Topics in Contemporary Mathematical Analysis and Applications 0367532662, 9780367532666

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
Editor
Contributors
Chapter 1 Certain Banach-Space Operators Acting on Free Poisson Elements Induced by Orthogonal Projections
1.1 Introduction
1.2 Preliminaries
1.3 Some Banach *-Algebras Induced by Projections
1.4 Weighted-Semicircular Elements Induced by Q
1.5 Semicircular Elements Induced by Q
1.6 The Semicircular Filterization (L[sub(Q)], Ί)
1.7 Free Poisson Elements of L[sub(Q)]
1.7.1 Free Poisson Elements
1.7.2 Certain Free Poisson Elements Induced by S
1.7.3 Some Free Poisson Elements Induced by S U X
1.8 Free Weighted-Poisson Elements of L[sub(Q)]
1.8.1 Free Weighted-Poisson Elements
1.8.2 Free Weighted-Poisson Elements Induced by S U X
1.8.3 Free Weighted-Poisson Elements Induced by X
1.9 Shifts on Z and Integer-Shifts on L[sub(Q)]
1.9.1 (±)-Shifts on Z
1.9.2 Integer-Shifts on L[sub(Q)]
1.9.3 Free Probability on L[sub(Q)] Under the Group-Action of B
1.10 Banach-Space Operators on L[sub(Q)] Generated by B
1.10.1 Deformed Free Probability of L[sub(Q)] by A
1.10.2 Deformed Semicircular Laws on L[sub(Q)] by A
1.11 Deformed Free Poisson Distributions on L[sub(Q)] by A
References
Chapter 2 Linear Positive Operators Involving Orthogonal Polynomials
2.1 Operators Based on Orthogonal Polynomials
2.1.1 Notations
2.1.2 Definitions
2.1.3 Appell Polynomials
2.1.4 Boas-Buck-Type Polynomials
2.1.5 Charlier Polynomials
2.1.6 Approximation by Appell Polynomials
2.1.7 Approximation by Operators Including Generalized Appell Polynomials
2.1.8 Szász-Type Operators Involving Multiple Appell Polynomials
2.1.9 Kantorovich-Type Generalization of K[sub(n)] Operators
2.1.10 Kantorovich Variant of Szász Operators Based on Brenke-Type Polynomials
2.1.11 Operators Defined by Means of Boas-Buck-Type Polynomials
2.1.12 Operators Defined by Means of Charlier Polynomials
2.1.13 Operators Defined by Using q-Calculus
Acknowledgment
References
Chapter 3 Approximation by Kantorovich variant of l??Schurer Operators and Related Numerical Results
3.1 Introduction
3.2 Auxiliary Results
3.3 Approximation Behavior of λ-Schurer-Kantorovich Operators
3.4 Voronovskaja-type Approximation Theorems
3.5 Graphical and Numerical Results
3.6 Conclusion
References
Chapter 4 Characterizations of Rough Fractional-Type Integral Operators on Variable Exponent Vanishing Morrey-Type Spaces
4.1 Introduction
4.2 Preliminaries and Main Results
4.2.1 Variable Exponent Lebesgue Spaces L[sup(P(·))]
4.2.2 Variable Exponent Morrey Spaces L[sup(P(·))],λ[sup(·)]
4.2.3 Variable Exponent Vanishing Generalized Morrey Spaces
4.2.4 Variable Exponent-Generalized Campanato Spaces C[sub(Π)][sup(q(·),ɣ(·))]
4.3 Conclusion
Funding
References
Chapter 5 Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices
5.1 Introduction
5.2 Preliminaries
5.3 pτ-Continuous and pτ-Bounded Operators
5.4 upτ-Continuous Operators
5.5 The Compact-Like Operators
Bibliography
Chapter 6 On Indexed Product Summability of an Infinite Series
6.1 Introduction
6.1.1 Historical Background
6.1.2 Notations and Definitions
6.2 Known Results
6.3 Main Results
6.4 Proof of Main Results
6.5 Conclusion
References
Chapter 7 On Some Important Inequalities
7.1 Concepts of Affinity and Convexity
7.1.1 Affine and Convex Sets and Functions
7.1.2 Effect of Affine and Convex Combinations in R[sup(n)]
7.1.3 Coefficients of Affine and Convex Combinations
7.1.4 Support and Secant Hyperplanes
7.2 The Jensen Inequality
7.2.1 Discrete and Integral Forms of the Jensen Inequality
7.2.2 Generalizations of the Jensen Inequality
7.3 The Hermite-Hadamard Inequality
7.3.1 The Classic Form of the Hermite-Hadamard Inequality
7.3.2 Generalizations of the Hermite-Hadamard Inequality
7.4 The Rogers-Hölder Inequality
7.4.1 Integral and Discrete Forms of the Rogers-Hölder Inequality
7.4.2 Generalizations of the Rogers-Hölder Inequality
7.5 The Minkowski Inequality
7.5.1 Integral and Discrete Forms of the Minkowski Inequality
7.5.2 Generalizations of the Minkowski Inequality
Bibliography
Chapter 8 Refinements of Young’s Integral Inequality via Fundamental Inequalities and Mean Value Theorems for Derivatives
8.1 Young’s Integral Inequality and Several Refinements
8.1.1 Young’s Integral Inequality
8.1.2 Refinements of Young’s Integral Inequality via Lagrange’s Mean Value Theorem
8.1.3 Refinements of Young’s Integral Inequality via Hermite-Hadamard’s and Čebyšev’s Integral Inequalities
8.1.4 Refinements of Young’s Integral Inequality via Jensen’s Discrete and Integral Inequalities
8.1.5 Refinements of Young’s Integral Inequality via H¨older’s Integral Inequality
8.1.6 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Lagrange’s Type Remainder
8.1.7 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Cauchy’s Type Remainder and H¨older’s Integral Inequality
8.1.8 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Cauchy’s Type Remainder and Čebyšev’s Integral Inequality
8.1.9 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Cauchy’s Type Remainder and Jensen’s Inequalities
8.1.10 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Cauchy’s Type Remainder and Integral Inequalities of Hermite-Hadamard Type for the Product of Two Convex Functions
8.1.11 Three Examples Showing Refinements of Young’s Integral Inequality
8.1.11.1 First Example
8.1.11.2 Second Example
8.1.11.3 Third Example
8.2 New Refinements of Young’s Integral Inequality via Pólya’s Type Integral Inequalities
8.2.1 Refinements of Young’s Integral Inequality in Terms of Bounds of the First Derivative
8.2.2 Refinements of Young’s Integral Inequality in Terms of Bounds of the Second Derivative
8.2.3 Refinements of Young’s Integral Inequality in Terms of Bounds of Higher-Order Derivatives
8.2.4 Refinements of Young’s Integral Inequality in Terms of L[sup(p)]-Norms
8.2.5 Three Examples for New Refinements of Young’s Integral Inequalities
8.2.5.1 First Example
8.2.5.2 Second Example
8.2.5.3 Third Example
8.3 More Remarks
Acknowledgments
Bibliography
Chapter 9 On the Coefficient Estimates for New Subclasses of Biunivalent Functions Associated with Subordination and Fibonacci Numbers
9.1 The Definition and Elementary Properties of Univalent Functions
9.1.1 Integral Operators
9.2 Subclasses of Analytic and Univalent Functions
9.3 The Class Σ
9.4 Functions with Positive Real Part
9.4.1 Subordination
9.5 Bi-univalent Function Classes S[sub(t,Σ)][sup(μ)] and K[sub(t,Σ)][sup(μ)] (P̃)
9.6 Inequalities for the Taylor-Maclaurin Coefficients
9.7 Concluding Remarks and Observations
Acknowledgment
Bibliography
Chapter 10 Fixed Point of Multivalued Cyclic Contractions
10.1 Multivalued Mappings in Metric Spaces
10.2 Multivalued Cyclic F-Contractive Mappings
10.3 Fixed Point Results of Multivalued Cyclic F-Contractive Mappings
10.4 Stability of Fixed Point Sets of Cyclic F-Contractions
10.5 Multivalued Mappings under Cyclic Simulation Function
10.6 Fixed Point Theorems under Cyclic Simulation Function
10.7 Stability of Fixed Point Sets under Cyclic Simulation Function
Bibliography
Chapter 11 Significance and Relevances of Functional Equations in Various Fields
11.1 Introduction
11.2 Application of Functional Equation in Geometry
11.3 Application of Functional Equation in Financial Management
11.4 Application of Functional Equation in Information Theory
11.5 Application of Functional Equation in Wireless Sensor Networks
11.6 Application of Rational Functional Equation
11.6.1 Geometrical Interpretation of Equation (11.17)
11.6.2 An Application of Equation (11.17) to Resistances Connected in Parallel
11.7 Application of RQD and RQA Functional Equations
11.8 Application of Other Multiplicative Inverse Functional Equations
11.8.1 Multiplicative Inverse Second Power Difference and Adjoint Functional Equations
11.8.2 Multiplicative Inverse Third Power Functional Equation
11.8.3 Multiplicative Inverse Fourth Power Functional Equation
11.8.4 Multiplicative Inverse Quintic Functional Equation
11.8.5 Multiplicative Inverse Functional Equation Involving Two Variables
11.8.6 System of Multiplicative Inverse Functional Equations with Three Variables
11.9 Applications of Functional Equations in Other Fields
11.10 Open Problems
Bibliography
Chapter 12 Unified-Type Nondifferentiable Second-Order Symmetric Duality Results over Arbitrary Cones
12.1 Introduction
12.2 Literature Review
12.3 Preliminaries and Definitions
12.3.1 Definition
12.3.2 Definition
12.3.3 Definition
12.3.4 Definition
12.4 Nondifferentiable Second-Order Mixed-Type Symmetric Duality Model Over Arbitrary Cones
12.4.1 Remarks
12.5 Duality Theorems
12.6 Self-Duality
12.7 Conclusion
Bibliography
Index

Citation preview

Topics in Contemporary Mathematical Analysis and Applications

Mathematics and its Applications Modelling, Engineering, and Social Sciences Series Editor: Hemen Dutta Department of Mathematics, Gauhati University Tensor Calculus and Applications Simplified Tools and Techniques Bhaben Kalita Discrete Mathematical Structures A Succinct Foundation Beri Venkatachalapathy Senthil Kumar and Hemen Dutta Methods of Mathematical Modelling Fractional Differential Equations Edited by Harendra Singh, Devendra Kumar, and Dumitru Baleanu Mathematical Methods in Engineering and Applied Sciences Edited by Hemen Dutta Sequence Spaces Topics in Modern Summability Theory Mohammad Mursaleen and Feyzi Bas¸ar Fractional Calculus in Medical and Health Science Devendra Kumar and Jagdev Singh Topics in Contemporary Mathematical Analysis and Applications Hemen Dutta Sloshing in Upright Circular Containers Theory, Analytical Solutions, and Applications Alexander Timokha and Ihor Raynovskyy

ISSN (online): 2689-0224 ISSN (print): 2689-0232 For more information about this series, please visit: https://www.routledge.com/Mathematics-and-its-applications/book-series/MES

Topics in Contemporary Mathematical Analysis and Applications Edited by

Hemen Dutta

First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. ISBN: 978-0-367-53266-6 (hbk) ISBN: 978-1-003-08119-7 (ebk) Typeset in Times by codeMantra

Contents Preface......................................................................................................................vii Editor.........................................................................................................................xi Contributors ............................................................................................................xiii Chapter 1

Certain Banach-Space Operators Acting on Free Poisson Elements Induced by Orthogonal Projections ................................. 1 Ilwoo Cho

Chapter 2

Linear Positive Operators Involving Orthogonal Polynomials...... 49 P. N. Agrawal and Ruchi Chauhan

Chapter 3

Approximation by Kantorovich variant of λ −Schurer Operators and Related Numerical Results..................................... 77 ¨ er, Kamil Demirci, and Sevda Yıldız Faruk Ozg

Chapter 4

Characterizations of Rough Fractional-Type Integral Operators on Variable Exponent Vanishing Morrey-Type Spaces ................. 95 Ferit Gurb ¨ uz, ¨ Shenghu Ding, Huili Han, and Pinhong Long

Chapter 5

Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices.................................................................. 125 Abdullah Aydın

Chapter 6

On Indexed Product Summability of an Infinite Series............... 143 B. P. Padhy and P. Baliarsingh

Chapter 7

On Some Important Inequalities.................................................. 165 Zlatko Pavic´

Chapter 8

Refinements of Young’s Integral Inequality via Fundamental Inequalities and Mean Value Theorems for Derivatives.............. 193 Feng Qi, Wen-Hui Li, Guo-Sheng Wu, and Bai-Ni Guo

v

vi

Chapter 9

Contents

On the Coefficient Estimates for New Subclasses of Bi-univalent Functions Associated with Subordination and Fibonacci Numbers...................................................................... 229 S¸ahsene Altınkaya

Chapter 10

Fixed Point of Multivalued Cyclic Contractions ......................... 249 Talat Nazir and Mujahid Abbas

Chapter 11

Significance and Relevances of Functional Equations in Various Fields .............................................................................. 281 B. V. Senthil Kumar and Hemen Dutta

Chapter 12

Unified-Type Nondifferentiable Second-Order Symmetric Duality Results over Arbitrary Cones ......................................... 299 Ramu Dubey and Vishnu Narayan Mishra

Index...................................................................................................................... 321

Preface This book covers certain aspects of contemporary mathematical analysis and its applications. It focuses on enriching the understanding of several methods, problems, and applications in the area of mathematical analysis having contemporary research and study significances. Each chapter of this book aims to offer readers the understanding of discussed research problems by presenting related developments in reasonable details. This book is expected to be a valuable resource for graduate students, researchers, educators, and teachers interested in contemporary mathematical analysis. There are 12 chapters in this book, and they are organized as follows. Chapter “Certain Banach-Space Operators Acting on Free Poisson Elements Induced by Orthogonal Projections” investigates the free (weighted-) Poisson elements in the Banach ∗-probability space generated by integer-many mutually free, semicircular elements, induced by mutually orthogonal integer-many projections in a fixed C∗-probability space. It constructs corresponding ∗-homomorphisms acting on the Banach *-probability space in order to study how they affect the free (weighted-) Poisson distributions and the free distributions of free (weighted-) Poisson elements. It also constructs certain Banach-space operators generated by the *-homomorphisms to examine how such operators deform free probability and to characterize how the free (weighted-) Poisson distributions are deformed by the action of those operators. Chapter “Linear positive operators involving orthogonal polynomials” aims to survey extensively on the available research work regarding approximations by the linear positive operators defined by using orthogonal polynomials. In the literature, several sequences of linear positive operators involving orthogonal polynomials have been defined, and their Durrmeyer and Kantorovich-type variants have been investigated. In recent years, there has been a significant increase in activities in the approximation of continuous functions on [0, ∞) by the linear positive operators based on orthogonal polynomials. Chapter “Approximation by Kantorovich variant of λ -Schurer operators and related numerical results” aims to construct Kantorovich variant of λ -Schurer operators using certain polynomials in order to study Voronovskaja-type theorems and obtain a global approximation formula for such operators. It presents a local direct estimate of the rate of convergence by Lipschitz-type function involving two parameters. The proposed operators reduce to the classical Schurer-Kantorovich operators as well as to the classical Bernstein-Kantorovich operators and λ -BernsteinKantorovich operators under certain special cases. Finally, it provides tables and graphs based on numerical experiments in order to justify the efficiency of the results in comparison with other available results. Chapter “Characterizations of rough fractional-type integral operators on variable exponent vanishing Morrey-type spaces” aims to apply relevant properties of

vii

viii

Preface

variable exponent to study on Adams and Spanne-type estimates for a class of fractional integral operators with variable orders on variable exponent vanishing generalized Morrey spaces. Finally, it claims to introduce the variable exponent-generalized Campanato spaces and then obtain the boundedness of the commutators of these operators on such spaces. Chapter “Compact-like operators in vector lattices normed by locally solid lattices” investigates continuous, bounded, and compact operators on locally solid Riesz spaces with respect to a concept of convergence, which was introduced and investigated on locally solid Riesz spaces. This chapter describes the generalization of several known classes of operators on Riesz spaces and Banach lattices such as norm continuous, order continuous, order bounded, bounded, and compact operators. It also presents the relations among such operators and examines the basic properties of the operators. Chapter “On indexed product summability of an infinite series” presents the find¨ ings based on the indexed product summability of an infinite series by using Norlund mean. In summability theory, indexed summability and indexed product summability of an infinite series have significant importance and relevance. Chapter “On some important inequalities” aims to present some useful math¨ ematical inequalities such as the Jensen, Hermite-Hadamard, Rogers-Holder , and Minkowski inequalities. The aforementioned four inequalities have been presented in discrete and integral forms, and their generalizations have also been discussed. The Jensen and Hermite-Hadamard inequalities have been considered in more detail. An expansion of the initial integral form of Jensen’s inequality is promoted for convex ¨ functions of several variables. The Rogers-Holder and Minkowski inequalities have been derived from the integral form of Jensen’s inequality for convex functions of several variables. The Minkowski inequality is realized independently of the Rogers¨ Holder inequality. Chapter “Refinements of Young integral inequality via fundamental inequalities and mean value theorems for derivatives” reviews several refinements of Young’s integral inequality via several mean value theorems, such as Lagrange’s and Taylor’s mean value theorems of Lagrange’s and Cauchy’s type remainders, and via ˇ ˇ v’s integral inequality, Hermiteseveral fundamental inequalities, such as Ceby se ¨ Hadamard’s type integral inequalities, Holder’ s integral inequality, and Jensen’s discrete and integral inequalities, in terms of higher-order derivatives and their norms. It also surveys several applications of several refinements of Young’s integral in´ equality and further refines Young’s integral inequality via Polya’ s type integral inequalities. Chapter “On the coefficient estimates for new subclasses of bi-univalent functions associated with subordination and Fibonacci numbers” investigates new subclasses of the bi-univalent function class associated with subordination and Fibonacci numbers in the open unit disc. It obtains estimates on the first two Taylor-Maclaurin coefficients, derives Fekete-Szego¨ inequalities for functions belonging to the newly defined classes, and also discusses relevant connections to some of the earlier known results.

Preface

ix

Chapter “Fixed point problems of multivalued mappings” aims to construct the fixed-point results of multivalued mappings subject to the satisfaction of generalized cyclic F-contractive conditions in metric spaces. It obtains the fixed points of multivalued mappings under cyclic simulation functions, and then establishes the stability of fixed-point sets of generalized cyclic F-contractive mappings and mappings under cyclic simulation function. The results have been obtained without using any form of continuity of multivalued mappings involved. Further, suitable examples have also been given to support the results. Chapter “Significance and relevance of functional equations in various fields” aims to impart the significant role of functional equations in various fields. The study of functional equations is a growing, and an important, area in mathematics. It covers many other areas of mathematics, and recently, their role in science and engineering has found to be very attractive. It also portrays a few applications of functional equations in geometry, finance, information theory, wireless sensor networks and electric circuits with parallel resistances, physics, and electromagnetism. Chapter “Unified type nondifferentiable second-order symmetric duality results over arbitrary cones” claims to formulate a new mixed-type second-order nondifferentiable symmetric duality in scalar-objective programming over arbitrary cones. It first discusses mixed-type primal-dual results based on available literature and constructs some numerical examples to justify the presented definitions. It derives the combined result with one model over arbitrary cones, and the duality theorems have been derived for some problems under bonvexity and pseudobonvexity with respect to η assumptions over arbitrary cones. The editor sincerely acknowledges the cooperation and patience of contributors during the entire process of handling their works submitted for this book. Reviewers deserve deep gratitude for selflessly offering their help for successfully bringing out this book. The editor also thankfully acknowledges the support of editorial staff at Taylor & Francis. Encouragement from several colleagues and friends is the primary source of inspiration to work for such book projects, and the editor gratefully acknowledges their mental support. Hemen Dutta Guwahati, India 26th April, 2020

Editor Hemen Dutta is a regular faculty member in Mathematics at Gauhati University, India. He did his Master of Science (MSc) in Mathematics, Post Graduate Diploma in Computer Application (PGDCA), Master of Philosophy (MPhil) in Mathematics, and Doctor of Philosophy (PhD) in Mathematics, respectively. His topics of primary research interest are in the area of mathematical analysis. He has contributed in authoring around 150 items so far as research papers for journals, chapters for books, and proceedings papers. He has already published 20 books as textbooks, reference books, edited books, and proceedings of conferences. He has organized many academic events for researchers and academicians, and also associated with several other such activities in different capacities. He has contributed several general and popular articles in newspaper, science magazines, science portals, popular books, etc. He has offered his service as a resource person/invited speaker in workshops, conferences, teachers’ training program, faculty development programs, etc. He has also visited foreign institutions on invitation for research collaboration and delivering talks.

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Contributors Mujahid Abbas Department of Mathematics Government College University Lahore Lahore, Pakistan

Kamil Demirci Department of Mathematics Sinop University Sinop, Turkey

P. N. Agrawal Department of Mathematics IIT Roorkee Roorkee, India

Ramu Dubey Department of Mathematics J.C. Bose University of Science and Technology Faridabad, India

S¸ahsene Altınkaya Department of Mathematics Bursa Uludag University Bursa, Turkey Abdullah Aydın Department of Mathematics Mus¸ Alparslan University Mus¸, Turkey P. Baliarsingh Department of Mathematics Gangadhar Meher University Sambalpur, Odisha, India Ruchi Chauhan Department of Mathematics IIT Roorkee Roorkee, India Ilwoo Cho Department of Mathematics and Statistics Saint Ambrose University Davenport, Iowa, U.S.A.

