reason, “random walk on balls“ algorithm is almost unusable for numerical modeling by Monte Carlo methods. The constructed "-displaced

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K. Shakenov

estimation of the solution pnC1 .x/ with the help of “random walk on balls“ algorithm has a uniformly bounded variance by " [5]. The algorithm “Random walk on lattices“ of Monte Carlo methods. At first we approximate the solution (22)–(24) with the help of finite difference method and construct Markov chain, its transition probabilities are defined with the help of coefficients and parameters of the difference problem (22)–(24). For this purpose we use the following approximation of the second derivative with respect to x, i.e. pnC1 .x C ei h/ C pnC1 .x  ei h/  2pnC1 .x/ pnC1 ; where h is step along x, xi xi .x/ D h2 2 ei is the unit vector along the axis xi . Obviously O.h / is a precision of the such approximation. Let’s denote approximation of a domain  by !h , and boundary @—by h . Now by time lowering superscripts n C 1; n; n  1 from (22), we obtain the following finite difference equation p.xi / D

1 1 h2  p.x C e h/ C  p.x  e h/   f .xi /: i i i i 2 C a1 h2 2 C a1 h2 2 C a1 h2

(25)

It’s obvious that 2 ! 1 for h ! 0;  !; m ! 0; 2 C a1 h2

(26)

where h is step along x,  is time step. That is realization of (26) correspond to convergence requirements of a difference schemes and relaxation process. Let’s 1 : As ˛.xi ; yi ; h; / > 0 and ˛ C ˛  1 on denote ˛.xi ; yi ; h; / D 2 C a1 h2 yi for 8xi , then ˛.xi ; yi ; h; / are transition probabilities of Markov chain. Here yi D xi ˙ ei h, ei is unit vector. Algorithm At first we play a coordinate axis with probability 1=3 for  2 R3 . Then the “particle“ moves (along the direction ei or Cei ) with identical probability ˛ from node xi into one of the neighboring node xi ˙ ei h. It is necessary to take into h2 account the “weight“ of node, it is proportional to  f .xi /. And so on until 2 C a1 h2 the “particle“ achieves the discrete boundary h . As soon as the “particle“ achieves the boundary h , boundary data p1 .xi / is add to a counter. Thus a random variable Nh h is defined along a discrete Markov chain with random length Nh . Then we average it on all trajectories, that is the estimation of the solution pnC1 .xi / in the node xi is M   1 P defined from pnC1 .xi /  Nh h i ; where M is trajectories amount of Markov M iD1 chain starting from the node xi [7–10]. Then we have the following Theorem 1 The Neumann–Ulam scheme is applicable to the finite difference problem for (22)– (24).

Numerical Modeling of the Linear Relaxational Filtration by Monte Carlo Methods

243

Proof Proof of the theorem follows from algorithm of the discrete solution of problem (22)–(24). The complete proof see in [10]. The theorem is proved. t u In this case variance of an estimation of the solution pnC1 .xi / will be bounded, it can be explicitly calculated [5, 8, 10]. Probability difference method. Let’s consider the finite difference problem (25) for a time layer˚n C 1 with a discrete boundary condition p.xi / D p1 .xi / xi 2 h :  Let’s denote by ih ; i D 0; 1; : : : value of transition chain. Let p1 .x/ is the arbitrary continuous function for˚x 2 h . Let Nh is a moment of the first way out of a discrete domain !h : Nh D min i W ih … !h . Combining (25) with a boundary condition we obtain p.x/ D Ex p.1h / C 4th ˛ f .x/; x 2 !h ;

p.x/ D p1 .x/; x 2 h

(27)

If Ex Nh < 1, then the problem (27) has a unique solution ph .x/ D Ex

h 1 n NX

o f .ih /  4tih C p1 .Nh h / :

(28)

iD0

n o Here 4tih D 4th .ih / is a process parameter. If f .x/ D 0, Px Nh < 1 D 1, then (27) has the unique solution n o ph .x/ D Ex p1 .Nh h /  IfNh