Hemen Dutta Department of Mathematics Gauhati University Guwahati, India Shenghu Ding School of Mathematics and Statistics Ningxia University Yinchuan, P. R. China ¨ uz ¨ Ferit Gurb Faculty of Education, Department of Mathematics Education Hakkari University Hakkari, Turkey Bai-Ni Guo School of Mathematics and Informatics Henan Polytechnic University Jiaozuo, China Huili Han School of Mathematics and Statistics Ningxia University Yinchuan, P. R. China

xiii

xiv

Wen-Hui Li Department of Fundamental Courses Zhengzhou University of Science and Technology Zhengzhou, China

Contributors

Zlatko Pavic´ Department of Mathematics Mechanical Engineering Faculty in Slavonski Brod University of Osijek Osijek, Croatia

Pinhong Long School of Mathematics and Statistics Ningxia University Yinchuan, P. R. China

Feng Qi Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China Vishnu Narayan Mishra College of Mathematics and Physics Department of Mathematics Inner Mongolia University for NationalIndira Gandhi National Tribal University ities Tongliao, China Lalpur, Amarkantak, India and School of Mathematical Sciences Talat Nazir Tianjin Polytechnic University Department of Mathematical Sciences Tianjin, China University of South Africa B. V. Senthil Kumar Johannesburg, South Africa Department of Information Technology and Department of Mathematics Nizwa College of Technology COMSATS University Islamabad Nizwa, Oman Abbottabad, Pakistan Guo-Sheng Wu ¨ Faruk Ozger School of Computer Science Department of Engineering Sciences Sichuan Technology and Business ˙Izmir Katip University ˆ C¸elebi University ˙Izmir, Turkey Chengdu, China B. P. Padhy School of Applied Sciences, Department of Mathematics KIIT Deemed to be University Bhubaneswar, India

Sevda Yıldız Department of Mathematics Sinop University Sinop, Turkey

Banach-Space 1 Certain Operators Acting on Free Poisson Elements Induced by Orthogonal Projections Ilwoo Cho Saint Ambrose University

CONTENTS 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction ....................................................................................................... 2 Preliminaries...................................................................................................... 3 Some Banach *-Algebras Induced by Projections ............................................ 3 Weighted-Semicircular Elements Induced by Q ............................................... 6 Semicircular Elements Induced by Q .............................................................. 11 The Semicircular Filterization (LQ , τ) ............................................................ 12 Free Poisson Elements of LQ .......................................................................... 15 1.7.1 Free Poisson Elements ......................................................................... 15 1.7.2 Certain Free Poisson Elements Induced by S ...................................... 19 1.7.3 Some Free Poisson Elements Induced by S ∪ X ................................. 21 1.8 Free Weighted-Poisson Elements of LQ .......................................................... 23 1.8.1 Free Weighted-Poisson Elements ........................................................ 23 1.8.2 Free Weighted-Poisson Elements Induced by S ∪ X ........................... 26 1.8.3 Free Weighted-Poisson Elements Induced by X ................................. 27 1.9 Shifts on Z and Integer-Shifts on LQ .............................................................. 29 1.9.1 (±)-Shifts on Z .................................................................................... 29 1.9.2 Integer-Shifts on LQ ............................................................................. 29 1.9.3 Free Probability on LQ Under the Group-Action of B ....................... 34 1.10 Banach-Space Operators on LQ Generated by B ........................................... 35 1.10.1 Deformed Free Probability of LQ by A ............................................... 37 1.10.2 Deformed Semicircular Laws on LQ by A .......................................... 40 1.11 Deformed Free Poisson Distributions on LQ by A.......................................... 43 References................................................................................................................ 46

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1.1

Topics in Contemporary Mathematical Analysis and Applications

INTRODUCTION

Let (A, ψ) be a C∗ -probability space containing mutually orthogonal |Z|-many projections {q j } j∈Z . Such projections q j ’s induce the corresponding free weightedsemicircular family {u j } j∈Z in a certain Banach ∗-probability space (LQ , τ), and under additional conditions, this free family {u j } j∈Z generates the free semicircular family {U j } j∈Z in (LQ , τ); see [6]. This free semicircular family {U j } j∈Z generates the Banach ∗-probabilistic substructure (LQ , τ) of (LQ , τ). The free probability on LQ was studied in [7,8,10,11]. There are ways to construct semicircular elements in earlier works, e.g., see [1,2,17,18,19,20,21,29,30]. However, our construction is different from those of them. Ours is motivated by the construction of the weighted-semicircular elements of [5,9,11]. In [5,9,11], (weighted-)semicircular elements are naturally constructed from analyses on the p-adic number fields Q p , for primes p (and their globalization, the analysis on the finite Adelic ring AQ ), from |Z|-many orthogonal projections induced by |Z|-many measurable p-adic characteristic functions of spheres in the p-adic number fields Q p (e.g., [26] and [27]). Motivated by these, we mimic the construction of [5,9,11], in the cases where a C∗ -probability space (A, ψ) has mutually orthogonal |Z|-many projections in [6] (also, see below). For more about details and applications, see [7,8,10] and [11]. The main purposes of this paper are (i) to construct free (weighted-)Poisson elements of the Banach ∗-probability space (LQ , τ), (ii) to consider the ∗-homomorphisms {β±n }n∈N0 acting on LQ induced by certain order-preserving bijective functions h± on the set Z of all integers, (iii) to study how these ∗-homomorphisms of (ii) affect the free probability on LQ , (iv) to consider how the free (weighted-)Poisson distributions, which are the free distributions of our free (weighted-)Poisson elements of (LQ , τ), are affected by the action of {β±n }n∈N0 , based on (iii), where N0 = N ∪ {0}, and (v) to construct a certain Banach ∗-algebra B generated by {β±n }n∈N0 , and to investigate how the operators of B distort the free probability on (LQ , τ). Our main results show that (I) typical types of free (weighted-)Poisson elements generated by (weighted-)semicircular elements are characterized by the construction of LQ , and the universality of the semicircular law; (II) our free weighted-Poisson distributions are completely characterized by free Poisson distributions; (III) the ∗-homomorphisms of (ii) are free-homomorphisms, i.e., ∗-homomorphisms of (ii) preserve the free probability on LQ , and hence, the free (weighted-)Poisson distributions of (i) are preserved by the action of these ∗-homomorphisms on LQ ; and (IV) the distortion of free probability on LQ , under the action of the Banach algebra B of (v), is characterized, and it shows how our free (weighted-)Poisson distributions are deformed by the action of B on LQ . In earlier works, free Poisson elements have been studied. Even though free Poisson elements and free Poisson distributions are well known (e.g., see [15,16] and [17]), our free “weighted-Poisson” elements are newly introduced here. Moreover,

Certain Banach-Space Operators

3

their free distributions, the free weighted-Poisson distributions, are completely characterized with respect to certain free Poisson distributions. In the long run, we show that free weighted-Poisson elements are factorized by certain operators and free Poisson elements. More interestingly, distorted free (weighted-)Poisson distributions are studied under the action of B on LQ .

1.2

PRELIMINARIES

Free probability is the noncommutative operator-algebraic version of classical measure theory and statistical analysis. The operator-algebraic freeness plays like the classical functional independence by replacing measures on sets to linear functionals on noncommutative algebras (e.g., [17,22,23,28] and [30]). It is one of the main subjects not only in pure mathematics (e.g., [3,4,20,21,24,25] and [29]), but also in related fields (e.g., [5] through [12]). In particular, here, the combinatorial free probability is used (e.g., [17,22] and [23]). In the text, without definitions and backgrounds, free moments and free cumulants of operators will be computed for analyzing free distributions of the operators. Also, we use free product ∗-probability spaces, without a detailed introduction. By the central limit theorem(s), studying semicircular elements whose free distributions are the semicircular law, is one of the main topics in free probability theory (e.g., [1,2,18,19,20,21] and [29]). As an application, free Poisson elements whose free distributions are the free Poisson distributions, have been considered (e.g., [15,16] and [17]).

1.3

SOME BANACH *-ALGEBRAS INDUCED BY PROJECTIONS

In this section, we establish backgrounds of our proceeding works. Let (B, ϕ) be a topological ∗-probability space (a C∗ -probability space, or a W ∗ -probability space, or a Banach ∗-probability space, etc.), where B is a topological ∗-algebra (a C∗ algebra, resp., a W ∗ -algebra, resp., a Banach ∗-algebra, etc.), and ϕ is a bounded linear functional on B. An operator a of B is said to be a free random variable whenever it is regarded as an element of (B, ϕ). As usual in operator theory, a free random variable a is said to be self-adjoint in (B, ϕ) if a∗ = a in B, where a∗ is the adjoint of a (e.g., [14]). Definition 1.1. A self-adjoint free random variable a is said to be weightedsemicircular in (B, ϕ) with its weight t0 ∈ C× = C \ {0} (or, in short, t0 semicircular), if a satisfies the free-cumulant computations,  k2 (a, a) = t0 if n = 2 kn (a, ..., a) = (1.1) 0 otherwise, for all n ∈ N, where k• (...) is the free cumulant on B in terms of ϕ under the Mobius ¨ inversion of [17,22,23]. If t0 = 1 in (1.1), the 1-semicircular element a is said to be semicircular in (B, ϕ), i.e., a is semicircular in (B, ϕ) if a satisfies

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 kn (a, ..., a) =

1 0

if n = 2 otherwise,

(1.2)

for all n ∈ N. ¨ By the Mobius inversion of [17,22] and [23], one can characterize the weightedsemicircularity (1.1) as follows: A self-adjoint operator a is t0 -semicircular in (B, ϕ) if and only if  n  ϕ(an ) = ωn t02 c n2 , (1.3) where de f

ωn =



1 0

if n is even if n is odd,

for all n ∈ N, and ck are the k-th Catalan numbers,     de f 1 2k (2k)! 1 ck = k+1 = k+1 k!(2k−k)! = k

(2k)! k!(k+1)! ,

for all k ∈ N0 . Similarly, a self-adjoint free random variable a is semicircular in (B, ϕ) if and only if a is 1-semicircular in (B, ϕ), and if and only if ϕ (an ) = ωn c n2 ,

(1.4)

by (1.2) and (1.3), for all n ∈ N, where ωn are in the sense of (1.3). So we use the t0 -semicircularity (1.1) (or, the semicircularity (1.2)) and its characterization (1.3) (resp., (1.4)) alternatively from below. If a is a self-adjoint free random variable in (B, ϕ), then the sequences consisting of the free moments (ϕ(an ))∞ n=1 , and the free cumulants (kn (a, ..., a))∞ n=1 provide equivalent free-distributional data of a in (B, ϕ), characterizing the free distribution of a (e.g., [17,22] and [23]). In the rest of this paper, we fix a C∗ -probability space (A, ψ) and assume that there are |Z|-many projections {q j } j∈Z in the C∗ -algebra A, i.e., the operators q j satisfy q∗j = q j = q2j in A, for all j ∈ Z (e.g., [14]). Assume further that these projections {q j } j∈Z are mutually orthogonal in A, in the sense that: qi q j = δi, j q j in A, for all i, j ∈ Z, where δ is the Kronecker delta. Now, we fix the family

(1.5)

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Certain Banach-Space Operators

Q = {q j } j∈Z

(1.6)

of mutually orthogonal projections (1.5) of A. Remark 1.1. One can have such a C∗ -algebraic structure A containing a family Q of (1.6), naturally or artificially. Clearly, in the settings of [5,9,12], one can naturally take such structures. Suppose there is a C∗ -algebra A0 containing a family QN = {q1 , ..., qN } of mutually orthogonal N-many projections q1 , ..., qN , for N ∈ N∞ = N ∪ {∞}. Then, under a suitable direct product, or tensor product, or free product of copies of A0 under product topology, one can construct a C∗ -algebra A containing a family Q with |Z|many mutually orthogonal projections, containing QN under unitarily equivalence. For example, see [10]. Let Q be the C∗ -subalgebra of A generated by the family Q of (1.6), de f

Q = C∗ (Q) ⊆ A.

(1.7)

Proposition 1.1. Let Q be a C∗ -subalgebra (1.7) of a fixed C∗ -algebra A. Then ∗-iso

∗-iso

Q = ⊕ (C · q j ) = C⊕|Z| , in (A, ψ) .

(1.8)

j∈Z

Proof. The proof of (1.8) is done by the orthogonality (1.5) of Q in A. Define now linear functionals ψ j on the C∗ -algebra Q of (1.7) by ψ j (qi ) = δi j ψ(q j ), for all i ∈ Z,

(1.9)

for all j ∈ Z, where ψ is the linear functional of the fixed C∗ -probability space (A, ψ). The linear functionals {ψ j } j∈Z of (1.9) are well defined on Q by (1.8). Assumption Let (A, ψ) be a fixed C∗ -probability space, and let Q be the ∗ C -subalgebra (1.7) of A. In the rest of this paper, we assume that ψ(q j ) ∈ C× , for all j ∈ Z.  Definition 1.2. The C∗ -probability spaces (Q, ψ j ) are called the j-th filters of Q in a given C∗ -probability space (A, ψ), where Q is in the sense of (1.7) and ψ j are the linear functionals of (1.9), for all j ∈ Z. Now, let’s define the bounded linear transformations c and a acting on the C∗ algebra Q, by linear morphisms satisfying c (q j ) = q j+1 , and a (q j ) = q j−1 ,

(1.10)

for all j ∈ Z. Then, c and a are the well-defined bounded linear operators “on Q.” One can understand that they are Banach-space operators in the operator space B(Q), consisting of all bounded linear transformations acting on Q, by regarding Q as a Banach space equipped with its C∗ -norm (e.g., [13]).

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Definition 1.3. We call these Banach-space operators c and a of (1.10) the creation and, respectively, the annihilation on Q. Define now a new Banach-space operator l on Q by l = c + a ∈ B(Q).

(1.11)

Definition 1.4. The Banach-space operator l ∈ B(Q) of (1.11) is called the radial operator on Q. Note that if l is in the sense of (1.11), then the powers ln are also contained in B(Q), for all n ∈ N0 , with axiomatization: l0 = 1Q , the identity operator of B(Q). Now, define a closed subspace L of the operator space B(Q) by de f

k.k

L = C[{l}]

k.k

= spanC {ln : n ∈ N0 } ,

(1.12)

generated by the radial operator l, where k.k is the operator norm on B(Q), defined to be kT k = sup{kT qkQ : kqkQ = 1}, k.k

for all T ∈ B(Q), where k.kQ is the C∗ -norm on Q, and X are the operator-norm closures of subsets X of the operator space B(Q) (e.g., [13]). It is not difficult to check that this subspace L forms an algebra in the vector space B(Q), by (1.12). So, it forms a Banach algebra in the topological vector space B(Q). On this Banach algebra L of (1.12), define a unary operation (∗) by ∞ n ∗ n (∑∞ n=0 tn l ) = ∑n=0 tn l in L,

(1.13)

where z are the conjugates of z ∈ C. Then, this operation (1.13) is a well-defined adjoint on the Banach algebra L (e.g., [6] and [12]), and hence, every element of L is adjointable in B(Q) in the sense of [13]. So the algebra L forms a Banach ∗-algebra in B(Q) with (1.13). Now, let L be the Banach ∗-algebra (1.12). Define the tensor product Banach ∗-algebra LQ by LQ = L ⊗C Q,

(1.14)

where ⊗C is the tensor product of Banach ∗-algebras. Definition 1.5. We call the Banach ∗-algebra LQ of (1.14) the radial projection (Banach ∗-)algebra on Q.

1.4

WEIGHTED-SEMICIRCULAR ELEMENTS INDUCED BY Q

We here construct the weighted-semicircular elements induced by the family Q of mutually orthogonal projections in a fixed C∗ -probability space (A, ψ), inducing the

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Certain Banach-Space Operators

radial projection algebra LQ of (1.14). Let (Q, ψ j ) be the j-th filters of Q in (A, ψ), where ψ j are the linear functionals (1.9), for all j ∈ Z. Remark that if u j are the operators of LQ , de f

u j = l ⊗ q j ∈ LQ , for all j ∈ Z,

(1.15)

then unj = (l ⊗ q j )n = ln ⊗ q j , for all n ∈ N, with axiomatization: axiom

u0j = l0 ⊗ q j = 1Q ⊗ q j , for all n ∈ N0 , for j ∈ Z. That is, the operators {u j } j∈Z of (1.15) generate LQ , by (1.8) and (1.12). One can construct a linear functional ϕ j on LQ by a linear morphism satisfying that de f

ϕ j (uni ) = ϕ j (ln ⊗ qi ) = ψ j (ln (qi )),

(1.16)

for all n ∈ N0 , for all i, j ∈ Z. These linear functionals {ϕ j } j∈Z of (1.16) are well defined by (1.8), (1.12), and (1.14). Definition 1.6. We call the Banach ∗-probability spaces (LQ , ϕ j ) , f or all j ∈ Z,

(1.17)

the j-th (Banach-∗-probability) spaces on Q. Observe that if c and a are the creation, and, respectively, the annihilation, then ca = 1Q = ac on Q. More generally, one has cn an = (ca)n = 1Q = (ac)n = an cn , ∀n ∈ N, and hence,

(1.18) cn1 an2 = an2 cn1 , ∀n1 , n2 ∈ N.

Thus, one obtains that ln

n

= (c + a) =

for all n ∈ N, by (1.18), where   n = k

∑nk=0



n! k!(n−k)! ,

n k



ck an−k ,

∀k ≤ n ∈ N0 .

(1.19)

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Note that, for any n ∈ N, l2n−1 = ∑2n−1 k=0



2n − 1 k



ck an−k ,

by (1.19). So formula (1.20) does not contain 1Q -terms by (1.18). Note also that for any N, one has   n ∈  2n 2n 2n 2n k n −k l = ∑k=0 ca = cn an + [Rest terms], k n   2n -many 1Q -terms by (1.18) and (1.21). by (1.19). So l2n contains n

(1.20)

(1.21)

Proposition 1.2. Let l be the radial operator on Q. Then for any n ∈ N, l2n−1 does not contain 1Q -terms in L, l2n contains



2n n

(1.22)

 · 1Q in L.

(1.23)

Proof. Statements (1.22) and (1.23) are proven by (1.20), respectively, by (1.21).

Remark that one has  
1 ϕ j u2n− = ψ j l2n−1 (q j ) = 0, j

(1.24)

for all n ∈ N, by (1.9) and (1.22). Similarly, we have     
2n 2n (q ) = ψ ϕ j u2n = ψ l q + [Rest terms](q ) j j j j j j n by (1.21)     2n 2n = ψ j (q j ) = ψ (q j ) , n n by (1.9) and (1.23). That is,     2n 2n ϕj uj = ψ (q j ) , n

(1.25)

for all n ∈ N. Proposition 1.3. Fix j ∈ Z, and let uk = l ⊗ qk be the k-th generating operators of the j-th space (LQ , ϕ j ), for all k ∈ Z. Then, ϕ j (unk ) = δ j,k ωn

 n 2

  + 1 ψ (q j ) c 2n ,

(1.26)

where ωn are in the sense of (1.3) for all n ∈ N, and ck are the k-th Catalan numbers for all k ∈ N0 .

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Certain Banach-Space Operators

Proof. First, take the j-th generating operator u j in (LQ , ϕ j ) . By (1.24) and (1.25), one can get that:   ϕ j u2n−1 = 0, j and

    2n ϕ j u2n = ψ (q j ) = j n

n+1 n+1






2n n

 ψ (q j )

= ((n + 1)ψ (q j )) cn , for all n ∈ N, where cn are the n-th Catalan numbers. Assume now that k = 6 j in Z. Then, by (1.9), (1.16), and (1.23),
ϕ j unk = 0, for all n ∈ N. Therefore, formula (1.26) holds. Motivated by (1.26), we define a linear morphism, E j,Q : LQ → LQ by a bounded linear transformation satisfying

de f E j,Q (uin ) =

   

ψ (q j )

  

0LQ , the zero operator of LQ

n−1

[ 2n ]+1

(

)

unj

if i = j (1.27) otherwise,

for all n ∈ N, i, j ∈ Z, where [ 2n ] mean the minimal integers greater than or equal to n 2 , e.g., [ 32 ] = 2 = [ 42 ]. The linear transformations E j,Q of (1.27) are the well-defined bounded linear transformations on LQ , because of the cyclicity (1.12) of the tensor factor L of LQ , and the structure theorem (1.8) of the other tensor factor Q of LQ , for all j ∈ Z. Define now the new linear functionals τ j on LQ by de f

τ j = ϕ j ◦ E j,Q on LQ , for all j ∈ Z,

(1.28)

where ϕ j are in the sense of (1.16), and E j,Q are in the sense of (1.27). Definition 1.7. The Banach ∗-probability spaces denote

LQ ( j) = (LQ , τ j )

(1.29)

are called the j-th filtered (Banach-∗-probability) spaces of LQ , where τ j are the linear functionals (1.28) on the radial projection algebra LQ , for all j ∈ Z.

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On the j-th filtered space LQ ( j) of (1.29), one can get that      τ j unj = ϕ j E j,Q unj  = ϕj

  ψ q n−1   ( j) unj = [ n ]+ ϕ un ([ n2 ]+1) ( 2 1) j j n−1

ψ (q j )

n−1

=

ψ (q j )

[ 2n ]+1

(

)

ωn

n 2



+ 1 ψ (q j ) c 2n ,

by (1.26), i.e.,   τ j unj = ωn ψ(q j )n c n2 ,

(1.30)

for all n ∈ N, for j ∈ Z, where ωn are in the sense of (1.3). Lemma 1.1. Let LQ ( j) = (LQ , τ j ) be the j-th filtered space of LQ , for j ∈ Z. Then   τ j (uni ) = δ j,i ωn ψ(q j )n c n2 , (1.31) for all n ∈ N, for all i ∈ Z. Proof. If i = j in Z, then formula (1.31) holds by (1.30), for all n ∈ N. If i 6= j in Z, then, by (1.16) and (1.27), τ j (uni ) = 0, for all n ∈ N. Therefore, the free-distributional data (1.31) holds true for all i ∈ Z. The following theorem is proven by the aforementioned free-distributional data (1.31). Theorem 1.1. Let LQ ( j) be the j-th filtered space (LQ , τ j ) of LQ for j ∈ Z. Then, the “ j-th” generating operator u j is ψ (q j )2 -semicircular in LQ ( j). Meanwhile, for all i 6= j ∈ Z, the i-th generating operators ui of LQ have the zero free distribution. Proof. First of all, the generating operators ui are self-adjoint in LQ , for all i ∈ Z, since u∗i = (l ⊗ qi )∗ = l ⊗ qi = ui in LQ , for all i ∈ Z, by (1.13). Let’s fix j ∈ Z, and let u j = l ⊗ q j be the j-th generating operator (1.15) of LQ ( j). Then, by (1.31), we have that    n 2 τ j unj = ωn ψ (q j )2 c n2 ,

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Certain Banach-Space Operators

for all n ∈ N, where ck are the k-th Catalan numbers, for all k ∈ N0 . Therefore, this self-adjoint element u j is ψ(q j )2 -semicircular in LQ ( j), by (1.3). Consider now the i-th generating operators ui of LQ ( j), for any i 6= j in Z. Since ui are self-adjoint in LQ , the free distributions of ui are completely characterized by the free-moment sequences, (τ j (uni ))∞ n=1 = (0, 0, 0, ...) , the zero sequence, by (1.31). So the free distributions of ui ∈ LQ ( j) are the zero free distribution, whenever i 6= j in Z. The above theorem characterizes the free-probabilistic information of the generators {ui }i∈Z in the j-th filtered space LQ ( j), for j ∈ Z. ¨ Note that, by the Mobius inversion of [17], if ui are the i-th generating operators of the j-th filtered space LQ ( j), then  δ j,i ψ (q j )2 if n = 2 j kn (ui , ..., ui ) = (1.32) 0 otherwise, for all n ∈ N, and i ∈ Z, by (1.31), where k•j (...) is the free cumulant on LQ with respect to the linear functional τ j , for j ∈ Z.

1.5

SEMICIRCULAR ELEMENTS INDUCED BY Q

As in Section 1.4, let LQ ( j) be the j-th filtered space (1.29) of Q for j ∈ Z. Then, the j-th generating operator u j is ψ(q j )2 -semicircular in LQ ( j), satisfying that   τ j unj = ωn ψ(q j )n c n2 , equivalently,

(1.33) knj (u j , ..., u j ) =



ψ(q j )2 0

if n = 2 otherwise,

for all n ∈ N, by (1.31) and (1.32). By the weighted-semicircularity (1.33), one may/can obtain the following semicircular element U j of LQ ( j) (under an additional condition), de f

Uj =

1 ψ (q j )

u j ∈ LQ ( j),

(1.34)

for j ∈ Z. Recall that we assumed ψ(qk ) ∈ C× , for all k ∈ Z, and hence, the above operator U j of (1.34) is well defined in LQ ( j). Theorem 1.2. Let U j = ψ (q1 j ) u j be a free random variable (1.34) of LQ ( j), for j ∈ Z, where u j is the j-th generating operator of LQ . If ψ(q j ) ∈ R× = R \ {0} in C× ,

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then U j is semicircular in LQ ( j). Proof. Fix j ∈ Z, and assume ψ(q j ) ∈ R× in C× . Then,  ∗ U j∗ = ψ(q1 j ) u j = U j , by the self-adjointness of u j in LQ . Observe that τ j U jn




n
1 = τ j unj ψ(q j )    1 n n = ω ψ(q ) c = ωn c n2 , n j 2 ψ (q j )n 

(1.35)

for all n ∈ N. So this operator U j is semicircular in LQ ( j), whenever ψ(q j ) ∈ R× , by (1.35) and (1.4).

Assumption 1.1 (in short, A 1.1, from below) We further assume that ψ(q j ) ∈ R× in C, for q j ∈ Q, for all j ∈ Z. 

1.6

THE SEMICIRCULAR FILTERIZATION (LQ , τ)

Let (A, ψ) be a fixed C∗ -probability space containing a family Q = {qk }k∈Z of mutually orthogonal projections satisfying ψ(qk ) ∈ R× , for all k ∈ Z, (under A 1.1). For the system {LQ ( j) : j ∈ Z} of j-th filtered spaces (1.29), define the free product Banach ∗-probability space LQ (Z) by LQ (Z)

denote

= (LQ (Z), τ )  = ? LQ ( j) = ? LQ, j , ? τ j .

de f

j∈Z

j∈Z

(1.36)

j∈Z

That is, our j-th filtered spaces LQ ( j) form the free blocks of LQ (Z), for all j ∈ Z (e.g., [17] and [30]). Definition 1.8. Let LQ (Z) be the free product Banach ∗-probability space (1.36) of the filtered spaces {LQ ( j)} j∈Z . Then, it is said to be the free filterization of Q.

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Certain Banach-Space Operators

Now, construct two subsets X and S of LQ (Z), X = {u j ∈ LQ ( j) : j ∈ Z}, and

(1.37) S = {U j ∈ LQ ( j) : j ∈ Z},

where u j are the j-th generating operators (1.15) of LQ ( j), and U j = ψ(q1 j ) u j are the operators (1.34) in LQ ( j) (under A 1.1), for all j ∈ Z. Recall that a subset Y of an arbitrary topological ∗-probability space (B, ϕ) is said to be a free family if all elements of Y are mutually free from each other in (B, ϕ). Also, a free family Y is called a free (weighted-)semicircular family in (B, ϕ) if this family Y is not only a free family in (B, ϕ), but also a subset of B whose elements are (weighted-)semicircular in (B, ϕ). (e.g., [6] and [30]). Theorem 1.3. Let X and S be the subsets (1.37) of the free filterization LQ (Z). (1.38) The family X is a free weighted-semicircular family in LQ (Z). (1.39) The family S is a free semicircular family in LQ (Z). Proof. Let X be in the sense of (1.37) in LQ (Z). All elements u j of X are taken from mutually distinct free blocks LQ ( j) of LQ (Z), for all j ∈ Z, and hence, they are free from each other in LQ (Z). Thus, this family X is a free family in LQ (Z). Moreover, the n-th powers unj of u j ∈ X are again contained in the free block LQ ( j) as free reduced words with their lengths-1, for all n ∈ N and for j ∈ Z. Thus,     τ unj = τ j unj = ωn ψ(q j )n c n2 , by (1.33), for all n ∈ N and for all j ∈ Z. It shows every element u j ∈ X is ψ(q j )2 semicircular in LQ (Z), for all j ∈ Z. Therefore, the family X is a free weightedsemicircular family in LQ (Z). Equivalently, statement (1.38) holds. Similarly, one can verify that the family S of (1.37) is a free semicircular family in LQ (Z) (under A 1.1). That is, statement (1.39) holds. By (1.31), (1.32), (1.36), (1.38), and (1.39), the only “ j-th” generating operators u j of the free blocks LQ ( j) provide possible nonzero free distributions on LQ (Z). More precisely, all free reduced words of LQ (Z) in ∪ {ui ∈ LQ ( j) : i ∈ Z},

j∈Z

are the free reduced words in X , having their nonzero free distributions in LQ (Z). So we now restrict our interests to the Banach ∗-subalgebra LQ of the free filterization LQ (Z), whose elements have possible nonzero free distributions in LQ (Z). Definition 1.9. Let LQ (Z) be the free filterization of Q. Define a Banach ∗-subalgebra LQ of LQ (Z) by

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LQ = C [X ],

(1.40)

where X is the free weighted-semicircular family (1.38) in LQ (Z), and Y are the Banach-topology closures of subsets Y of LQ (Z). Construct the Banach ∗-probability space,   denote LQ = LQ , τ = τ |LQ , (1.41) as a free-probabilistic substructure of LQ (Z) = (LQ (Z), τ) . We call the Banach ∗-algebra LQ of (1.40), or the Banach ∗-probability space LQ of (1.41), the semicircular (free-sub-)filterization of LQ (Z). The semicircular filterization LQ satisfies the following structure theorem. Theorem 1.4. Let LQ be the semicircular filterization (1.40) of LQ (Z). Then, de f

LQ

= C [X ] = C[S]

∗-iso

∗-iso

= ? C[{u j }] = C j∈Z



 ? {u j } ,

(1.42)

j∈Z

∗-iso

in LQ (Z), where “ = ” means “being Banach-∗-isomorphic,” and where (?) in the first ∗-isomorphic relation of (1.42) is the free-probabilistic free product of [17,30], and (?) in the second ∗-isomorphic relation of (1.42) is the pure-algebraic free product inducing noncommutative free words in X . Proof. The free weighted-semicircular family X of (1.38) can be rewritten as X = {ψ(q j )U j ∈ LQ ( j) : U j ∈ S} in the free filterization LQ (Z) of Q, where S is the free semicircular family (1.39). Therefore, C[X ] = C[S] in LQ (Z), by (1.40). It shows that the set equality (=) of (1.42) holds. By definition (1.40) of LQ , it is generated by the free family X by (1.38), and hence, the first ∗-isomorphic relation of (1.42) holds in the free filterization LQ (Z) by (1.36), because C [{u j }] ⊂ LQ ( j) in LQ (Z), for all j ∈ Z. Since ∗-iso

LQ =

? C[{u j }] in LQ (Z),

j∈Z

every element T of LQ is a limit of linear combinations of free reduced words in X . Also, all (pure-algebraic) free words in X have their unique free-reduced-word forms under operator-multiplication on LQ (Z). Therefore, the second ∗-isomorphic relation of (1.42) holds, too.

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Certain Banach-Space Operators

1.7

FREE POISSON ELEMENTS OF LQ

Let (A, ψ) be a fixed C∗ -probability space, containing a family Q = {q j } j∈Z of mutually orthogonal projections, and let LQ be the semicircular filterization (1.41) of the free filterization LQ (Z) of the C∗ -subalgebra Q = C∗ (Q) of A. Throughout this section, we also assume A 1.1, and hence, the family S of (1.37) is a welldetermined free semicircular family in LQ (Z), generating LQ by (1.39) and (1.42). From the generating free family S of LQ , we here study free Poisson elements. 1.7.1

FREE POISSON ELEMENTS

Let (B, ϕ) be a topological ∗-probability space. If x ∈ (B, ϕ) is a free random variable, then the free distribution of x is characterized by the joint free moments of {x, x∗ }, ϕ (xr1 xr2 ...xrn ) , or, by the joint free cumulants of {x, x∗ }, knB (xr1 , xr2 , ..., xrn ) , for all (r1 , ..., rn ) ∈ {1, ∗}n , and for all n ∈ N, where knB (..) is the free cumulant on B ¨ in terms of ϕ under the Mobius inversion of [17,22,23]. And they provide equivalent free-distributional data of x ∈ (B, ϕ), representing its free distribution. Thus, if x is a self-adjoint free random variable of (B, ϕB ), satisfying x = x∗ in B, then the free distribution of x is characterized by the free-moment sequence
2 3 (ϕ(xn ))∞ n=1 = ϕ(x), ϕ(x ), ϕ(x ), ... , or, by the free-cumulant sequence
∞
knB (x, ..., x) n=1 = k1B (x) = ϕ (x), k2B (x, x), ... . For example, every semicircular element s ∈ (B, ϕ) has its free distribution, the semicircular law, characterized by the free-moment sequence (0, c1 , 0, c2 , 0, c3 , ...) , or, by the free-cumulant sequence (0, 1, 0, 0, 0, ...), where ck are the k-th Catalan numbers for all k ∈ N. Notation From below, we will write “ a free random variable x ∈ (B, ϕ) has its n ∞ free distribution
∞ (ϕ(x ))n=1 ,” and equivalently, “x ∈ (B, ϕ) has its free distribution B kn (x, ..., x) n=1 ,” if (i) x is self-adjoint in B and (ii) the free distribution of x is characterized by

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∞ B (ϕ(xn ))∞ n=1 , or, kn (x, ..., x) n=1 .  Definition 1.10. Let s ∈ (B, ϕ) be a semicircular element, and let a be a self-adjoint free random variable of (B, ϕ), having its free distribution (ϕ(an ))∞ n=1 . Assume that a and s are free in (B, ϕ). Then, a new free random variable Wsa = sas ∈ (B, ϕ)

(1.43)

is called the free Poisson element generated by s and a. By definition (1.43), a free Poisson element Wsa is a free reduced word with its length-3 in (B, ϕ). Also, by (1.43), (Wsa )∗ = (sas)∗ = s∗ a∗ s∗ = sas = Wsa

(1.44)

in B, and hence, it is a self-adjoint free random variable of (B, ϕ), too. Let Ω = {e1 , ..., en } be a finite set with its cardinality n ∈ N. Then, the lattice NC(Ω) of noncrossing partitions of Ω is well defined with its partial ordering ≤, θ1 ≤ θ2 ⇐⇒ ∀V1 ∈ θ1 , ∃V2 ∈ θ2 , s.t., V1 ⊆ V2 , where ⊆ is the usual set inclusion, where “V ∈ θ ” means “V is a block of θ .” For example, if Ω5 = {1, 2, ..., 5}, and if θ1 = {(1, 4), (2, 3), (5)}, and θ2 = {(1, 2, 3, 4), (5)} in NC(Ω5 ), then θ1 ≤ θ2 . Notation From below, if a given finite set Ω is a subset {1, 2, ..., n} of N, for some n ∈ N, then we denote Ω by Ωn .  Under (≤), the lattice NC(Ω) has its maximal element, 1Ω = {(e1 , ..., en )}, the 1-block partition, and its minimal element, 0Ω = {(e1 ), (e2 ), ..., (en )}, the n-block partition. Suppose Ω = Ω1 t Ω2 , with Ωl ⊂ Ω, ∀l = 1, 2.

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Certain Banach-Space Operators


where t is the disjoint union, and let NC Ωl be the noncrossing partition lattices for Ωl , for l = 1, 2. Then, for θl ∈ NC(Ωl ), for l = 1, 2, one can construct a noncrossing partition θ ∈ NC(Ω) as the join of θ1 and θ2 , θ = θ1 ∨ θ2 ∈ NC(Ω), as in [17,22,23]. For example, if θ1 = {(2, 5), (3)} ∈ NC({2, 3, 5}), and θ2 = {(4), (1, 6, 7)} ∈ NC({1, 4, 6, 7}), then, we obtain θ1 ∨ θ2 = {(1, 6, 7), (2, 5), (3), (4)}, in NC (Ω7 ). Now, fix n ∈ N, and let NC(Ω3n ) be the noncrossing partition lattice, and let Ω13n = {1, 3, 4, 6, 7, 9, 10, ..., 3n − 3, 3n − 2, 3n}, and

(1.45) Ω23n = Ω3n \ Ω13n = {2, 5, 8, 11, ..., 3n − 1},

satisfying Ω3n = Ω13n t Ω23n .
And then take θo ∈ NC Ω13n , θo = {(1, 3n), (3, 4), (6, 7), ..., (3n − 3, 3n − 2)},

(1.46)

Ω13n

where is in the sense of (1.45). The following lemma is already shown (e.g., in the page 207 of [17]), but we provide a sketch of the proof for our future works. Lemma 1.2. Let a ∈ (B, ϕ) be a self-adjoint free random variable having its free distribution (ϕ(an ))∞ n=1 , and let s ∈ (B, ϕ) be a semicircular element, free from a in (B, ϕ). If Wsa = sas is a free (1.43), then  Poisson element knB Wsa , Wsa , ...., Wsa  = ϕ(an ), | {z } n-times

for all n ∈ N.

(1.47)

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Proof. By the semicircularity (1.2) of s,  1 B kn (s, ..., s) = 0

if n = 2 otherwise,

(1.48)

for all n ∈ N. So, knB (Wsa , ..., Wsa ) = knB (sas, sas, ..., sas) =

∑ θ ∈NC(Ω23n ),

θ ∨π0 ≤13n

kθB (sas, ..., sas)

where Ω23n is the set (1.45), and kθB (...) are the block-depending free cumulants of [27], and 13n is the maximal 1-block partition of NC(Ω3n ), and π0 = {(1, 2, 3), (4, 5, 6), ..., (3n − 2, 3n − 1 , 3n)} (see [17,22] and [23] for details), and hence, it goes to =

∑ θ ∈NC(Ω23n ), θo ∨θ ∈NC(Ω3n )

kθBo ∨θ (sas, ..., sas)

by (1.48), where Ω23n is in the sense of (1.45), and θo is the partition (1.46)  =







kθBo s, s, s, ..., s kθB a, a, ..., a ∑ | {z } | {z } 2 θ ∈NC(Ω3n ) 2n-times n-times

since all “mixed” free cumulants of s and a vanish of s and a  by the freeness  =

∑ θ ∈NC(Ω23n )


n
B k2B (s, s) kθ a, a, ..., a | {z } n-times

 =

∑ θ ∈NC(Ω23n )



kθB a, a, ..., a | {z } n-times

by (1.2) =

∑

B (a, ..., a) kπ

π∈NC(Ωn )


since the sublattice NC Ω23n of NC (Ω3n ) is equivalent to the lattice NC (Ωn ) = ϕ(an ) ¨ by the Mobius inversion of [17], for all n ∈ N. Therefore, the free-distributional data (1.47) is obtained.

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Certain Banach-Space Operators

The free-cumulant formula (1.47) shows that a free Poisson distribution, and the free distribution of a free Poisson element Wsa , is characterized by the free distribution (ϕ(an ))∞ n=1 of a fixed self-adjoint free random variable a ∈ (B, ϕ). Theorem 1.5. Let Wsa = sas ∈ (B, ϕ) be a free Poisson element (1.43). Then, 



ϕ ((Wsa )n ) = knB a, a, ...., a, | {z }

(1.49)

n-times

for all n ∈ N, where |V | are the cardinalities of blocks V . Proof. Observe that ϕ ((Wsa )n ) =

∑

kθB (Wsa , ..., Wsa )

θ ∈NC(Ωn )

¨ by the Mobius inversion  =





 B W a , ......., W a   Π k|V s  | | s {z } V ∈θ θ ∈NC(Ωn ) ∑

|V |-times

 =

∑ θ ∈NC(Ωn )

Π ϕ a|V |




V ∈θ

by (1.47) = knB (a, ..., a), ¨ by the Mobius inversion, for all n ∈ N. So formula (1.49) holds. 1.7.2

CERTAIN FREE POISSON ELEMENTS INDUCED BY S

Let LQ = (LQ , τ) be the semicircular filterization (1.41). In this section, we fix a semicircular element U j ∈ S in LQ , for a fixed j ∈ Z. By Section 1.7.1, if T ∈ LQ is a self-adjoint operator, and if T is free from U j in LQ , then one obtains the corresponding free Poisson element W jT = U j TU j in LQ , satisfying that   kn W jT , ..., W jT = τ (T n ) , and

(1.50)  n  τ W jT = kn (T, ..., T ) ,

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by (1.47) and (1.49), for all n ∈ N, where k• (...) is the free cumulant on LQ in terms of the linear functional τ of (1.41). U Define free Poisson elements W jk = W j k of (1.50) by de f

W jk = U jUkU j ∈ LQ ,

(1.51)

where j 6= k in Z. Then, one has that  
kn W jk , ..., W jk = τ Ukn = ωn c n2 , and

(1.52)  n  τ W jk = kn (Uk , ..., Uk ) = δn,2

by (1.50) and (1.2), for all n ∈ N, where δ is the Kronecker delta. More generally, for k 6= j ∈ Z, define a free Poisson element, W jk,N = U jUkN U j ∈ LQ , for N ∈ N.

(1.53)

Theorem 1.6. Let W jk,N be a free Poisson element (1.53) for N ∈ N. Then,   kn W jk,N , ..., W jk,N = ωnN c nN , 2

and

(1.54)

 n  τ W jk,N =

      

 Π c N |B|

∑ π∈NC(Ωn )

      ωn



B∈θ

if N is even

2

!

 ∑ θ ∈NCe (Ωn )

Π c N|V |

V ∈θ

if N is odd,

2

for all n ∈ N, where NCe (Ωn ) = {π ∈ NC(Ωn ) : |V | is even, ∀V ∈ π}. Proof. Observe first that   
n  kn W jk,N , ..., W jk,N = τ UkN
= τ UknN = ωnN c nN , 2

by (1.47), for all n ∈ N. Thus, the free-cumulant formula of (1.54) holds. Consider now that  n 
τ W jk,N = kn U N , ..., U N by (1.50)

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Certain Banach-Space Operators

 =

Π

∑ π∈NC(Ωn )

V ∈π

   N|V | τ Uk

¨ by the Mobius inversion  =

∑ π∈NC(Ωn )

  Π ωN|V | c N|V |

V ∈π

2

by the semicircularity (1.4) of Uk ∈ S in LQ

=

      



if N is even

Π c N |B|

∑ π∈NC(Ωn )

      ωn



B∈θ

2

!

 ∑ θ ∈NCe (Ωn )

Π c N|V |

V ∈θ

if N is odd,

2

because if N is even, then lN is even for all l ∈ N, respectively, if N is odd, then lN is even if and only if l is even in N. Therefore, the free-moment formula of (1.54) holds true. Trivially, if N = 1 in (1.54), then relation (1.52) is automatically obtained. 1.7.3

SOME FREE POISSON ELEMENTS INDUCED BY S ∪ X

As in Section 1.7.2, let’s fix a semicircular element U j ∈ S, and let X be the free weighted-semicircular family (1.38) in the semicircular filterization LQ . For any k 6= j in Z, we have the corresponding free Poisson elements Y jk,N = U j uNk U j in LQ ,

(1.55)

where uk ∈ X is a ψ(q j )2 -semicircular element of LQ , for all N ∈ N. Since uk = ψ(qk )Uk ∈ X , in LQ , by (1.55) Y jk,N = ψ(qk )N U jUkN U p, j = ψ (qk )N W jk,N ,

(1.56) (1.57)

in LQ , by (1.56), where W jk,N is the free Poisson element (1.55). Theorem 1.7. Let Y jk,N be a free Poisson element (1.55) in LQ , for N ∈ N. Then   kn Y jk,N , ..., Y jk,N = ωnN ψ (qk )nN c nN , 2

and

(1.58)

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 n  τ Y jk,N =

    ψ(qk )nN    

!

 Π c N|B|

∑ π∈NC(Ωn )

    nN    ωn ψ(qk )

B∈θ

if N is even

2

!

 Π c N|V |

∑ θ ∈NCe (Ωn )

V ∈θ

if N is odd,

2

for all n ∈ N. Proof. Consider that     k,N kn Y j , ..., Y jk,N = kn ψ (qk )N W jk,N , ..., ψ (qk )N W jk,N by (1.57)   = ψ(qk )nN kn W jk,N , ..., W jk,N by the bimodule-map property of free cumulants (e.g., [17,22] and [23])  
= ψ(qk )nN τ UknN = ψ(qk )nN ωnN c nN 2

= ωnN ψ(qk )nN c nN 2


= τ unN , k

for all n ∈ N, by (1.54). Similarly, for any n ∈ N, one obtains that τ

 n    n  = τ ψ(qk )nN W jk,N Y jk,N

by (1.57)   = ψ(qk )nN τ (W jk,N )n

=

   nN    ψ (qk )  

!



    nN    ωn ψ (qk )

if N is even

Π c N |B|

∑ π∈NC(Ωn )

B∈θ

2

!

 ∑ θ ∈NCe (Ωn )

Π c N|V |

V ∈θ

if N is odd,

2

by (1.54). Therefore, the free-distributional data (1.58) hold in LQ . The above theorem shows that the free Poisson elements Y jk,N of (1.55) induced by the free weighted-semicircular family X of (1.38) have their free Poisson distributions, affected by the weights of fixed weighted-semicircular elements of X in LQ .

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Certain Banach-Space Operators

Corollary 1.1. Let Y jk,1 = U j ukU j be a free Poisson element (1.55) in LQ . Then,   kn Y jk,1 , ..., Y jk,1 = ωn ψ(qk )n c n2 , and

(1.59) τ

 n  Y jk,1 = δn,2 ψ(qk )2 ,

for all n ∈ N. Proof. The free-distributional data (1.59) is directly obtained by (1.58), by replacing N to 1.

1.8

FREE WEIGHTED-POISSON ELEMENTS OF LQ

In this section, we are interested in free-Poisson-like free random variables in our semicircular filterization LQ . 1.8.1

FREE WEIGHTED-POISSON ELEMENTS

Let (B, ϕ) be an arbitrary topological ∗-probability space, and let x ∈ (B, ϕ) be a t0 -semicircular element for some t0 ∈ C× , satisfying n

ϕ(xn ) = ωn t02 c n2 , and

(1.60) knB (x, ..., x) = δn,2t0

for all n ∈ N, where knB (...) is the free cumulant on B in terms of ϕ. Definition 1.11. Let x ∈ (B, ϕ) be a t0 -semicircular element (1.60). Suppose a ∈ (B, ϕ) is a self-adjoint free random variable, which is free from x in (B, ϕ). A free random variable Txa = xax ∈ (B, ϕ)

(1.61)

is called a free weighted-Poisson element with its weight t0 (in short, free t0 -Poisson element) of (B, ϕ). Let’s consider free-distributional data of a free t0 -Poisson element Txa of (1.61). Theorem 1.8. Let Txa = xax be a free t0 -Poisson element (1.61), where x is a fixed t0 -semicircular element (1.60) in (B, ϕ), and a has its free distribution (ϕ(an ))∞ n=1 . Then,

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Topics in Contemporary Mathematical Analysis and Applications





knB Txa , Txa , ..., Txa  = t0n (ϕ(an )), | {z }

(1.62)

n-times

for all n ∈ N. Proof. Observe that knB (Txa , ..., Txa ) = knB (xax, xax, ..., xax) =

kθB (xax, ..., xax)

∑

θ ∈NC(Ω3n ), θ ∨π0 ≤13n

where Ω3n is in the sense of (1.45), and kθB (...) are the block-depending free cumulants of [17], and π0 = {(1, 2, 3), (4, 5, 6), ..., (3n − 2, 3n − 1 , 3n)}, and hence, it goes to =

∑ θ ∈NC(Ω23n ), θo ∨θ ∈NC(Ω3n )

kθBo ∨θ (xax, ..., xax)

by (1.48), where θo is in the sense of (1.46)  =

∑ θ ∈NC(Ω23n )

=





kθBo x, x, x, ..., x kθB a, a, ..., a | {z } | {z }

∑ θ ∈NC(Ω23n )

2n-times

n-times


n
B k2B (x, x) kθ (a, a, ..., a)

 = t0n 



 kθB (a, a, ..., a)

∑

θ ∈NC(Ω23n )

by (1.48), because x is a t0 -semicircular element satisfying (1.60) ! = t0n

∑

B (a, ..., a) kπ

π∈NC(Ωn )


since the sublattice NC Ω23n of NC (Ω3n ) is equivalent to the lattice NC(Ωn ) = t0n (ϕ(an )) ,

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Certain Banach-Space Operators

¨ by the Mobius inversion, for all n ∈ N. Therefore, the free-distributional data (1.62) is obtained. The above theorem illustrates that the free distributions of our free weightedPoisson elements Txa of (1.61) are depending not only on the free distributions of a, but also on the weights of fixed weighted-semicircular elements x in (B, ϕ), by ¨ (1.62). By the Mobius inversion, we obtain the following equivalent result. Theorem 1.9. Let Txa be a free t0 -Poisson element (1.61) in (B, ϕ). Then, 





ϕ ((Txa )n ) = t0n knB a, a, ......, a, | {z }

(1.63)

n-times

for all n ∈ N. Proof. Let Txa ∈ (B, ϕ) be a free t0 -Poisson element (1.61). Then, ϕ ((Txa )n ) =

 ∑ π∈NC(Ωn )

 Π k|BV | (Txa , ..., Txa )

V ∈π

¨ by the Mobius inversion  =

Π

∑ π∈NC(Ωn )

V ∈π

 
 |V | t0 ϕ a|V |

by (1.62)  =

∑ π∈NC(Ωn )

|V |

V ∈π

V ∈π

∑ π∈NC(Ωn )

=

∑ π∈NC(Ωn )

!

t0

t0n




Π ϕ a|V |




V ∈π

∑ |V |

=



Π t0

Π ϕ a|V |




V ∈π



Π ϕ a|V |




V ∈π

since ∑ |V | = |Ωn | = n, for all π ∈ NC(Ωn ) V ∈π

=

t0n




 ∑ π∈NC(Ωn )

Πϕ

V ∈π

a|V |




! ,

¨ for all n ∈ N. Therefore, the free-momental data (1.63) is obtained by the Mobius inversion.

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The aforementioned free-distributional data (1.62) and (1.63) provide the following free-distributional information for the free weighted-Poisson elements. Theorem 1.10. Let Txa be a free t0 -Poisson element (1.61) in (B, ϕ), and let Wsa be a free Poisson element (1.43) of (B, ϕ), where s is an arbitrary semicircular element of (B, ϕ), which is free from a fixed self-adjoint free random variable a of (B, ϕ). Then, knB (Txa , ..., Txa ) = t0n knB (Wsa , ..., Wsa ) = t0n ϕ (an ), and

(1.64) 



ϕ ((Txa )n ) = t0n ϕ ((Wsa )n ) = t0n kn a, ........, a , | {z } n-times

for all n ∈ N. Proof. Under hypothesis, one has that knB (Txa , ..., Txa ) = t0n ϕ (an ) = t0n knB (Wsa , ..., Wsa ) , for all n ∈ N, by (1.47) and (1.62). Similarly, we have ϕ ((Txa )n ) = t0n (kn (a, ..., a)) = t0n ϕ ((Wsa )n ) , for all n ∈ N, by (1.49) and (1.63). Therefore, relation (1.64) holds in (B, ϕ). The above theorem characterizes the difference between the free Poisson distributions and the free weighted-Poisson distributions by (1.64). It also shows how our weights act free-probabilistically. 1.8.2

FREE WEIGHTED-POISSON ELEMENTS INDUCED BY S ∪ X

Let LQ be the semicircular filterization, and let S be the free semicircular family (1.39) in LQ . Let’s fix a ψ (q j )2 -semicircular element u j ∈ X in LQ , where X is the free weighted-semicircular family (1.38) of LQ . Then, as in (1.61), one can define the free ψ (q j )2 -Poisson elements T jk,N = u jUkN u j ∈ LQ ,

(1.65)

for Uk ∈ S, any N ∈ N, whenever k 6= j in Z. By (1.62), (1.63), and (1.64), we obtain the following free-distributional data of free ψ(q j )2 -Poisson elements (1.65) in LQ . Theorem 1.11. Let T jk,N = u jUkN u j be a free ψ(q j )2 -Poisson element (1.65) in LQ , for N ∈ N. If W jk,N = U jUkN U j is a free Poisson element (1.53) in LQ , then

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Certain Banach-Space Operators

    kn T jk,N , ..., T jk,N = ψ (q j )2n kn W jl,N , ..., W jl ,N   = ψ(q j )2n ωnN c nN , 2

and

(1.66) τ

 n   n  T jk,N = ψ (q j )2n τ W jl,N

=

    ψ(q j )2n    

!

 Π c N |V |

∑ π∈NC(Ωn )

    2n    ωn ψ(q j )

V ∈π

if N is even

2

!

 ∑ θ ∈NCe (Ωn )

Π c N|V |

V ∈θ

if N is odd,

2

for all n ∈ N, where NCe (Ωn ) are the subsets (1.54) of the lattices NC(Ωn ). Proof. The proof of (1.66) is done by (1.54) and (1.64). The following corollary is obtained immediately by (1.66). Corollary 1.2. Let T jk,1 be a ψ(q j )2 -free Poisson element (1.65) with N = 1. Then   kn T jk,1 , ..., T jk,1 = ωn ψ(q j )2n c n2 , and

(1.67) τ

 n  T jk,1 = δn,2 ψ(q j )2 ,

for all n ∈ N. Proof. The free-distributional data (1.67) of T jk,1 holds by (1.52) and (1.66). The above corollary shows the connections between our ψ(q j )2 -semicircular laws, and the ψ(q j )2 -free Poisson distributions on LQ , by (1.67). 1.8.3

FREE WEIGHTED-POISSON ELEMENTS INDUCED BY X

Let LQ be the semicircular Adelic filterization, and X , the free weightedsemicircular family (1.38) of LQ , and let’s fix u j ∈ X in LQ . For any k 6= j in Z, define the corresponding free ψ(q j )2 -Poisson elements for N ∈ N.

X jk,N = u j uNk u j ∈ LQ ,

(1.68)

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Since Uj =

1 ψ(q j )

u j in S ⇐⇒ u j = ψ(q j )U j in X ,

in LQ , for all j ∈ Z, our free ψ(q j )2 -Poisson elements X jk,N of (1.68) are also understood as X jk,N = ψ(q j )2 ψ(qk )N U jUkN U j
= ψ (q j )2 ψ(qk )N W jk,N ,

(1.69)

in LQ , where W jk,N are the free Poisson element (1.53) of LQ , for all N ∈ N. Theorem 1.12. Let X jk,N = u j uNk u j be a free ψ (q j )2 -Poisson element (1.68) of LQ , for N ∈ N. Then,     kn X jk,N , ..., X jk,N = βn kn W jl,N , ..., W jl,N   = βn ωnN c nN , 2

and

(1.70)  n   n  τ X jk,N = βn τ W jl,N

=

    β    n 

!



       ωn βn

if N is even

Π c N|V |

∑ π∈NC(Ωn )

V ∈π

2

!

 Π c N|V |

∑ θ ∈NCe (Ωn )

V ∈θ

if N is odd,

2

with βn = ψ (q j )2 ψ (qk )N


n

= ψ (q j )2n ψ(qk )nN ,

in R× , for all n ∈ N. Proof. The proof of the free-distributional data (1.70) is done by (1.58) and (1.69). The above theorem shows that the free weighted-Poisson distributions induced by the free weighted-semicircular family X are characterized by the free Poisson distributions induced by the free semicircular family S in the semicircular filterization LQ up to the weights {ψ(q j )2 } j∈Z from X , where the quantities {ψ(q j )} j∈Z represent

29

Certain Banach-Space Operators

the free distributions of our mutually orthogonal projections {q j } j∈Z in the fixed C∗ probability space (A, ψ). That is, the study of our free weighted-Poisson elements of LQ is to investigate the free Poisson elements of LQ up to certain scalar multiples, characterized by (1.70). Therefore, from below, we concentrate on studying the free Poisson elements of LQ .

1.9

SHIFTS ON Z AND INTEGER-SHIFTS ON LQ

As before, let (A, ψ) be a fixed C∗ -probability space containing a family Q = {q j } j∈Z of mutually orthogonal projections q j ’s having ψ(q j ) ∈ R× , for all j ∈ Z, (under A 1.1), and let LQ be the semicircular filterization. 1.9.1

(±)-SHIFTS ON Z

Define bijections h+ and h− on the set Z of all integers by h± ( j) = j ± 1,

(1.71)

for all j ∈ Z. By definition (1.71), indeed, these functions are bijective on Z, since −1 means the inverses of bijections f . h−1 + = h− , where f Then, for these bijections h± of (1.71), one can construct the following bijections (n) h± on Z, (n) h± = h± ◦ h± ◦ · · · ◦ h± , (1.72) } {z | n-times

(1)

for all n ∈ N, with identities, h± = h± , where (◦) is the composition. Then, (n)

h± ( j) = j ± n, for all j ∈ Z,  −1 (n) (n) satisfying h+ = h− on Z, for all n ∈ N0 , with axiomatization: (0)

h± = idZ , the identity function on Z. (n)

Definition 1.12. Let h± be the bijections (1.72) on Z, for all n ∈ N0 . Then, we call (n) h± , the n-(±)-shifts on Z. 1.9.2 (n)

INTEGER-SHIFTS ON LQ (n)

Let h± be n-(±)-shifts (1.72) on Z, for n ∈ N0 . In this section, by using h± , certain (n) ∗-isomorphisms β± on LQ are constructed, and we study how these ∗-isomorphisms act on LQ , for n ∈ N0 .

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Topics in Contemporary Mathematical Analysis and Applications

Define “multiplicative” bounded linear transformations β± on LQ by the morphisms satisfying that: β± (U j ) = Uh± ( j) = U j±1 , (1.73) for U j ∈ S, and for all j ∈ Z, where S is the free semicircular family (1.39). By the structure theorem (1.42), the aforementioned multiplicative linear transformations β± of (1.73) are well defined on LQ . More precisely, one obtains the following computations. N

n

Lemma 1.3. Let Y = Π U jll be a free reduced word of LQ with its length-N, for l =1

U j1 , ..., U jN ∈ S , and n1 , ..., nN ∈ N, where ( j1 , ..., jN ) is alternating in ZN , in the sense that: j1 6= j2 , j2 6= j3 , ..., jN−1 6= jN in Z, for N ∈ N. Then, β± (Y ) become new free reduced words of LQ with their lengths-N in S, N

n

β± (Y ) = Π U jll±1 , in LQ .

(1.74)

l=1

Proof. Let Y be given as above in LQ . Then, by the multiplicativity of the linear transformations β± of (1.73), one has that   N N N

n n n β± (Y ) = Π β± U jll = Π β± U jl l = Π Uh±l ( j ) . l=1

l=1

l =1

l

Therefore, formula (1.74) holds. Moreover, it is easily checked that if ( j1 , ..., jN ) is an alternating N-tuple of ZN , then the N-tuples ( j1 ± 1, j2 ± 1, ..., jN ± 1) are alternating in ZN , too. Therefore, the images β± (Y ) form new free reduced words of LQ with their lengths-N in S. As one can see in (1.74), the morphisms β± of (1.73) preserve the freeness on LQ , by (1.42). Lemma 1.4. The morphisms β± of (1.73) are ∗-isomorphisms on LQ . Proof. By (1.40), (1.41), and (1.42), all elements of the semicircular filterization LQ are the limits of linear combinations of free reduced words in the free semicircular family S of (1.39). So, let’s focus on the free reduced words of LQ in S. Let ( j1 , ..., jN ) be an alternating N-tuple of ZN for N ∈ N, and N

n

Y = Π U jll , for n1 , ..., nN ∈ N. l =1

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Certain Banach-Space Operators

Then, by (1.74), N

n

β± (Y ) = Π Uh±l ( j ) ,

(1.75)

l

l=1

where h± are the (±)-shifts (1.71) on Z. By the bijectivity of h± , relation (1.75) implies the bijectivity of β± on LQ . That is, these multiplicative linear transformations β± of (1.73) are generator-preserving (bijective), and bounded on LQ . Consider now that if Y is as above, then   N N nN −l+1 n β± (Y ∗ ) = β± Π U jN−l+1 = Π Uh±N(−l+1 j ) l =1

l=1

by (1.74)  =

N

∗

n

Π Uh±l ( j )

l=1

l

N−l+1

= (β± (Y ))∗ .

(1.76)

So β± (S∗ ) = (β± (S))∗ , for all S ∈ LQ , by (1.76). Therefore, the bounded, bijective, multiplicative linear transformations β± are adjoint-preserving on LQ , by (1.77). That is, they are ∗-isomorphisms on LQ . The above lemma shows that the (±)-shifts h± on Z induce the corresponding ∗-isomorphisms β± on LQ . Let β± be the ∗-isomorphisms (1.73). Then, one can construct ∗-isomorphisms, β±n = β± β± · · · · · ·β± on LQ , | {z }

(1.77)

n-times

β±0

for all n ∈ N0 , with identity: = 1LQ , the identity map on LQ . Since β± and 1LQ are ∗-isomorphisms, the morphisms β±n of (1.77) are welldefined ∗-isomorphisms on LQ , too, for all n ∈ N0 . Definition 1.13. The ∗-isomorphisms β±n of (1.77) are called the n-(±)(integer-)shifts on LQ , for all n ∈ N0 . These ∗-isomorphisms {β±n }n∈N0 satisfy the following identity relation on LQ . Proposition 1.4. Let β±n be the n-(±)-integer-shifts (1.77) on LQ . Then, (β+ β− )n = β+n β−n = 1LQ = β−n β+n = (β− β+ )n on LQ ,

(1.78)

for all n ∈ N0 . Moreover, β+n1 β−n2 = β−n2 β+n1

  1LQ = β n1 −n2  +n2 −n1 β−

if n1 = n2 if n1 > n2 if n1 < n2 ,

(1.79)

on LQ , for all n1 , n2 ∈ N0 . Proof. As we discussed above, it suffices to consider the cases where we have the free reduced words

32

Topics in Contemporary Mathematical Analysis and Applications N

n

Y = Π U jll of LQ , for n1 , ..., nN ∈ N, l=1

for N ∈ N, by (1.74) and (1.42). Observe that   N N nl n β+ β− (Y ) = β+ Π U jl −1 = Π U( jl −1)+1 N

l

l =1

l=1

n

= Y = Π U( jl +1)−1 l l=1   N n = β− Π U jll+1 = β− β+ (Y ). l=1

Therefore, β+ β− = 1LQ = β− β+ on LQ .

(1.80)

By (1.80), one can get that β+n β−n = (β+ β− )n = 1LQ = (β− β+ )n = β−n β+n , on LQ , for all n ∈ N0 . Therefore, formula (1.78) holds. By (1.78), one has that if n1 > n2 in N0 , then β+n1 β−n2 = β+n1 −n2 β+n2 β−n2 = β+n1 −n2 on LQ , and similarly, if n1 < n2 in N0 , then β+n1 β−n2 = β+n1 β−n1 β−n2 −n1 = β−n2 −n1 . Therefore, formula (1.79) holds. The above relations (1.78) and (1.79) can be re-expressed simply by |e n +e2 n2 | 1 n1 +e2 n2 )

1 1 βen11 βen22 = βen22 βen11 = βsgn(e

on LQ ,

(1.81)

for all e1 , e2 ∈ {±}, and n1 , n2 ∈ N0 , where sgn(·) is the sign-map on Z,  + if j ≥ 0 sgn( j) = − if j < 0 for all j ∈ Z, and where |.| is the absolute value on Z. Now, consider the system B of all n-(±)-shifts β±n on LQ , i.e., B = {β±n }n∈N0 . Let Aut(LQ ) be the automorphism group    α is a   Aut (LQ ) =  α : LQ → LQ ∗-isomorphism , ·   on LQ where the operation (·) is the product of ∗-isomorphisms.

(1.82)

(1.83)

33

Certain Banach-Space Operators

By definition (1.82), the system B is contained in the automorphism group Aut(LQ ) of (1.83). Note that the operation (·) is closed on B, in the sense that:
βen11 , βen22 ∈ B × B 7−→ βen11 βen22 ∈ B, for all e1 , e2 ∈ {±}, and n1 , n2 ∈ N0 , by (1.81). Clearly, by (1.84), one can get that
n n +n +n n
βen1 βen2 βe 3 = βe 1 2 3 = βen1 βen2 βe 3 ,

(1.84)

(1.85)

for all e ∈ {±}, and n1 , n2 , n3 ∈ N0 . Observe now that
n |e1 n1 e2 n2 | |e1 n1 e2 n2 e3 n3 | n βen11 βen22 βe33 = βsgn(e β 3 = βsgn(e , n e n ) e3 n e n e n ) 1 1 2 2

1 1 2 2 3 3

and

(1.86) n3


|e n e3 n3 | 2 n2 e3 n3 ))

2 2 βen11 βen22 βe3 = βen11 βsgn(e

|e n e2 n2 e3 n3 )| . 1 n1 e2 n2 e3 n3 ))

1 1 = βsgn(e

Thus, by (1.85) and (1.86),
n n
βen11 βen22 βe33 = βen11 βen22 βe33 on LQ ,

(1.87)

for all e1 , e2 , e3 ∈ {±}, and n1 , n2 , n3 ∈ N0 . Theorem 1.13. If B is the set (1.82), then B is an abelian subgroup of Aut(LQ ). Proof. By (1.81), the operation (·) is closed on B. So the algebraic pair B = (B, ·) is a well-defined algebraic substructure of Aut(LQ ). By (1.87), this operation (·) is associative on B, and hence, the pair forms a semigroup. Since β+0 = 1LQ = β−0 ∈ B, and since βen · 1LQ = βen = 1LQ · βen on LQ , for all e ∈ {±}, and n ∈ N0 , the semigroup B contains its (·)-identity 1LQ , and hence, it forms a monoid. By (1.78), all elements β±n ∈ B have their unique (·)-inverses β∓n ∈ B, for all n ∈ N0 . So this monoid B is a group. The commutativity on B is clear by (1.79). Therefore, the system B is an abelian subgroup of the automorphism group Aut(LQ ). The above theorem characterizes the system B of (1.82) as an abelian subgroup of Aut(LQ ). As a group, B satisfies the following group-property. Theorem 1.14. Let B be the abelian group (1.82). Then, Group

B = (Z, +), Group

where “ = ” means “being group-isomorphic to.”

(1.88)

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Topics in Contemporary Mathematical Analysis and Applications

Proof. Define now a function Φ : Z → B by | j|

Φ : j ∈ Z 7−→ βsgn( j) ∈ B,

(1.89)

with identity: Φ(0) = 1LQ in B. By (1.79), this map Φ is a well-defined bijection from Z onto B. Observe that |j +j |

|j |

|j |

1 2 1 Φ( j1 + j2 ) = βsgn( = βsgn( β 2 j +j ) j ) sgn( j 1

2

1

2)

= Φ( j1 )Φ( j2 ),

(1.90)

in B, by (1.81), for all j1 , j2 ∈ Z. So the bijection Φ of (1.89) is a group-homomorphism by (1.90). Therefore, the group-isomorphic relation (1.88) holds true. The above theorem characterizes the group-structure of the subgroup B = {β±n }n∈N0 in Aut(LQ ). That is, B is an infinite abelian cyclic group hβ+ i = hβ− i. Definition 1.14. Let B be the abelian group (1.82). We call B the integer-shift (sub)group (of the automorphism group Aut(LQ ) acting) on LQ . 1.9.3

FREE PROBABILITY ON LQ UNDER THE GROUP-ACTION OF B

Let B be the integer-shift group (1.84) acting on the semicircular filterization LQ , which is an infinite abelian cyclic subgroup of the automorphism group Aut(LQ ), by (1.88). In this section, we consider how our ∗-isomorphisms β±n ∈ B affect the free n probability on LQ . Throughout this section, let’s fix n0 ∈ N0 and β±0 ∈ B. Lemma 1.5. Let u j ∈ X be the ψ(q j )2 -semicircular element, and U j ∈ S, a semicircular element of LQ . Then,  
n
n τ β±0 (u j ) = ωn ψ (q j )n c n2 = τ unj , and

(1.91) τ

 
n
n = ωn c n2 = τ U jn , β±0 (U j )

for all n ∈ N. Proof. Observe that, for any n ∈ N, one has that τ


n
n β±0 (u j ) =τ


n
n ψ(q j )β±0 (U j )

n

since β±0 are linear on LQ and u j = ψ(q j )U j in LQ     n n ω cn = ψ(q j )n τ U j±n = ψ(q ) j n 0 2

(1.92)

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Certain Banach-Space Operators

by the semicircularity (1.39) of U j±n0 ∈ S = ωn ψ (q j )2


n2

  c n2 = τ unj ,

(1.93)

by (1.38). Therefore, the first free-distributional data of (1.91) hold by (1.93). As one can see in (1.92), we have that  
n
n τ β±0 (U j ) = ωn c n2 = τ U jn , for all n ∈ N, by (1.39), for all j ∈ Z. So the second free-distributional data of (1.91) hold. The above lemma shows how the original free-distributional data on LQ is affected by the group-action of B. That is, the action preserves the free distributions of generating operators of LQ . Definition 1.15. Let (B1 , ϕ1 ) and (B2 , ϕ2 ) be arbitrary topological ∗-probability spaces. We say that they are free-(∗-)isomorphic if (i) B1 and B2 are ∗-isomorphic via a ∗-isomorphism Ω : B1 → B2 , and (ii) ϕ2 (Ω(a)) = ϕ1 (a), ∀a ∈ (B1 , ϕ1 ). In such a case, we call the ∗-isomorphism Ω a free-isomorphism. By (1.91), one obtains the following theorem. Theorem 1.15. All integer-shifts of B are free-isomorphisms on LQ . Proof. Let βen ∈ B, for e ∈ {±} and n ∈ N0 . Since B is a subgroup of Aut(LQ ), a group-element βen ∈ B is a ∗-isomorphisms on LQ . Moreover, by (1.42) and (1.91), τ (βen (T )) = τ (T ) , for all T ∈ LQ . Therefore, the ∗-isomorphism βen ∈ B is a free-isomorphism. Since βen is arbitrary in B, all integer-shifts of B are free-isomorphisms on LQ . The above theorem illustrates that the free probability on the semicircular filterization LQ is preserved by the action of the integer-shift group B.

1.10

BANACH-SPACE OPERATORS ON LQ GENERATED BY B

Throughout this section, let LQ be the semicircular filterization, and let B be the integer-shift group acting on LQ . In this section, we consider certain Banach-space operators acting “on LQ ,” generated by B, by regarding LQ as a Banach space (e.g., [13]). Let B (LQ ) be the operator space consisting of all Banach-space operators on the Banach space LQ equipped with its operator norm,

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Topics in Contemporary Mathematical Analysis and Applications

  v ∈ LQ , and kT k = sup kT vkLQ kvk , LQ = 1

(1.94)

for all T ∈ B (LQ ) , where k.kLQ is the norm on the Banach ∗-algebra LQ (e.g., [13] and [14]). Define now a (closed) subspace A of the vector space B(LQ ) by k.k

de f

A = spanC (B) = C [B] ,

(1.95)

k.k

where B is the integer-shift group, and Y are the operator-norm topology closures of the subsets Y of B(LQ ), where k.k is in the sense of (1.94). Note that since B is a group, the set equality of (1.95) holds. That is, A is not only a closed subspace, but also an algebra of B (LQ ) . On this Banach algebra A, define a unary operation (∗) by !∗ ∑ (e,n)∈{±}×N0

de f

t(e,n) βen

=

∑ (e,n)∈{±}×N0

n , t(e,n) β−e

(1.96)

on A, for all t(e,n) ∈ C with their conjugates t(e,n) ∈ C. Proposition 1.5. The subspace A of (1.95) forms a Banach ∗-algebra in B(LQ ). Proof. First of all, for any βen ∈ B ⊂ A, there always exists (βen )∗ = β−n e ∈ B ⊂ A, for all e ∈ {±} and n ∈ N0 . So for any A ∈ A, there exists a unique A∗ in A. So the operation (∗) of (1.96) is closed on A. Observe that
n ∗ = βn n ((βen )∗ )∗ = β−e −(−e) = βe in A, by (1.96), for all e ∈ {±}, and n ∈ N0 . So if A ∈ A, then A∗∗ = A, in A.

(1.97)

By the very definition (1.96), one immediately obtain that (zA)∗ = z A∗ , ∀z ∈ C, and A ∈ A.

(1.98)

Now, observe that βen11 + βen22


∗

n1 n2 = β−e + β−e = βen11 1 2


∗

+ βen22


∗

,

in A, by (1.96), for all e1 , e2 ∈ {±}, and n1 , n2 ∈ N0 . Thus, if A1 , A2 ∈ A, (A1 + A2 )∗ = A∗1 + A∗2 in A, by (1.98). Also, one can get that βen11 βen22


∗

n , = (βen )∗ = β−e

and βen22


∗

βen11


∗

n2 n1 n , = β−e β n1 = β−e β n2 = β−e 2 −e1 1 −e2

(1.99)

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Certain Banach-Space Operators

where e = sgn(e1 n1 + e2 n2 ) ∈ {±}, and n = |e1 n1 + e2 n2 | ∈ N0 . And hence,
∗
∗ n1
∗ βen11 βen22 = βen22 βe1 , in A, for all e1 , e2 ∈ {±}, and n1 , n2 ∈ N0 . So (A1 A2 )∗ = A∗2 A1∗ in A,

(1.100) for all A1 , A2 ∈ A. Therefore, the operation (∗) of (1.96) is a well-defined adjoint on A, by (1.97), (1.98), (1.99), and (1.100). So the Banach algebra A forms a Banach ∗-algebra. The above proposition shows that every element of A is an adjointable Banachspace operator on LQ in the sense of [13]. Definition 1.16. Let A be the Banach ∗-algebra (1.95) embedded in the operator space B(LQ ). Then, we call A the integer-shift operator algebra (on LQ ). All elements of A are said to be integer-shift operators on LQ . 1.10.1

DEFORMED FREE PROBABILITY OF LQ BY A

Let LQ be the semicircular filterization, and B, the integer-shift group acting on LQ , and let A be the integer-shift-operator algebra (1.95) generated by B in the operator space B(LQ ). In this section, we act operators of A on LQ and study how the original free-distributional data on LQ are deformed by the action. By (1.40) and (1.42), we focus on how our (weighted-)semicircularity induced by (X ∪)S is affected by the action of A on LQ . Since we already considered how the integer-shift group B, the generator set of A, affects the free probability on LQ in Section 1.9, we here are interested in the distorted (weighted-)semicircular law(s) by the action of nongenerator operators A of A, formed by A = ∑ t(e,n) βen ∈ A. (1.101) (e,n)∈N0±

Note again that if A ∈ B ⊂ A, then such an integer-shift operator A is an integershift, which is free-isomorphism on LQ , and hence, it preserves the free probability on LQ . However, if A ∈ A \ B, then we cannot guarantee that A preserves the freedistributional data on LQ . Notation From below, for convenience, we let denote

N± = {±} × N0 . 0  Lemma 1.6. Let A = tβek ∈ A, for t ∈ C× , and (e, k) ∈ N± , and let u ∈ X be a j 0 ψ(q j )2 -semicircular element in LQ , for j ∈ Z. Then, τ ((A(u j ))n ) = ωn (tψ(q j ))n c n2 ,

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Topics in Contemporary Mathematical Analysis and Applications

(1.102)

and τ ((A(u j )∗ )n ) = ωn (tψ(q j ))n c n2 , for all n ∈ N. × Proof. Let A = tβek ∈ A, for (e, k) ∈ N± 0 , and let t ∈ C . Then,
n

n
τ (A(u j )n ) = τ t βek (u j ) = τ tψ(q j )U jek n = ωnt n ψ(q j )n c n2 , = t n ψ(q j )n τ U jek

and τ ((A(u j )∗ )n ) = τ




∗ n  tβek (u j )

= ωnt n ψ(q j )n c n2 , for all n ∈ N, and j ∈ Z, implying the formulas of (1.104). By (1.102), we obtain the following result. Theorem 1.16. Let A = tβen ∈ A, for t ∈ C× , and (e, n) ∈ N± 0 . Then, if t ∈ R× , then tβen (u j ) ∈ LQ are (tψ(q j ))2 -semicircular,

(1.103)

in LQ , for all (e, n) ∈ N± 0 and for all j ∈ Z. Proof. Let t ∈ R× in C, and let A = tβen , for (e, n) ∈ N± 0 , then, for any j ∈ Z, one has that     τ (A(u j ))k = ωk (tψ(q j ))k c k = τ (A(u j )∗ )k , 2

for all k ∈ N, by (1.102). Indeed, since t ∈ R× , we have (A(u j ))∗ = tψ(q j )U jek


∗

= tψ(q j )U jek = A(u j ),

in LQ (under A 1.1); equivalently, A(u j ) is self-adjoint in LQ , for all j ∈ Z. Therefore, tβen (u j ) is (tψ(q j ))2 -semicircular in LQ , for all j ∈ Z, whenever t ∈ R× , for all (e, n) ∈ N± 0. The above theorem shows that the integer-shift operators tβen ∈ A distort the original ψ (q j )2 -semicircular laws to the (tψ(q j ))2 -semicircular laws on LQ , for all (e, n) × ∈ N± 0 , and j ∈ Z, whenever t ∈ R \ {1}, by (1.103). Corollary 1.3. Let A = tβen ∈ A, for t ∈ R× , and (e, n) ∈ N± 0 , and let U j ∈ S be a semicircular element in LQ , for j ∈ Z. Then, if t ∈ R× , then tβen (U j ) is t 2 -semicircular in LQ , for all (e, n) ∈ N± 0 and for all j ∈ Z.

(1.104)

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Certain Banach-Space Operators

Proof. Statement (1.104) is shown by the proof of (1.103). Now, let A ∈ A be in the sense of (1.101), and assume further that A has more than one summand. Then, for any u j ∈ X and U j ∈ S, one can have that   n in L , A U jn = ∑ t(e,k)U jek Q (e,k)∈N0

and hence,

(1.105) !

  A unj = ψ(q j )n

∑

(e,k)∈N± 0

n t(e,k)U jek

in LQ ,

for all n ∈ N. Theorem 1.17. Let A ∈ A be an integer-shift operator (1.101), and u j ∈ X , U j ∈ S in LQ . Then,    

τ A unj = ωn ψ(q j )n c n  ∑ t(e,k)  2

(e,k)∈N± 0






= τ unj 

t(e,k)  ,

∑

(e,k)∈N± 0

(1.106)

and 

 τ A U jn





  = ωn c n2 

∑

t(e,k) 

(e,k)∈N± 0


n

= τ Uj 



∑

t(e,k)  ,

(e,k)∈N± 0

for all n ∈ N. Proof. Remark that if A ∈ A is in the sense of (1.101), then

A unj = ∑ t(e,k) βek unj (e,k)∈N0±

=

∑

  n t(e,k) ψ(q j )nU jek ,

(e,k)∈N0±

by (1.105). So    τ A unj = ψ(q j )n

∑

(e,k)∈N± 0

  n t(e,k) τ U jek

!

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Topics in Contemporary Mathematical Analysis and Applications

= ψ(q j

)n

∑

(e,k)∈N0±



(1.107)

!



= ωn ψ(q j )n c n2

   = τ unj

!   t(e,k) ωn c 2n

∑

(e,k)∈N± 0

t(e,k)

! ∑

(e,k)∈N± 0

t(e,k) ,

(1.108)

for all n ∈ N. By (1.107), one obtains that      τ A U jn = ωn c n2

  = τ U jn

! ∑

(e,k)∈N± 0

t(e,k) !

∑

(e,k)∈N± 0

t(e,k) ,

for all n ∈ N. Therefore, by (1.107) and (1.109), the free-distributional data (1.106) hold. In the formulas of (1.106), the quantity de f

tA =

t(e,k) ∈ C

∑

(1.109)

(e,k)∈N± 0

is the sum of all coefficients {t(e,k) }(e,k)∈N± of the operator A ∈ A. So 0

 
τ A(U jn ) = tA ωn c n2 , ∀n ∈ N, for all j ∈ Z, by (1.106) and (1.109). And, similar to (1.110),  
τ A(unj ) = (tA ψ(q j )n ) ωn c n2 , ∀n ∈ N,

(1.110)

(1.111)

for all j ∈ Z, where tA is in the sense of (1.109). Formulas (1.110) and (1.111) show how the integer-shift-operator algebra A deforms the (weighted-)semicircular law(s) on LQ . 1.10.2

DEFORMED SEMICIRCULAR LAWS ON LQ BY A

We, here, concentrate on the distortions of the semicircular law on LQ by acting the integer-shift-operator algebra A. Now, observe that if A is an integer-shift operator, A=

∑

(e,k)∈N± 0

t(e,k) βek ∈ A,

(1.112)

41

Certain Banach-Space Operators

then A2 =

∑

t(e1 ,k1 )t(e2 ,k2 ) βek11 βek22

∑

1 1 t(e1 ,k1 )t(e2 ,k2 ) βsgn(e

± ((e1 ,k1 ),(e2 ,k2 ))∈N± 0 ×N0

=

((e1 ,k1 ),(e2 ,k2 ))∈N0± ×N± 0

|e k +e2 k2 | , 1 k1 +e2 k2 )

and hence, ! A2

=

∑

(e,k)∈N± 0

∑ ((e1 ,k1 ),(e2 ,k2 ))∈[e,k]2

t(e1 ,k1 )t(e2 ,k2 ) βek ,

where

(1.113)   (e1 , k1 ), (e2 , k2 ) ∈ N± ,   0 [e, k]2 = ((e1 , k1 ), (e2 , k2 )) e = sgn(e1 k1 + e2 k2 ), ,  and k = |e1 k1 + e2 k2 | 

for all (e, k) ∈ N0± , because the integer-shift group B, the generator set of A, is abelian. Lemma 1.7. Let A ∈ A be an integer-shift operator (1.112). Then,  ! An =

n

∑

(e,k)∈N0±

∑

((el ,kl ))nl=1 ∈[e,k]n

Π t(el ,kl )

l =1

βek ∈ A,

with

(1.114)   (el , kl ) ∈ N± , ∀l = 1, ..., n,   0 [e, k]n = ((el , kl ))nl=1 e = sgn (∑nl =1 el kl ) ∈ {±}, ,  and k = |∑n el kl | ∈ N0  l=1

for all n ∈ N. Proof. The operator equality (1.114) is obtained by the induction on (1.113). Notation Let An be as in (1.114) for an integer-shift operator A ∈ A of (1.112), for n ∈ N. Then, we denote the quantities   n denote (1.115) Π t = t[e,k]n , (el ,kl ) ∑ ((el ,kl ))nl =1 ∈[e,k]n

l =1

for all (e, k) ∈ N± 0 , where [e, k]n are in the sense of (1.114). By notation (1.115), one can re-express (1.115) by An =

∑

(e,k)∈N± 0

for all n ∈ N.

t[e,k]n βek in A,



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Topics in Contemporary Mathematical Analysis and Applications

Theorem 1.18. Let U j ∈ S in LQ , and let A ∈ A be an integer-shift operator (1.112). Then, !      n2 n 1 τ A Uj = ωn2 c n2 (1.116) ∑ t[e,k]n1 2

(e,k)∈N± 0

where [e, k]n1 are in the sense of (1.115), for all (e, k) ∈ N± 0 , and n1 , n2 ∈ N. Proof. Now, let n1 , n2 ∈ N, and j ∈ Z. Then, !      n n n k 2 2 τ A 1 Uj =τ ∑ t[e,k]n βe U j 1

(e,k)∈N0±

by (1.115) =

∑

(e,k)∈N0±

  n2 t[e,k]n τ U jek 1

where [e, k]n1 are in the sense of (1.115) for all (e, k) ∈ N± 0 =

∑

(e,k)∈N± 0

t[e,k]n

1

  ωn2 c n2 2

by the semicircularity of U jek ∈ S in LQ , for all j ∈!Z, and (e, k) ∈ N0±   = ωn2 c n2 ∑ t[e,k]n . 2

(e,k)∈N± 0

1

Therefore, formula (1.116) holds. Similar to (1.116), we also have the following free-distributional data on LQ . Theorem 1.19. Let u j ∈ X in LQ , and let A be an integer-shift operator (1.112) in A. Then, !     n τ An1 (u j 2 ) = ωn2 ψ(q j )n2 c n2 (1.117) ∑ t[e,k]n , 2

(e,k)∈N± 0

1

for all n1 , n2 ∈ N. Proof. The proof of (1.117) is similarly done by (1.114) and (1.116). The above two theorems generalize the distorted free-distributional data (1.106) of the semicircular law on LQ . Similar to the notation tA of (1.109), one can define the following quantities. Notation Let’s denote the quantity denote (n) ∑ t[e,k]n = tA in C, (e,k)∈N± 0

(1.118)

43

Certain Banach-Space Operators

where {t[e,k]n }(e,k)∈N± are the coefficients (1.115) of An ∈ A of the integer-shift 0 operator A of (1.112). Remark that, by (1.118), (1)

tA of (1.109) = tA of (1.118).  By using the above new notation (1.118), formulas (1.116) and (1.117) can be re-expressed simply by        (n ) (n ) τ An1 U jn2 = tA 1 ωn2 c n2 = tA 1 τ U jn2 , 2

and

(1.119)        (n ) (n ) τ An1 unj 2 = tA 1 ωn2 ψ(q j )n2 c n2 = tA 1 τ unj 2 , 2

for all n1 , n1 ∈ N, and for all j ∈ Z.

1.11

DEFORMED FREE POISSON DISTRIBUTIONS ON LQ BY A

In this section, as an application of Section 1.10, we study the deformed free (weighted-)Poisson distributions on LQ under the action of the integer-shift-operator algebra A. Throughout this section, we fix a free Poisson element W jm,N = U jUmN U j ∈ LQ ,

(1.120)

for U j , Um ∈ S, and a free ψ(q j )2 -Poisson element Y jm,N = u jUmN u j ∈ LQ ,

(1.121)

for u j ∈ X , where j 6= m in Z, for N ∈ N. By (1.54) and (1.66), we have the following free Poisson distribution of W jm,N and the free ψ(q j )2 -Poisson distribution of Y jm,N :      Π c if N is even ∑ | | N V   π∈NC(Ωn ) V ∈π 2  n   τ W jm,N =       Π c N|V | if N is odd,  ωn ∑ π∈NCe (Ωn )

and for all n ∈ N.

V ∈π

(1.122)

2

 n   n  τ Y jm,N = ψ(q j )2n τ W jm,N ,

(1.123)

44

Topics in Contemporary Mathematical Analysis and Applications

Corollary 1.4. Let W jm,N be the free Poisson element (1.120) of LQ . If A ∈ A, an integer-shift operator (1.112), then   (n1 )     tA  

  n2  τ An1 W jm,N =

!

 Π c n2 |V |

∑

V ∈π

π∈NC(Ωn2 )

     ωnt (n1 )  A 

if N is even

2

!

 Π c n2 |V |

∑

V ∈π

π∈NCe (Ωn2 )

if N is odd,

2

(1.124) with 

(n )

tA 1 =

∑

(e,k)∈N± 0

!

n

Π t(el ,kl )

∑

l =1

((el ,kl ))nl=1 ∈[e,k]n1

,

(1.125)

for all n1 , n2 ∈ N, where

[e, k]n1

  (el , kl ) ∈ N± , ∀l = 1, ..., n1 ,   0 n1 n = ((el , kl ))l=1 . e = sgn (∑l=1 el nl ) ∈ {±},  and k = |∑n el nl | ∈ N0  l=1

Proof. Fix n1 , n2 ∈ N, and let Tn1 ,n2 = An1

 n2  W jm,N ∈ LQ .

Then Tn1 ,n2 = An1 =



∑

U jUmN U j

(e,k)∈N± 0


n2 

t[e,k]n βek



U jUmN U j

1


n2 

with t[e,k]n = 1

∑

n

1 ∈[e,k] ((el ,kl ))l=1 n1

n  1 Π t(el ,kl ) l =1

by (1.115), where [e, k]n1

  (el , kl ) ∈ N± , ∀l = 1, ..., n1 ,   0 n1 n el nl ) ∈ {±}, = ((el , kl ))l=1 , e = sgn (∑l=1  and k = |∑n el nl | ∈ N0  l=1

by (1.114), for all (e, k) ∈ N± 0 , and hence, it goes to =

∑

(e,k)∈N± 0

in LQ .

N U t[e,k]n U jekUmek jek 1


n2

,

(1.126)

45

Certain Banach-Space Operators

N U Note that in (1.126), the summands U jekUmek jek are the free Poisson elements of LQ , because the freeness of semicircular elements U jek and Umek is preserved from the freeness of U j and Um in LQ (by the action of the integer-shift group B), for all j 6= m ∈ Z. Thus,

τ (Tn1 ,n2 ) =

t[e,k]n τ

N U U jekUmek jek

∑

t[e,k]n τ

N U U jekUmek jek

∑

t[e,k]n τ

(e,k)∈N± 0

=

(e,k)∈N± 0

=


n2


∑

(e,k)∈N0±

1

1

1

U jUmN U j


n2



n2


by Theorem 1.15 ! =τ


n
U jUmN U j 2

(n )

= tA 1 τ

∑

(e,k)∈N± 0

t[e,k]n

1

 n2  W jm,N ,

(1.127)

(n )

where tA 1 is in the sense of (1.125). Therefore, by (1.54) and (1.127), the freedistributional data (1.124) hold true. The aforementioned free-distributional data (1.124) shows how our integer-shift operators A ∈ A of (1.112) distort the free Poisson distribution of W jm,N on LQ . As a special case of (1.124), one obtains the following fact. Corollary 1.5. Let W jm,1 be the free Poisson element (1.120) of LQ . If A ∈ A, an integer-shift operator (1.112), then   n2  (n ) τ An1 W jm,1 = δn2 ,2 tA 1 with (n ) tA 1

 =

∑

(e,k)∈N± 0

∑

((el ,kl ))nl =1 ∈[e,k]n1

for all n1 , n2 ∈ N.

n

Π t(el ,kl )

l =1

! , 

Also, we obtain the following corollary. Corollary 1.6. Let Y jm,N be the free ψ(q j )2 -Poisson element (1.121) of LQ , and let A ∈ A be an integer-shift operator (1.112). Then,

46

Topics in Contemporary Mathematical Analysis and Applications

  n2  τ An1 Y jm,N =

   (n )  tA 1 ψ (q j )2n2    

!



    (n )   ωntA 1 ψ (q j )2n2 

if N is even

Π c N|V |

∑ π∈NC(Ωn )

V ∈π

2

!

 ∑ π∈NCe (Ωn )

Π c N|V |

V ∈π

if N is odd,

2

(1.128) (n )

for all n1 , n2 ∈ N, where tA 1 are in the sense of (1.125). Proof. The free-distributional data (1.128) is proven by (1.66), (1.126), and (1.127). The free-distributional data (1.128) illustrates how the integer-shift operator A ∈ A distorts the free ψ(q j )2 -Poisson distribution of Y jm,N on LQ , in general. As a special case of (1.128), one has the following corollary. Corollary 1.7. Let Y jm,1 be the free ψ(q j )2 -Poisson element (1.121) of LQ , and let A ∈ A be an integer-shift operator (1.112). Then,    n2   (n ) τ An1 Y jm,1 = δn2 ,2 tA 1 ψ(q j )4 , (n )

for all n1 , n2 ∈ N, where tA 1 are in the sense of (1.125).



REFERENCES [1] M. Ahsanullah, Some Inferences on Semicircular Distribution, J. Stat. Theo. Appl., 15, no. 3, (2016) 207–213. [2] H. Bercovici, and D. Voiculescu, Superconvergence to the Central Limit and Failure of the Cramer Theorem for Free Random Variables, Probab. Theo. Related Fields, 103, no. 2, (1995) 215–222. [3] M. Bozejko, W. Ejsmont, and T. Hasebe, Noncommutative Probability of Type D, Internat. J. Math., 28, no. 2, (2017) 1750010, 30. [4] M. Bozheuiko, E. V. Litvinov, and I. V. Rodionova, An Extended Anyon Fock Space and Non-commutative Meixner-Type Orthogonal Polynomials in the InfiniteDimensional Case, Uspekhi Math. Nauk., 70, no. 5, (2015) 75–120. [5] I. Cho, Semicircular Families in Free Product Banach ∗-Algebras Induced by p-Adic Number Fields over Primes p, Compl. Anal. Oper. Theo., 11, no. 3, (2017) 507–565. [6] I. Cho, Semicircular-Like Laws and the Semicircular Law Induced by Orthogonal Projections, Compl. Anal. Oper. Theo., 12, no.1, DOI:10.1007/s11785-018-0781-x, (2018). [7] I. Cho, Banach-Space Operators Acting on Semicircular Elements Induced by Orthogonal Projections, Compl. Anal. Oper. Theo., 13, Issue 8, (2019) 4065–4115. [8] I. Cho, Acting Semicircular Elements Induced by Orthogonal Projections on von Neumann Algebras, Mathematics, 5, 74, DOI:10.3390/math5040074, (2017).

Certain Banach-Space Operators

47

[9] I. Cho, Semicircular-Like and Semicircular Laws on Banach ∗-Probability Spaces Induced by Dynamical Systems of the Finite Adele Ring, Adv. Oper. Theo., Special Issue: Trends in Operators on Banach Spaces (dedicated to Prof. S. Banach), Adv. Oper. Theo., 4, no. 1, (2019) 24–70. [10] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by Projections on Separable Hilbert Spaces, Monograph Ser., Operator Theory: Advances & Applications, Published by Birkhauser, Basel, (2019) To Appear. [11] I. Cho, and P. E. T. Jorgensen, Banach ∗-Algebras Generated by Semicircular Elements Induced by Certain Orthogonal Projections, Opuscula Math., 38, no. 4, (2018) 501– 535. [12] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by p-Adic Number Fields, Opuscula Math., 35, no. 5, (2017) 665–703. [13] A. Connes, Noncommutative Geometry, ISBN: 0-12-185860-X, (1994) Academic Press (San Diego, CA). [14] P. R. Halmos, Hilbert Space Problem Books, Grad. Texts in Math., 19, ISBN: 9780387906850, (1982) Published by Springer. [15] I. Kaygorodov, and I. Shestakov, Free Generic Poisson Fields and Algebras, Comm. Alg., 46, issue 4, DOI:10.1080/00927872.2017.1358269, (2018). [16] L. Makar-Limanov, and I. Shestakov, Polynomials and Poisson Dependence in Free Poisson Algebras and Free Poisson Fields, J. Alg., vol. 349, issue 1, (2012) 372–379. [17] A. Nica, and R. Speicher, Lectures on the Combinatorics of Free Probability (1-st Ed.), London Math. Soc. Lecture Note Ser., 335, ISBN-13:978-0521858526, (2006) Cambridge Univ. Press. [18] I. Nourdin, G. Peccati, and R. Speicher, Multi-Dimensional Semicircular Limits on the Free Wigner Chaos, Progr. Probab., 67, (2013) 211–221. [19] V. Pata, The Central Limit Theorem for Free Additive Convolution, J. Funct. Anal., 140, no. 2, (1996) 359–380. [20] F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the C∗ Algebra of a Free Group of Nonsingular Index, Invent. Math., 115, (1994) 347–389. [21] F. Radulescu, Free Group Factors and Hecke Operators, notes taken by N. Ozawa, Proceed. 24-th Conference in Oper. Theo., Theta Advanced Series in Math., (2014) Theta Foundation. [22] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Amer. Math. Soc. Mem., vol 132, no. 627, (1998). [23] R. Speicher A Conceptual Proof of a Basic Result in the Combinatorial Approach to Freeness, Infinit. Dimention. Anal. Quant. Prob. & Related Topics, 3, (2000) 213–222. [24] R. Speicher, and T. Kemp, Strong Haagerup Inequalities for Free R-Diagonal Elements, J. Funct. Anal., 251, Issue 1, (2007) 141–173. [25] R. Speicher, and U. Haagerup, Brown’s Spectrial Distribution Measure for R-Diagonal Elements in Finite Von Neumann Algebras, J. Funct. Anal., 176, Issue 2, (2000) 331–367. [26] V. S. Vladimirov, p-Adic Quantum Mechanics, Comm. Math. Phy., 123, no. 4, (1989) 659–676. [27] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol 1, ISBN: 978-981-02-0880-6, (1994) World Scientific.

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[28] D. Voiculescu, Aspects of Free Analysis, Jpn. J. Math., 3, no. 2, (2008) 163–183. [29] D. Voiculescu, Free Probability and the Von Neumann Algebras of Free Groups, Rep. Math. Phy., 55, no. 1, (2005) 127–133. [30] D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1., ISBN-13: 978-0821811405, (1992) Published by Amer. Math. Soc.

Linear Positive Operators 2 Involving Orthogonal Polynomials P. N. Agrawal and Ruchi Chauhan IIT Roorkee

CONTENTS 2.1 Operators Based on Orthogonal Polynomials ................................................. 49 2.1.1 Notations.............................................................................................. 50 2.1.2 Definitions ........................................................................................... 50 2.1.3 Appell Polynomials ............................................................................. 51 2.1.4 Boas-Buck-Type Polynomials ............................................................. 53 2.1.5 Charlier Polynomials ........................................................................... 54 2.1.6 Approximation by Appell Polynomials............................................... 54 2.1.7 Approximation by Operators Including Generalized Appell Polynomials ......................................................................................... 55 ´ 2.1.8 Szasz-T ype Operators Involving Multiple Appell Polynomials .......... 55 2.1.9 Kantorovich-Type Generalization of Kn Operators ............................. 57 ´ Operators Based on Brenke-Type 2.1.10 Kantorovich Variant of Szasz Polynomials ......................................................................................... 60 2.1.11 Operators Defined by Means of Boas-Buck-Type Polynomials.......... 63 2.1.12 Operators Defined by Means of Charlier Polynomials........................ 66 2.1.13 Operators Defined by Using q-Calculus.............................................. 73 Acknowledgment ..................................................................................................... 74 References................................................................................................................ 75

2.1

OPERATORS BASED ON ORTHOGONAL POLYNOMIALS

In recent years, there has been a significant increase in activities in approximation of continuous functions on the semi-real axis by the linear positive operators based on orthogonal polynomials. Jakimovski and Leviatan [21] initiated the study in this ´ operators based on Appell polynodirection by defining the generalization of Szasz ´ operators by means of Sheffer polynomials. mials. Ismail [19] generalized the Szasz

49

50

Topics in Contemporary Mathematical Analysis and Applications

´ Varma and Tasdelen [42] proposed Szasz-type operators involving Charlier poly´ nomials. Kajla and Agrawal [24] introduced the Szasz-Durrme yer-type operators in´ volving Charlier polynomials. Varma et al. [41] proposed Szasz-type operators based ´ on Brenke-type polynomials. Sucu et al. [37] defined Szasz-type operators involving Boas-Buck polynomials that include Brenke-type polynomials, Sheffer polynomials, and Appell polynomials. In the literature, several sequences of linear positive operators have been defined, involving orthogonal polynomials, and their Durrmeyer and Kantorovich-type variants have been investigated. Our aim in this chapter is to make a survey of the research work available on approximation by the linear positive operators defined by using orthogonal polynomials. Throughout this chapter, we shall use the following notations and definitions. 2.1.1

NOTATIONS

R+ C[0, ∞) CB [0, ∞) E1 C˜ [0, ∞) C˜ r [0, ∞) C˜B [0, ∞)

(0, ∞), the space of all continuous functions on [0, ∞), the space of all continuous and bounded functions n o on [0, ∞), f : [0, ∞) → R : x ∈ [0, ∞), limx→∞

f (x) 1+x2

exists ,

the space of all uniformly continuous functions on [0, ∞), the space of r-times differentiable functions such that f (r) is uniformly continuous on [0, ∞), the space of all real-valued bounded and uniformly continuous functions f on [0, ∞), endowed with the norm k f k = sup | f (x)|, x∈[0,∞)

C2∗ [0, ∞)

{ f ∈ C[0, ∞) : | f (x)| ≤ M f (1 + xγ ), γ > 0}, the space of all real-valued functions on [0, ∞) satisfying the condition | f (x)| ≤ M f (1 + x2 ), where M f is a positive constant depending only on f and 1 + x2 is a weight function, the space of all continuous functions in B2 [0, ∞) with the norm k f k2 := | f (x)| sup 1 + x2 x∈[0,∞)   f (x)| f ∈ C2 [0, ∞) : limx→∞ |1+x is finite , 2

E2 E3

{ f : [0, ∞) → R : | f (x)| ≤ MeAx , for some A, M ∈ R+ },  R f : [0, ∞) → R : | f (x)| = | 0x f (s)ds| ≤ KeBx , for some B, K ∈ R+ .

Cγ [0, ∞) B2 [0, ∞)

C2 [0, ∞)

2.1.2

DEFINITIONS

Let f ∈ C˜ [0, ∞) and δ > 0. The modulus of continuity ω( f ; δ ) of the function f is defined by ω( f ; δ ) = sup sup | f (x + h) − f (x)|. (2.1) 0 0 such that n o √ K2 ( f , δ ) ≤ M ω2 ( f , δ ) + min(1, δ ) k f kC˜B [0,∞) , (2.3) where the second-order modulus of smoothness is defined as √ ω2 ( f , δ ) = sup√ sup | f (x + 2h) − 2 f (x + h) + f (x)|. 0 0 , (t + x) 2

(2.5)

for some M f > 0. For f ∈ C2 [0, ∞), Ispir and Atakut [20] introduced the weighted modulus of continuity as follows: Ω( f ; δ ) =

f (x + h) − f (x)| . 2 2 0≤|h| 1, and g(1) 6= 0, and established several approximation properties of these operators. ´ operators by means of ShefLater, Ismail [19] gave another generalization of Szasz ∞ k 6 0), and H(z) = ∑k=1 hk zk (hk = 6 0 fer polynomials. Let A(z) = ∑∞ k=0 ak z , (a0 = ∀k be the analytic functions in the disk |z| ≤ R (R > 1), where ak and hk are real. The Sheffer polynomials pk (x) have the generating functions of the form ∞ A(t)exH(t) = ∑k=0 pk (x)t k , |t | < R. By assuming pk (x) ≥ 0, ∀x ∈ [0, ∞), A(1) 6= 0, 0 and H (1) = 1, Ismail introduced and studied the convergence properties of the positive linear operators defined by   k e−nx ∞ Tn ( f ; x) = p (nx) f . ∑ k A(1) k=0 n In [8], Ciupa considered the following operators of general form Dn ( f ; x) =

e−nx ∞ ∑ pk (nx)Ak,n ( f ), x ≥ 0, g(1) k=0

where pk (x) are the Appell polynomials defined by the relation g(u)eux =

∞

∑ pk (u)uk , g(1) 6= 0. k=0

The main result of this paper is that the rth derivative of Dn ( f ; x) is given by   r nr ∞ −nx (r) i r (Dn ( f )) (x) = ∑ e pk (nx) ∑ (−1) i Ak+r−i,n ( f ). g(1) k=0 i=0 Choosing different expressions for Ak,n ( f ), the known formulas for the cases such as ´ operators, the Durrmeyer variant of the Jakimovski-Leviatan operators, Favard-Szasz

53

Linear Positive Operators

´ operators Jakimovski-Leviatan operators, and the Durrmeyer variant of Favard-Szasz may be obtained. Brenke-type polynomials [7] have the generating functions of the type ∞

A(t)B(xt) =

∑ pk (x)t k ,

(2.10)

k=0

where A(t) and B(t) are the analytic functions given by ∞

A(t) = ∑ ait i , a0 = 6 0

(2.11)

i=0

∞

B(t) =

6 0, ∀ j ∑ b jt j , b j =

(2.12)

j=0

and pk (x) = ∑ki=0 ak−i bi xi , k = 0, 1, 2..., which includes Appell polynomials for ak−i bi B(t) = et , as a particular case. Assuming that A(1) 6= 0, ≥ 0, 0 ≤ i ≤ k, A(1) k = 0, 1, 2..., B : [0, ∞) → (0, ∞) and (2.10)–(2.12) converge for |t| < R (R > 1), Verma et al. [40] proposed the following operators:   ∞ 1 k Ln ( f ; x) = p (nx) f , x ∈ [0, ∞) (2.13) k ∑ A(1)B(nx) k=0 n and studied some results concerning the rate of convergence of these operators. In particular, if B(t ) = et and A(t) = g(t), then the operators given by (2.13) reduce to the operators defined by (2.8). 2.1.4

BOAS-BUCK-TYPE POLYNOMIALS

´ operators based on Boas-BuckFor f ∈ C[0, ∞), Sucu et. al [37] considered the Szasz type polynomials as follows:   ∞ 1 k Bn ( f ; x) = p (nx) f , x ≥ 0, n ∈ N, ∑ k A(1)G(nxH(1)) k=0 n where the generating function of the Boas-Buck-type polynomials is given by ∞

A(u)G(xH (u)) =

∑ pk (x)uk , k=0

and A(u), G(u), and H(u) are the analytic functions described by A(u) = ∞ ∞ k k k ∑∞ k=0 ak u , (a0 6= 0, G(u) = ∑k=0 gk u , (gk 6= 0, ∀k), and H(u) = ∑k=1 hk u , (h1 6= 0, and studied some approximation properties of these operators. As a particular case, Boas-Buck polynomials include Sheffer polynomials and Appell polynomials. Yilik et al. [44] estimated the approximation degree of the operators Bn for bounded variation functions using some results of the probability theory. The au¨ thors also investigated the Voronovskaja and Gruss-V oronovskaja-type theorems in the quantitative form.

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2.1.5 CHARLIER POLYNOMIALS ´ [38] introduced the following linear positive operators: In 1950, Szasz −nx

Sn ( f ; x) = e

(nx)k ∑ k! f k=0 ∞

  k , n

(2.14)

where x ∈ [0, ∞) and f (x) is a continuous function on [0, ∞) whenever the above sum converges uniformly. These polynomials [18] have the generating functions of the form   ∞ tk t u (a) et 1 − = ∑ Ck (u) , |t| < a, a k! k=0

(2.15)

   r k 1 (−u) and (m)0 = 1, (m) j = m(m + 1)· · · (m + j − 1) r ∑ r a r=0 ´ for j ≥ 1. For CA [0, ∞), Varma and Tas¸delen [42] defined the following Szasz-type operators involving Charlier polynomials (a)

k

where Ck (u) =

−1

Ln ( f ; x, a) = e



1 1− a

(a−1)nx

(a)

C (−(a − 1)nx) f ∑ k k! k=0 ∞

  k , n

(2.16)

where a > 1 and x ∈ [0, ∞), and studied the degree of approximation of these operators. 2.1.6

APPROXIMATION BY APPELL POLYNOMIALS

For f ∈ CB [0, ∞), Karaisa [26] introduced a Durrmeyer-type modification based on Appell polynomials as follows: Ln ( f ; x) =

e−nx g(1) +

∞

pk (nx)

Z ∞

∑ B(n + 1, k)

k=1 −nx e

g(1)

0

bn,k (t) f (t)dt

a0 f (0), x ≥ 0,

(2.17) k−1

t where B(k + 1, n) is the beta function, bn,k (t ) = (1+t) n+k+1 , and studied the local approximation properties and the convergence in a weighted space of functions. A Voronovskaja-type theorem for these operators was also proved. Gupta and Agrawal [14] established the rate of convergence for a Lipschitz-type space and obtained the degree of approximation in terms of Lipschitz-type maximal function for the Durrmeyer-type modification of Jakimovski-Leviatan operators based on Appell polynomials.

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Linear Positive Operators

2.1.7 APPROXIMATION BY OPERATORS INCLUDING GENERALIZED APPELL POLYNOMIALS For f ∈ C[0, ∞), Icoz et al. [17] considered the linear operators   ∞ 1 k Mn ( f ; x) := ∑ pk (nx) f n , A(g(1))B(nxg(1)) k=0

(2.18)

where pk (x) are the generalized Appell polynomials [4] having the generating functions of the following form: ∞

A(g(t))B(xg(t)) =

∑ pk (x)t k

(2.19)

k=0

and A, B, and g are the analytic functions such that ∞

A(t) =

∞

∞

6 0), B(t) = ∑ akt k (a0 =

6 0), g(t) = ∑ bkt k (bk =

∑ gkt k

k=0

k=0

k=1

6 0). (g1 = (2.20)

Assuming that the generalized Appell polynomials (2.19) satisfy 6 0, g0 (1) = 1, pk (x) ≥ 0 k = 0, 1, 2, . . . , 1. A(g(1)) = 2. B : R → (0, ∞), 3. (2.19) and the power series (2.20) converges for |t| < R (R > 1), it turns out that the linear operators Mn defined by (2.18) are positive. Remark 2.1. Let g(t) = t, the operators (2.18) reduce to the operators Ln involving Brenke-type polynomials. Remark 2.2. Let g(t) = t and B(t ) = et , one can get the operators (2.8). In addition, if we choose A(t) = 1, we meet the well-known Szasz ´ operators (2.7). Icoz et al. [17] proved the qualitative and quantitative approximation theorems. Icoz et al. [17] obtained the qualitative convergence result by means of Mn operators with the help of universal Korovkin-type property with respect to positive linear operators. Next, they stated the quantitative results for estimating the error of approximation using the classical approach, the second modulus of continuity, and Peetre’s K-functional in the continuous functions space and the Lipschitz class. Neer et al. [32] defined Baskakov-Durrmeyer-type operators based on the generalized Appell polynomials and established some results in weighted approxima¨ tion besides the quantitative Voronovskaja and Gruss-V oronovskaja-type theorems. The approximation for functions having derivatives of bounded variation was also studied. 2.1.8

´ SZASZ-TYPE OPERATORS INVOLVING MULTIPLE APPELL POLYNOMIALS

First, let us recall the definition of multiple Appell polynomials [28]. A set of polynomials {pk1 ,k2 (x)}∞ k1 ,k2 =0 with degree (k1 + k2 ) for k1 , k2 ≥ 0, is called multiple

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polynomial system (multiple PS), and a multiple PS is called multiple Appell if it is generated by the relation ∞

A(t1 , t2 )ex(t1 +t2 ) =

∞

∑ ∑

k1 =0 k2 =0

pk1 ,k2 (x) k1 k2 t t , k1 !k2 ! 1 2

(2.21)

where A(t1 ,t2 ) is given by ∞

A(t1 , t2 ) =

∞

ak1 ,k2 k1 k2 t1 t2 , =0 k1 !k2 !

∑ ∑

k1 =0 k2

(2.22)

with A(0, 0) = a0,0 6= 0. Theorem 2.1. For multiple PS, { pk1 ,k2 (x)}∞ k1 ,k2 =0 , the following statements are equivalent: a. {pk1 ,k2 (x)}∞ k1 ,k2 =0 is a set of multiple Appell polynomials. b. There exists a sequence {ak1 ,k2 }∞ k1 ,k2 =0 with a0,0 6= 0 such that ∞ ∞    k1 k2 pk1 ,k2 (x) = ∑ ∑ ak1 −r1 ,k2 −r2 xr1 +r2 . r r 2 1 r1 =0 r2 =0 c.

For every k1 + k2 ≥ 1, we have p0k1 ,k2 (x) = k1 pk1 −1,k2 (x) + k2 pk1 ,k2 −1 (x).

Utilizing these polynomials, for any f ∈ C[0, ∞), Varma [40] defined a sequence of linear positive operators as   nx   p k ,k ∞ ∞ −nx 1 2 e k1 + k2 2 Kn ( f ; x) = f , (2.23) ∑ k1 !k2 ! A(1, 1) k∑ n =0 k =0 1

2

ak ,k

1 2 provided A(1, 1) 6= 0, A(1,1) ≥ 0 for k1 , k2 ∈ N, and series (2.21) and (2.22) converge for |t1 | < R1 , |t2 | < R2 (R1 , R2 > 1), respectively.

Remark 2.3. Let us note the following special cases: 1. For t2 = 0, the generating functions given by (2.21) reduce to the generating functions for the Appell polynomials given by (2.9). 2. For t2 = 0 and A(t1 , 0) = 1, from the generating function given by (2.21), we easily find pk (x) = xk . From this fact, one can obtain Szasz ´ operators (2.7). We use the following notation for the partial derivatives: ∂A ∂ 2A = Ati and Ati t j = i, j = 1, 2, 3, 4. ∂ti ∂ti ∂t j From definition (2.23) and relation (2.21), we obtain the following auxiliary result

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Linear Positive Operators

Lemma 2.1. [40] The operator (2.23) satisfies the following inequalities: 1. Kn (1; x) = 1; At1 (1,1)+At2 (1,1) ; nA(1,1)2(A (1,1)+A (1,1))  t1 t2 x 2 2 Kn (s ; x) = x + n 1 + A(1,1) 1 {At1 (1, 1) + At2 (1, 1) + At1 t1 (1, 1) + 2At1 t2 (1, 1) + At2 t2 (1, 1)}, + n2 A(1,1)

2. Kn (s; x) = x + 3.

where x ≥ 0. The following theorem shows that the operator defined by (2.23) is an approximation process for continuous functions contained in E. Theorem 2.2. [40] Let f ∈ C[0, ∞) ∩ E1 . Then, lim Kn ( f ; x) = f (x),

n→∞

the convergence being uniform in each compact subset of [0, ∞). Theorem 2.3. [40] Let f ∈ C˜ [0, ∞) ∩ E1 , we have the following inequality for the operator (2.23) |Kn ( f ; x) − f (x)| s   1 ≤ 1+ x+ {At (1, 1) + At2 (1, 1) + At1 t1 (1, 1) + 2At1 t2 (1, 1) + At2 t2 (1, 1)} nA(1, 1) 1   1 √ ×ω f; . n The following theorem shows that the operators (2.23) possess the simultaneous approximation. Theorem 2.4. [40] For f ∈ C˜ r [0, ∞) ∩ E1 , the following inequality holds: (r) |Kn ( f ; x) − f (r) (x)| s   1 ≤ 1+ x+ {At1 (1, 1) + At2 (1, 1) + At1 t1 (1, 1) + 2At1 t2 (1, 1) + At2 t2 (1, 1)} nA(1, 1)    r r (r) 1 ×ω f ; √ + + ω f (r) ; . n n n 2.1.9

KANTOROVICH-TYPE GENERALIZATION OF Kn OPERATORS

Also, Varma [40] gave a Kantorovich-type generalization of the operators Kn (2.23) with the same restrictions
Z k1 +k2 +1 n ne−nx ∞ ∞ pk1 ,k2 nx ∗ 2 Kn ( f ; x) = f (s)ds. (2.24) ∑ ∑ k1 +k2 A(1, 1) k =0 k =0 k1 !k2 ! n 1

2

In order to study the approximation properties of the operators Kn∗ , the first author gave a lemma for the moments of these operators:

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Lemma 2.2. [40] The operators (2.24) satisfy the following equalities: 1. Kn∗ (1; x) = 1; At1 (1,1)+At2 (1,1) 1 + 2n ; nA(1,1)   ( A ( 1 , 1 )+ A t t2 (1,1)) 1 Kn∗ (s2 ; x) = x2 + 2x 1 + n A(1,1) 1 {2(A (1, 1) + A (1, 1) + At1 t2 (1, 1))At1 t1 (1, 1) + At2 t2 (1, 1)}+ 3n12 ; + n2 A(1, t t 1 1)  2

2. Kn∗ (s; x) = x + 3.

1 4. Kn∗ (s3 ; x) = 4n3 A(1,1 )

36At1 (1, 1)+36At2 (1, 1)+14A(1, 1)+12At2 t2 (1, 1)+    2 2 12At1 t1 (1, 1)+24At1 t2 (1, 1) nx+ 48At1 (1, 1)+48At2 (1, 1)+ 72A(1, 1) n 4x  n3 x3 +32A(1, 1) 8 + 14At1 (1, 1)+14At2 (1, 1)+ 18At2 t2 (1, 1)+18At1 t1 (1, 1)+  36At1 t2 (1, 1)+4At2 t2 (1, 1)+4At1 t1 t1 (1, 1)+12At1 t2 t2 (1, 1)+12At1 t1 t2 (1, 1) ;   1 ∗ 4 5. Kn (s ; x) = n4 A(1,1) nx 2 12A(1, 1)+60At1 (1, 1)+60At2 (1, 1)+ 48At1 t1 (1, 1)+ 48At2 t2 (1, 1) +8At1 t1 t1 (1,  1) + 8At2 t2 t2 (1, 1) + 96At1 t2 (1, 1) + 12At1 t1 t2 (1, 1) + 2 2

12At1 t2 t2 (1, 1) + n 4x

54A(1, 1)+78At2 (1, 1)+78At2 (1, 1)+18At1 t1 (1, 1)+    n3 x 3 18At2 t2 (1, 1)+24At1 t2 (1, 1) + 8 64A(1, 1)+32At1 (1, 1)+32At2 (1, 1) + n4 x4 A(1, 1) + 4At1 (1, 1) + 4At2 (1, 1) + 13At1 t1 (1, 1) + 13At2 t2 (1, 1) + 9At1 t1 t1 (1, 1)+9A t2 t2 t2 (1, 1)+6At1 t1 t2 t2 (1, 1)+6At1 t1 t2 (1, 1)+6At1 t2 t2 +16At1 t1 +  6At1 t2 t2 + 15 .

In the following, the authors show that the uniform convergence of Kn∗ ( f ) to f is established. Theorem 2.5. [40] Let f ∈ C[0, ∞) ∩ E1 . Then, lim Kn∗ ( f ; x) = f (x),

n→∞

the convergence being uniform in each compact subset of [0, ∞). In the next theorem, the authors obtained the order of approximation Kn∗ ( f ) to f . Theorem 2.6. [40] Let f ∈ C˜ [0, ∞) ∩ E1 , we have the following inequality for the operator (2.24)

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Linear Positive Operators

|Kn∗ ( f ; x) − f (x)| 

s

1 1 {2(At1 (1, 1) + At2 (1, 1) + At1 t2 (1, 1)) + At1 t1 (1, 1) + At2 t2 (1, 1)} + nA(1, 1) 3n   1 ×ω f; √ . n

≤ 1+



x+

Lemma 2.3. [15] For all x ≥ 0 and n > 2, we have p Kn∗ (|s − x|; x) ≤ γn (x), where γn (x) = Kn∗ ((s − x)2 ; x). Theorem 2.7. [15] Let 0 < r ≤ 1 and f ∈ Lip∗M (r). Then for all x > 0 and n > 2, we have  r γn (x) 2 ∗ |Kn ( f ; x) − f (x)| ≤ M , x where γn (x) is defined as in Lemma 2.3. In order to study the estimate of the error in terms of the second-order modulus of continuity via Peetre’s K-functional, we define an auxiliary operator   At1 (1, 1) + At2 (1, 1) 1 ∗∗ ∗ + , (2.25) Kn ( f ; x) = Kn ( f ; x) + f (x) − f x + nA(1, 1) 2n Applying Lemma 2.2, we have Kn∗∗ (1; x) = 1 and Kn∗∗ (t; x) = x. Theorem 2.8. [15] Let f ∈ C˜B [0, ∞). Then for all x ≥ 0, the following inequality holds:

|Kn∗ ( f ; x) −

f (x)| ≤ Cω 2 ( f ,

p



 2(At1 (1, 1) + At2 (1, 1)) + A(1, 1) ψn (x)) + ω f ; , 2A(1, 1)n

  2  2(At1 (1,1)+At2 (1,1))+A(1,1) where ψn (x) = γn (x) + . 2A(1,1)n In order to obtain the next result, the authors adopt an approach of Steklov mean. For f ∈ C˜B [0, ∞), the Steklov mean is defined as fh (x) =

4 h2

Z 0

h 2

Z

h 2

[2 f (x + ξ + η) − f (x + 2(ξ + η))]d ξ dη.

0

The Steklov mean verifies the following inequalities: 1. k fh − f k ≤ ω2 ( f , h); 2. fh0 , fh00 ∈ C˜B [0, ∞) and k fh0 k ≤ 5n ω ( f , h), k fh00 k ≤

9 ω ( f , h). h2 2

(2.26)

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Topics in Contemporary Mathematical Analysis and Applications

Theorem 2.9. [15] Let f ∈ C˜B [0, ∞). Then, for every x ≥ 0, the following inequality holds: p p 13 |Kn∗ ( f ; x) − f (x)| ≤ 5ω( f , γn (x)) + ω2 ( f , γn (x)). 2 Theorem 2.10. [15] For any f ∈ CB1 [0, ∞) and x ∈ [0, ∞), we have p 2(At1 (1, 1) + At2 (1, 1)) + A(1, 1) + 2ω( f 0 , δ ) γn (x). |Kn∗ ( f , x) − f (x)| ≤ | f 0 (x)| 2A(1, 1)n Theorem 2.11. [15] Let f ∈ C˜B [0, ∞), then for every x ∈ (0, ∞), we have  r  µ ∗ |Kn ( f ; x) − f (x)| ≤ Cωτ f ; . n    2At1 (1,1)+2At2 (1,1)+2At1 t2 (1,1)At1 t1 (1,1)+At2 t2 (1,1) 1 where µ ≥ max 1, +3 is any real A(1,1) constant. In the next result, the authors establish a quantitative Voronovskaja-type theorem by means of the weighted modulus of continuity defined in (2.6). Theorem 2.12. Let f ∈ C2∗ [0, ∞) such that f 00 ∈ C2∗ [0, ∞). Then, the inequality   00 n(Kn∗ ( f ; x) − f (x)) − f 0 (x) At2 (1, 1) + At1 (1, 1) + 1 − x f (x) A(1, 1) 2 2   00 f (x) 2At1 (1, 1) + 2At2 (1, 1) + 2At1 t2 t2 (1, 1) + At1 t2 (1, 1) + At1 t1 t2 t2 (1, 1) 1 − + 2n A(1, 1) 3 √ = O(1)Ω( f 00 ; 1/ n), as n → ∞ holds true. ¨ Now the Gruss-V oronovskaja-type theorem for the operators Kn∗ , we state. Theorem 2.13. Let f 0 , g0 , f 00 , g00 ∈ C2∗ [0, ∞), then for every x ∈ [0, ∞) lim n(Kn∗ ( f g)(x) − Kn∗ ( f ; x)Kn∗ (g; x)) = x f 0 (x)g0 (x).

n→∞

Ansari et al. [3] introduced Jakimovski-Leviatan-Durrmeyer-type operators involving multiple Appell polynomials, and investigated Korovkin-type theorem and the order of convergence by means of the modulus of continuity. Some results in the weighted approximation were also discussed in this paper. 2.1.10

´ OPERATORS BASED ON KANTOROVICH VARIANT OF SZASZ BRENKE-TYPE POLYNOMIALS

´ operaFor f ∈ C[0, ∞), Varma et al. [41] defined another generalization of the Szasz tors by using Brenke-type polynomials, which are as follows:   ∞ 1 k Ln ( f ; x) = p (nx) f , (2.27) ∑ k A(1)B(nx) k=0 n

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Linear Positive Operators

where x ≥ 0 and n ∈ N. Tas¸delen et al. [39] introduced the Kantorovich modification of the operators (2.27), which is as follows: Kn ( f ; x) =

∞ n pk (nx) ∑ A(1)B(nx) k=0

Z (k+1)/n

f (t) dt,

(2.28)

k/n

where n ∈ N, x ≥ 0, and f ∈ C[0, ∞). If we take B(t) = et and A(t) = 1, then it ´ Mirakyan-Kantorovich operators defined by reduces to the Szasz (nx)k k=0 k! ∞

Kn ( f ; x) = ne−nx ∑

Z (k+1)/n

f (t) dt.

(2.29)

k/n

´ Mursaleen and Ansari [30] introduced a Chlodowsky-type generalization of Szasz operators defined by means of the Brenke-type polynomials, proved basic convergence theorem, and also established the order of approximation with the aid of the second-order modulus of continuity and the Peetre’s K-functional. ¨ uzer ¨ Oks et al. [33] estimated the rate of convergence of the operators (2.28) for the functions of bounded variations using the techniques of probability theory. The authors rewrite the operator (2.28) as: Z ∞

Kn ( f ; x) =

0

f (t) Mn (x,t)dt,

(2.30)

where Mn (x,t) =

∞ n pk (nx)χn,k (t) ∑ A(1)B(nx) k=0

χn,k (t) is the characteristic function of the interval [k/n, (k + 1)/n] with respect to I = [0, ∞). Lemma 2.4. [39] For the operator Kn , there hold the equalities 1. Kn (1; x) = 1; B0 (nx) aA0 (1) + A(1) 2. Kn (t; x) = + ; B(nx) 2nA(1) 00 0 0 B (nx) 2 2B (nx)[A (1) + A(1)] 3. Kn (t 2 ; x) = x + x nA(1)B(nx  B(nx)  ) 1 A(1) + 2 A00 (1) + 2A0 (1) + ; n A(1) 3  00  B (nx) − 2B0 (nx) + B(nx) 2 4. Kn ((t − x)2 ; x) = x + B(nx)  0  2A (1)[B0 (nx) − B(nx)] + A(1)[2B0 (nx) − B(nx)] +

1 n2 A(1)



A00 (1) + 2A0 (1) +



A(1) . 3

x

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Theorem 2.14. [39] Let B0 (y) = 1, y→∞ B(y) lim

B00 (y) = 1. y→∞ B(y) lim

If f ∈ C[0, ∞) ∩ E1 , then lim Kn ( f ; x) = f (x),

n→∞

and the operators Kn converge uniformly in each compact subset of [0, ∞). Next theorem gives the rate of convergence for the operator Kn . Theorem 2.15. [39] Let f ∈ C˜ [0, ∞) ∩ E, the operators Kn satisfy the following inequality:  p  |Kn ( f ; x) − f (x)| ≤ 2ω f ; δn (x) ,
where δn (x) = Kn (t − x)2 ; x . Theorem 2.16. [39] Let f ∈ CB2 [0, ∞). There holds the inequality |Kn ( f ; x) − f (x)| ≤ ζ || f ||C2 , B

where   0 B00 (nx) − 2B0 (nx) + B(nx) 2A (1) (B0 (nx) − B(nx)) + ζ = ζn (x) = 2B(nx) 2nA(1)B(nx)  0 A(1) (2(n + 1)B (nx) − (2n + 1)B(nx)) 2A0 (1) + A(1) + x+ 2nA(1)B(nx) 2nA(1)   1 A(1) + 2 A00 (1) + 2A(1) + . 2n A(1) 3 

Theorem 2.17. [39] Let f ∈ C˜B [0, ∞). There holds n o √ |Kn ( f ; x) − f (x)| ≤ 2 M ω2 ( f ; δ ) + min(1, δ )|| f ||CB , where

1 δ = δn (x) = ζn (x), 2 and M > 0 is a constant, which is independent of f and δ . Atakut and Buyukyazici [6] considered a generalization of the Kantorovich´ Szasz-type operators involving Brenke-type polynomials as follows: Lnαn ,βn ( f ; x) =

∞ βn ∑ pk (αn x) A(1)B(αn x) k=0

Z

k+1 βn k βn

f (t)dt,

(2.31)

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Linear Positive Operators

where αn and βn are strictly sequence of positive numbers such that  increasing  1 αn 1 limn→∞ = 0, = 1+O , as n → ∞ and pk (x), k = 1, 1, 2... are Brenke-type βn βn βn polynomials. B(k) (x) Assuming that limx→∞ = 1 as k = 1, 2, 3, 4, Atakut and Buyukyazici obB(x) tained the order of convergence of the operators given by (2.31). Subsequently, Garg et al. [12] investigated the degree of approximation of these operators by means of Peetre’s K-functional and the Ditizian-Totik modulus of smoothness. The rate of approximation of functions having derivatives equivalent with a function of bounded variation was also obtained. 2.1.11

OPERATORS DEFINED BY MEANS OF BOAS-BUCK-TYPE POLYNOMIALS

´ Sidharth et al. [36] introduced the Szasz-Durrme yer-type polynomials based on Boas-Buck-type polynomials and established a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. The approximation of functions whose derivative are locally of bounded variation are also discussed. For a function f ∈ Cγ [0, ∞), Sidharth et al. [36] defined the Sza´sz-Durrmeyer-type operators based on Boas-Buck-type polynomials as follows: Mn ( f ; x) =

∞ pk (nx) 1 ∑ A(1)G(nxH (1)) k=1 B(k, n + 1)

+

Z ∞ 0

t k−1 f (t)dt (1 + t)n+k+1

a0 b0 f (0), A(1)G(nxH(1))

(2.32)

where B(k, n + 1) is the beta function and x ≥ 0, n ∈ N. Lemma 2.5. [36] For the operator Mn , the following equalities hold: 1. Mn (1; x) = 1;   1 G0 (nxH(1)) A0 (1) 2. Mn (t; x) = nx + ; n G(nxH(1)) A(1)  00  0  1 G (nxH(1)) 2 2 A (1) 00 2 3. Mn (t ; x) = n x + 2 + H (1) + 2 n(n − 1) G(nxH(1))  A(1) 0 0 00 G (nxH(1)) A (1) A (1) × nx + 2 + ; G(nxH(1)) A(1) A(1)  000  0  1 G (nxH(1)) 3 3 A (1) 4. Mn (t 3 ; x) = n x + 3 + 6 + 3H 00 (1) n(n − 1)(n− 2) G(nxH (1)) A(1)  0 (1) G00 (nxH(1)) 2 2 A0 (1) A 00 00 000 × n x + 12 + H (1) + 3 H (1) + H (1) + 4 G(nxH(1)) A(1) A(1)  0 00 0 G (nxH(1)) A (1) A (1) × nx + 7 +6 ; G(nxH(1)) A(1) A(1)

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 0  iv 1 G (nxH(1)) 4 4 A (1) 5. Mn n x + 4 + 6H 00 (1) n(n − 1)(n − 2)(n − 3) G(nxH(1)) A(1)  000  00 G (nxH(1)) 3 3 A (1) A0 (1) 00 A0 (1) H 00 (1) + 12 n x + 6 + 12 H (1) + 21 +3 G(nxH(1)) A(1) A(1)  A(1)  00 A(1) G00 (nxH(1)) 2 2 A (1) 000 00 00 2 + 4H (1) + 18H (1) + 3(H (1)) + 21 n x + 4 G(nxH(1)) A(1) A00 (1) 00 A00 (1) 00 A0 (1) A0 (1) 000 A00 (1) +6 H (1) + 36 H (1) + 42 +4 H (1) + 36 A(1) A(1) A(1) A(1)  A0(1) iv (1) G ( nxH (1)) A A00 (1) + H iv (1) + 12H 000 (1) + 36H 00 (1) + 24 nx + + 48 G(nxH(1)) A(1) A(1)  A0 (1) + 13 + 11 . A(1) (t 4 ; x) =

Lemma 2.6. [36] For the operator (2.32), the following results hold:  0  G (nxH(1)) A0 (1) 1. Mn ((t − x); x) = −1 x+ ; G(nxH(1)) nA(1)   n G00 (nxH(1)) G0 (nxH(1)) 2. Mn ((t − x)2 ; x) = −2 + 1 x2 n − 1 G(nxH(1)) G(nxH (1))    0 1 A (1) G0 (nxH(1)) 2 A0 (1) 00 + 2 + H (1) + 2 − x n − 1  A(1) G(nxH (1)) n A(1)  1 A0 (1) A00 (1) + 2 + ; n(n − 1) A(1) A(1) n3 Giv (nxH(1)) (n − 1)(n − 2)(n − 3) G(nxH(1))  4n2 G000 (nxH(1)) 6n G00 (nxH(1)) G0 (nxH(1)) − + −4 + 1 x4 (n − 1)(n − 2) G(nxH(1)) (n − 1)G(nxH(1)) G(nxH(1   )) n2 G000 (nxH(1)) A0 (1) 00 + 4 + 6H (1) + 12 (n − 1)(n − 2)(n − 3) G(nxH(1)) A(1)  0  00 4n G (nxH(1)) A (1) 6 G0 (nxH(1)) 00 − 3 +3H (1)+6 + (n − 1)(n − 2) G(nxH (n − 1) G(nxH(1))  (1)) 0 A(1)  A0 (1) 4 A (1) n G00 (nxH(1)) × 2 +H 00 (1)+2 − x3 + n A(1) (n − 1)(n − 2)(n − 3) G(nxH(1))  A(1) A0 (1) 00 A0 (1) H 00 (1) A00 (1) × 6 + 12 H (1) + 21 +3 + 4H 000 (1) + 18H 00 (1) A(1) A(1 ) A(1) A(1)   4 G0 (nxH(1)) A0 (1) A0 (1) 00 +3(H 00 (1))2 +21 − 12 +H 00 (1)+3 H − 1)(n − 2) G(nxH(1)) A(1)  A(1)  (n  1 G0 (nxH(1)) A00 (1) (1) + H 00 (1) + 4 x2 + 4 (n − 1)(n − 2)(n − 3) G(nxH(1)) A(1) 00 00 0 0 A (1) 00 A (1) 00 A (1) A (1) 000 A00 (1) +6 H (1) + 36 H (1) + 42 +4 H (1) + 36 A(1) A(1) A(1) A(1) A(1)

3. Mn

((t − x)4 ; x) =



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Linear Positive Operators

  00  4 A (1) A0 (1) +H iv (1)+12H 000 (1)+36H 00 (1)+24 − 7 +6 x n(n − 1)(n − 2) A(1) A(1)  iv 1 A (1) A00 (1) A0 (1) + + 48 + 13 + 11 . (n − 1)(n − 2)(n − 3) A(1) A(1) A(1) Lemma 2.7. [36] For the operator (2.32), the equalities hold: 1.

0

(1) limn→∞ nMn ((t − x); x) = l1 (x)x + AA(1) ;

lim nMn ((t − x)2 ; x) = l2 (x)x2 + x(H 00 (1) + 2) = η(x), (say);  00 A (1) A0 (1) H 00 (1) 3. lim n2 Mn ((t − x)4 ; x) = l4 (x)x4 + l3 (x)x3 + 6 − 27 + + n→∞ A(1) A(1) A(1) 14H 00 (1) + 3(H 00 (1))2 + 5 = ν(x), (say).

2.

n→∞

The following theorem shows that the operators defined by (2.32) are an approximation process for f ∈ Cγ [0, ∞), using the Bohman-Korovkin theorem. Theorem 2.18. [36] Let f ∈ Cγ [0, ∞). Then, lim Mn ( f ; x) = f (x),

n→∞

holds uniformly in x ∈ [0, a], a > 0. Proof. From Lemma 2.5, it follows that lim Mn (t i ; x) = xi , i = 0, 1, 2

n→∞

uniformly in x ∈ [0, a]. Hence by Bohman-Korovkin theorem, the required result is immediate. Theorem 2.19. [36] Let f ∈ Cγ [0, ∞), admitting a derivative of second order at a point x ∈ [0, ∞), then there holds    A0 (1) 0 lim n(Mn ( f ; x) − f (x)) = l1 (x)x + f (x) + l2 (x)x2 (2.33) n→∞ A(1)  00 f (x) + x(H 00 (1) + 2) . (2.34) 2 If f 00 is continuous on [0, ∞), then the limit in (2.33) holds uniformly in x ∈ [0, a] ⊂ [0, ∞), a > 0. In 2018, Mursaleen et al. [31] considered a Chlodowsky variant of general´ ized Szasz-type operators involving Boas-Buck-type polynomials and studied some approximation properties of these operators in a weighted space of continuous functions on [0, ∞).

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2.1.12 OPERATORS DEFINED BY MEANS OF CHARLIER POLYNOMIALS ´ [38] introduced the following linear positive operators In 1950, Szasz   ∞ (nx)k k −nx Sn ( f ; x) = e ∑ k! f n , k=0

(2.35)

where x ∈ [0, ∞) and f (x) is a continuous function on [0, ∞) whenever the above sum converges uniformly. These polynomials [18] have the generating functions of the form   ∞ tk t u (a) et 1 − = ∑ Ck (u) , |t| < a, (2.36) a k! k=0    r k 1 (−u) and (m)0 = 1, (m) j = m(m + 1)· · · (m + j − 1) r ∑ r a r=0 ´ for j ≥ 1. For f ∈ C[0, ∞)∩E2 , Varma and Tas¸delen [42] defined the following Szasztype operators involving Charlier polynomials (a)

k

where Ck (u) =

−1

Ln ( f ; x, a) = e



1 1− a

(a−1)nx

(a)

C (−(a − 1)nx) f ∑ k k! k=0 ∞

  k , n

(2.37)

where a > 1 and x ∈ [0, ∞). For the special case, a → ∞ and x − 1n instead of x, these operators reduce to the operators (2.35). Lemma 2.8. [42] The operators (2.37) satisfy the following equalities: 1. Ln (1; x, a) = 1; 2. Ln (t; x, a) = x + 1n ;
3. Ln (t 2 ; x, a) = x2 + nx 3 + a−1 1 + n22 where x ≥ 0. Theorem 2.20. [42] Let f ∈ C[0, ∞) ∩ E2 , then lim Ln ( f ; x, a) = f (x),

n→∞

the sequence of operators given by (2.37) converges uniformly in each compact subset of [0, ∞). Theorem 2.21. [42] Let f ∈ C˜ [0, ∞) ∩ E2 . For the operators Ln given by (2.37), there holds the inequality: s  ( )    1 2 1 |Ln ( f ; x, a) − f (x)| ≤ 1 + x 1 + + ω f; √ . a−1 n n

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Linear Positive Operators

Also, Varma and Tas¸delen [42] considered Kantorovich-type generalization of the operators Ln ( f ; x, a) for a function f ∈ C¯ [0, ∞) :=



Z x  f ∈ C[0, ∞) : |F (x)| = f (s)ds ≤ KeBx , B ∈ R and K ∈ R+ 0

as follows : ∗ Ln,a ( f ; x) = ne−1



1 1− a

(a−1)nx

(a)

Ck (−(a − 1)nx) k! k=0 ∞

∑

k+1 n

Z k n

f (t)dt, (2.38)

∗ ( f ; x) to f on each where a > 1 and x ≥ 0, and studied the uniform convergence of Ln,a compact subset of [0, ∞) and the degree of approximation in terms of the classical modulus of continuity.

Lemma 2.9. [42] The operators given by (2.38) satisfy the following equalities: 1. Ln∗ (1; x, a) = 1; 3 2. Ln∗ (t; x, a) = x + 2n ;
10 ∗ 2 2 3. Ln (t ; x, a) = x + nx 4 + a−1 1 + 3n 2. Theorem 2.22. [42] Let f ∈ C[0, ∞) ∩ E3 , then lim L∗ ( f ; x, a) = n→∞ n

f (x),

holds uniformly in each compact subset of [0, ∞). Theorem 2.23. [42] Let f ∈ C˜ [0, ∞) ∩ E3 . For the operators Ln ∗, there holds the following inequality: s ( )     1 10 1 ∗ |Ln ( f ; x, a) − f (x)| ≤ 1 + x + 1 + + ω f; √ . a−1 3n n Further, for the operator Ln (2.37), Kajla and Agrawal [23] proved some approximation theorems. Let ei (x) = xi , i = 0, 1, 2, · · · Lemma 2.10. [23] For the operators Ln (es (t); x, a), s = 3, 4, we have     3 2x 1 3 5 x2 1. Ln (e3 (t ); x, a) = x3 + 6+ + 2 + + 5 + 3; n a − 1  n (a − 1)2 a − 1 n  x3 6 x2 30 11 4 2. Ln (e4 (t); x, a) = x + 10 + + 31 + + n a − 1 n2 a − 1 (a − 1)2  x 31 20 6 15 + 3 67 + + + + 4. 2 3 n a − 1 (a − 1) (a − 1) n Let exi (t ) = (t − x)i , i = 0, 1, 2 · · ·

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Lemma 2.11. [23] For the operators Ln ( f ; x, a), we have 1 1. Ln (ex1 (t); x, a) = ; n ax 2 2. Ln (ex2 (t); x, a) = + 2; n(a− 1) n  x 49 20 6 x 3. Ln (e4 (t); x, a) = 3 17 + − + n (a −1) (a − 1)2 (a − 1)3  2 x 46 3 15 + 2 19 − + + 4. n (a − 1) (a − 1)2 n Proof. Using ([42], p. 119, Lemma 1) and Lemma 2.10, the proof of this lemma easily follows. Hence, the details are omitted. To study the rate of convergence of functions having a derivative of bounded variation, let us rewrite the operators (2.37) as Ln ( f ; x, a) = where

  

Z ∞

f (w) 0

∂ {Kn (x, w, a)} dw, ∂w

(2.39)

(a)

(a−1)nx

Ck (−(a − 1)nx) , if 0 < w < ∞, ∑e Kn (x, w, a) = k!   k≤nw 0, if w = 0. From Lemma 2.11, for x ∈ (0, ∞) and sufficiently large n, we have r
1/2 λ (a)x x 2 Ln (|e1 (t)|; x, a) ≤ Ln ((t − x) ; x, a) ≤ , (2.40) n −1



1 1− a

where λ (a) is some positive constant depending on a. Also the authors get, for r ≥ 2 and fixed x ∈ [0, ∞), Ln (ex2r (t); x, a) = O(n−r ); n → ∞.

(2.41)

Lemma 2.12. [23] For all x ∈ (0, ∞) and sufficiently large n, we have

2.

Z t ∂

1 λ (a)x , 0 ≤ t < x; (x − t)2 n 0 Z ∞ ∂ 1 λ (a)x {Kn (x, w, a)} dw ≤ 1 − ϑn,a (x, z) = , x < z < ∞, 2 (z − x) n z ∂w

1. ϑn,a (x,t) =

∂w

{Kn (x, w, a)} dw ≤

where λ (a) is a constant as described in (2.40). Proof. First, we prove (i). ϑn,a (x,t) =

Z t ∂ 0

∂w

{Kn (x, w, a)} dw ≤

Z t 0

x−w x−t

2

∂ {Kn (x, w, a)} d w ∂w

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Linear Positive Operators

1 Ln ((w − x)2 ; x, a) (x − t)2 1 λ (a)x ≤ . (x − t)2 n

≤

The proof of (ii) is similar, and hence, it is omitted. Theorem 2.24. [23] Let f ∈ Lip∗M (r) and r ∈ (0, 1]. Then, for all x ∈ (0, ∞), we have 

ζn,a (x) | Ln ( f ; x, a) − f (x) |≤ M a1 x2 + a2 x

r

2

,

where ζn,a (x) = Ln ((t − x)2 ; x, a). 2 2 and q = , we find that r 2−r   2 r/2    (a) r 1 (a−1)nx ∞ Ck (−(a − 1)nx) k | Ln ( f ; x, a) − f (x) |≤ e−1 1 − f − f (x) ∑ a k! n k=0  2 k   (a−1)nx ∞ (a) r/2 −x Ck (−(a − 1)nx) 1 n −1
. ≤M e 1− ∑ k 2 a k! k=0 n + a1 x + a2 x Proof. Applying the Ho¨lder’s inequality with p =

Since f ∈ Lip∗M (r) and q

1 k n

+ a1 x2 + a2 x

| Ln ( f ; x, a) − f (x) | ≤

=

1

0, we have |Ln ( f ; x, a) − f (x)| = 0. n→∞ x∈[0,∞) (1 + x2 )1+β lim sup

Proof. For any fixed x0 > 0, |Ln ( f ; x, a) − f (x)| |Ln ( f ; x, a) − f (x)| |Ln ( f ; x, a) − f (x)| ≤ sup + sup 2 )1+β 2 )1+β (1 + x (1 + x (1 + x2 )1+β x≤x0 x≥x0 x∈[0,∞) sup

|Ln (1 + t 2 ; x, a)| (1 + x2 )1+β x≥x0

≤k Ln ( f ) − f kC[0,x0 ] + k f k2 sup

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Linear Positive Operators

| f (x)| , (1 + x2 )1+β x≥x0

+ sup

=: I1 + I2 + I3 , say.

(2.44)

Since | f (x)| ≤ || f ||2 (1 + x2 ), we have I3 = sup x≥x0

| f (x)| || f ||2 || f ||2 ≤ sup ≤ . (1 + x2 )1+β x≥x0 (1 + x2 )β (1 + x02 )β

Let ε > 0 be arbitrary. Mf ε < , 3 (1 + x02 )γ

(2.45)

In view of ( [42], Theorem 1), there exists n1 ∈ N such that   |Ln (1 + t 2 ; x, a)| 1 ε 2 k f k2 < k f k2 (1 + x ) + , ∀n ≥ n1 3|| f ||2 (1 + x2 )1+β (1 + x2 )1+β k f k2 ε < + , ∀n ≥ n1 . (2.46) 2 β 3 (1 + x ) |Ln (1 + t 2 ; x, a)| || f ||2 ε < + , ∀n ≥ n1 . 2 2 1+β β 3 (1 + x ) (1 + x0 ) x≥x0 2|| f ||ϕ ε Thus, I2 + I3 < + , ∀n ≥ n1 . (1 + x02 )β 3 k f kϕ ε Now, let us choose x0 to be so large that < . 2 β 6 (1 + x ) Then,

Hence, || f ||2 sup

2ε , ∀n ≥ n1 . 3 By Theorem (2.28), there exists n2 ∈ N such that ε I1 =k Ln ( f ) − f kC[0,x0 ] < , ∀n ≥ n2 . 3 I2 + I3
0. δ Hence applying Cauchy-Schwarz inequality, we get |Ln ( f ; x, a) − f (x)| ≤ 4M f (1 + x2 )Ln ((t − x)2 ; x, a)   1 + ωb+1 ( f , δ ) 1 + Ln (|t − x|; x, a) δ   q 1 ≤ 4M f (1 + x2 )ζn,a (x) + ωb+1 ( f , δ ) 1 + ζn,a (x) δ   q 1 ≤ 4M f (1 + b2 )ζn,a (b) + ωb+1 ( f , δ ) 1 + ζn,a (b) . δ p Choosing δ = ζn,a (b), we get the desired result. Further, Kajla and Agrawal [25] considered Kantorovich-type generalization of the operators L n ( f ; x, a) for a function R f ∈ C˜ [0, ∞) := f ∈ C[0, ∞) : |F(x)| = | 0x f (s)ds| ≤ KeBx , B ∈ R and K ∈ R+ , and ∗ established some more approximation properties of the operators Ln,a such as weighted approximation, A-statistical convergence, and approximation of functions with a derivative of bounded variation. They also presented some moment estimates and a result needed to study approximation of functions with derivatives of bounded variation. They discussed the main results of this paper wherein we establish approximation in a Lipschitz-type space, weighted approximation theorems, and A∗ . Lastly, they obtained the statistical convergence properties for the operators Ln,a rate of convergence for functions having a derivative of bounded variation on every finite subinterval of [0, ∞), for these operators. In 2016, Agrawal and Ispir [2] introduced a new bivariate operator associated ´ with a combination of Chlodowsky and generalized Szasz-Charlier -type operators as follows:     n ∞ x kαn j Tn,m ( f ; x, y, a) = ∑ ∑ pn,k sm, j (βmy , a) f , . αn n γm k=0 j=0 for all n, m ∈ N, f ∈ C(Aαn ) with Aαn = {(x, y) : 0 ≤ x ≤ αn , 0 ≤ y ≤ and C(Aαn ) = { f : Aαn → R is continuous}. Here, αn is an unbounded
quence of positive numbers such that limn→∞ αnn = 0, and also γm and βm

∞} sede

note the unbounded sequences of positive numbers such that βγmm = 1 + O γ1m .   n−k
  Also, the basic elements are pn,k αxn = nk αxn 1 − αxn and sm, j (βm y, a) = (a )
(a−1)βm y C j (−(a−1)βm y) (a) e−1 1 − a1 , where C j (u) is Charlier polynomial and a > 1. j!
For the special case, a → ∞ and y − mx instead of y, the operators Tn,m reduce to

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Linear Positive Operators

the operators studied by Gazanfer and Buyukyazici [13]. They give the degree of approximation for these bivariate operators by means of the complete and partial modulus of continuity, and also by using weighted modulus of continuity. Furthermore, they construct a GBS (generalized Boolean sum) operator of bivariate Chlodowsky´ Szasz-Charlier type and estimate the order of approximation in terms of mixed modulus of continuity. Motivated by the above development, Wafi et al. [43] defined a new sequence ´ of Kantorovich-Szasz-type operators which preserves constant and quadratic test functions, i.e., e0 (x) and e2 (x), where ei (x) = xi , i = 0,2 ∗   1 (a−1)nrn,a (x) ∞ ∗ ∗ Kn,a ( f ; rn,a (x)) = ne−1 1 − ∑ a k=0

(a)

∗ (x)) Z k+1 Ck (−(a − 1)nrn,a n f (t)dt k k! n q

2

(2.49)  q 10

 3 = and ≥ 0 for x ∈ , ∞ . 2n n q  10  ∗ (x) → x, as n → ∞, and the operators (2.49) reduce to For a fixed x ∈ n 3 , ∞ , rn,a

where

∗ (x) rn,a

1 + (−4+ a−1 ) (4+ a−1 1 )

+4(n2 x2 − 10 3 )

∗ (x) rn,a

operators (2.38). Wafi et al. [43] discussed the rate of convergence by determining better error estimates. Further, the authors investigated the order of approximation by means of local approximation results with the help of Ditzian-Totik modulus of smoothness, second-order modulus of continuity, Peetre’s K-functional, and Lipschitz class. 2.1.13

OPERATORS DEFINED BY USING Q-CALCULUS

First, we shall give some definitions. Following [5,22], for any fixed real number q > 0, satisfying the condition 0 < q < 1, the q-integer [k]q , for k ∈ N, and q-factorial [k]q ! are defined as   (1 − qk ) 6 , if q =1 [k]q = (1 − q)  k, if q = 1, and  [k]q ! =

[k]q [k − 1]q . . . 1, if k ≥ 1 1, if k = 0,

respectively. For any integers n, k satisfying 0 ≤ k ≤ n, the q-binomial coefficient is given by   [n]q ! n = . k q [n − k]q ![k]q !

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Topics in Contemporary Mathematical Analysis and Applications

The q-analogue of (1 − x)n is given by   n−1 1 − q j x
, n = 1, 2, ... ∏ n (1 − x)q =  j=0 1, n = 0. The q-integration in the interval [0,a] is defined by Z a 0

∞

f (t)dqt = a(1 − q) ∑ f (aqn )qn

0 < q < 1,

n=0

provided the series converges. ´ operator Karaisa [27] introduced a Stancu-type generalization of the q-Favard-Szasz as follows: −[n] t ∞

α,β Tn,t ( f ; q, x) =

Eq q A(1)

  Pk (q; [n]qt) [k]q + α ∑ [k]q ! f x + [n]q + β , k=0

where Pk (q; .) ≥ 0 for each k is a q-Appell generated by B(u)e[n]q tu =

Pk (q; [n]qt )uk [k]q ! k=0 ∞

∑

and B(u) is defined by ∞

B(u) =

∑ ak uk , k=0

and studied Korovkin-type statistical approximation properties and rate of convergence using modulus of continuity. He also obtained some local approximation results for these operators. For f ∈ Cγ [0, ∞), Agrawal and Gupta [1] defined the q-analogue of the operators (2.17) defined in [26] as follows: Ln ( f ; q, x) =

k(k−1) Pk (q; [n]q x) E −[n]q x ∞ q 2 ∑ g(1) k=0 [k]q !Bq (n + 1, k)

Z ∞ 0

t k−1 f (qk t)dqt, (1 + t )qn+k+1

and obtained the rate of convergence in terms of the weighted modulus of continuity and a Lipschitz-type maximal function for these operators.

ACKNOWLEDGMENT Many thanks to our TEX-pert for developing this class file.

Linear Positive Operators

75

REFERENCES ´ operators based on [1] P. N. Agrawal and P. Gupta, Durrmeyer variant of q-Favard-Szasz Appell polynomials. Creat. Math. Inform. 26(1) (2017) 9–7. ´ [2] P. N. Agrawal, and N. Ispir, Degree of approximation for bivariate Chlodowsky-SzaszCharlier type operators. Results Math., 69(3–4) (2016) 369–385. [3] K. J. Ansari, M. Mursaleen and S. Rahman, Approximation by Jakimovski-Leviatan operators of Durrmeyer type involving multiple Appell polynomials. RACSAM 113 (2019) 1007–1024. [4] P. E. Appell, Sur une classe de polynomes. Annales scientifique de 1’E.N.S. 2(9) (1880) 119–144. [5] A. Aral, V. Gupta and R. P. Agarwal, Application of q-Calculus in Operator Theory. Springer, New York (2013). ¨ uk ¨ yazici, Approximation by Kantorovich-Szasz ´ type operators [6] C¸. Atakut and I. Buy based on Brenke type polynomials. Numer. Funct. Anal. Optim. 37(12) (2016), 1488– 1502. [7] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, NewYork, 1978. [8] A. Ciupa, Positive linear operators obtained by means of Appell polynomials. in Approximation and optimization, Vol. II (Cluj-Napoca, 1996), 63–68, Transilvania, Cluj. [9] R. A. Devore and G. G. Lorentz, Constructive Approximation. Springer, Berlin (1993). [10] A. D. Gadjiev., On P. P. Korovkin type theorems. Math. Zametki 20(5) (1976) 781–786. [11] A. D. Gadjiev, R. O. Efendiyev and E. Ibikli, On Korovkin type theorems in space of locally integrable functions. Czechoslovak Math. J. 53(128)(1) (2003) 45–53. ´ type [12] T. Garg, P. N. Agrawal and S. Araci, Rate of convergence by Kantorovich-Szasz operators based on Brenke type polynomials. J. Inequal. Appl. 2017 (156). ¨ uk ¨ yazici, Approximation by certain linear positive operators [13] A. F. Gazanfer and I. Buy of two variables. Abstr. Appl. Anal. 2014, Art. ID 782080 (2014). [14] P. Gupta and P. N. Agrawal, Jakimovski-Leviatan operators of Durrmeyer type involving Appell polynomials. Turk. J. Math. 42 (2018) 1457–1470. ¨ Voronovskaja-type [15] P. Gupta and P. N. Agrawal, Quantitative Voronovskaja and Gruss theorems for operators of Kantorovich type involving multiple Appell polynomials. Iran. J. Sci. Technol. Trans. Sci. 43 (2019) 1679–1687. [16] E. Ibikli and E. A. Gadjiev, The order of approximation of some unbounded function by the sequences of positive linear operators. Turk. J. Math. 19(3) (1995) 331–337. ¨ S. Varma and S. Sucu, Approximation by operators including generalized [17] G. ˙Ic¸oz, Appell polynomials, Filomat 30 (2) (2016) 429–440. [18] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005). ´ operators. Mathematica (Cluj), 39 (1974) [19] M. E. H. Ismail, On a generalization of Szasz 259–267. ´ [20] N. Ispir and C. ¸ Atakut, Approximation by modified Szasz-Mirakjan operators on weighted space. Proc. Indian Acad. Sci. Math. Sci. 112(4) (2002) 571–578. ´ operators for the approximation in [21] A. Jakimovski and D. Leviatan, Generalized Szasz the infinite interval. Mathematica (Cluj), 34 (1969) 97–103. [22] V. Kac and P. Cheung, Quantum Calculus. Universitext, Springer, New York (2002). ´ type operators based [23] A. Kajla and P. N. Agrawal, Approximation properties of Szasz on Charlier polynomials. Turk. J. Math. 39 (2015) 990–1003.

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´ [24] A. Kajla and P. N. Agrawal, Szasz-Durrme yer type operators based on Charlier polynomials. Appl. Math. Comput. 268 (2015) 1001–1014. ´ [25] A. Kajla and P. N. Agrawal, Szasz-Kantoro vich type operators based on Charlier polynomials. Kyungpook Math. J. 56(3) (2016) 877–897. [26] A. Karaisa, Approximation by Durrmeyer type Jakimovski-Leviatan operators. Math. Meth. Appl. Sci. 39(9) (2016) 2401–2410. ´ oper[27] A. Karaisa, D. Turgut and Y. Asar, Stancu type generalization of q-Favard-Szasz ators. Appl. Math. Comput. 264 (2015) 249–257. [28] D. Lee, On multiple Appell polynomials. Proc. Amer. Math. Soc. 139(6) (2011), 2133–2141. [29] B. Lenze, On Lipschitz type maximal functions and their smoothness spaces. Nederl. Akad. Indag. Math. 50 (1) (1988) 53–63. ´ operators by Brenke [30] M. Mursaleen and K. J. Ansari, On Chlodowsky variant of Szasz type polynomials. Appl. Math. Comput. 271 (2015) 991–1003. [31] M. Mursaleen, A. H. Al-Abied and A. M. Acu, Approximation by Chlodowsky type ´ operators based on Boas-Buck type polynomials. Turk. J. Math. 42 (2018) of Szasz 2243–2259. [32] T. Neer, A. M. Acu and P. N. Agrawal, Baskakov-Durrmeyer type operators involving generalized Appell polynomials. Math. Meth. Appl. Sci. 2019: 1–13, http://doi.org/10.1002/mma.6089. ¨ Oks ¨ uzer ¨ , H. Karsli and F. Tas¸delen, Approximation by Kantorovich variant of Szasz ´ [33] O. operators based on Brenke type polynomials. Mediterr. J. Math. 13 (2016) 3327–3340. ¨ [34] M. A. Ozarslan and O. Duman, Local approximation behaviour of modified SMK operators. Miskolc Math. Notes. 11(1) (2010) 87–99. [35] I. M. Sheffer, Some properties of polynomial sets of type zero. Duke Math. J. 5 (1939) 590–622. ´ [36] M. Sidharth, P. N. Agrawal and S. Araci, Szasz-Durrme yer operators involving BoasBuck polynomials of blending type. J. Inequal. Appl. 2017, 127. ¨ and S. Varma, On some extension of Szasz [37] S. Sucu, G. ˙Ic¸oz ´ operators including BoasBuck type polynomials, Absr. Appl. Anal. volume 2012. Hindawi Publishing Corporation, (2012). ´ Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. [38] O. Szasz, Nat. Bur. Standards 45 (1950) 239–245. ´ operators includ[39] F. Tas¸delen, R. Aktas¸, and A. Altin, A Kantorovich type of Szasz ing Brenke type polynomials , Abs. Appl. Anal. Artical Number 867203 (2012) DOI: 10.1155/2012/867203. ´ operators by multiple Appell polynomials. Stu[40] S. Varma, On a generalization of Szasz dia Universitatis Babes-Bolyai, Mathematica 58(3) (2013) 361–369. ´ operators involving Brenke [41] S. Varma, S. Sucu and G. Icoz, Generalization of Szasz type polynomials, Comput. Math. Appl. 64(2) (2012) 121–127. ´ type operators involving Charlier polynomials. Math. [42] S. Varma and F. Tas¸delen, Szasz Comput. Modelling, 56 (2012) 118–122. ´ type oper[43] A. Wafi, N. Rao and Deepmala, On Kantorovich form of generalized Szasz ators using Charlier polynomials. Korean J. Math. 25(1) (2017) 99–116. ¨ O ¨ Yilik, T. Garg and P. N. Agrawal, Convergence rate of Szasz ´ operators involving [44] O Boas-Buck type polynomials. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., DOI 10.1007/s40010-020-00663-3.

by 3 Approximation Kantorovich Variant of λ −Schurer Operators and Related Numerical Results ¨ er Faruk Ozg ˙Izmir Katip C ¸ elebi University

Kamil Demirci and Sevda Yıldız Sinop University

CONTENTS 3.1 Introduction ..................................................................................................... 77 3.2 Auxiliary Results ............................................................................................. 80 3.3 Approximation Behavior of λ -Schurer-Kantorovich Operators ..................... 82 3.4 Voronovskaja-type Approximation Theorems................................................. 87 3.5 Graphical and Numerical Results.................................................................... 90 3.6 Conclusion....................................................................................................... 93 References................................................................................................................ 93

3.1

INTRODUCTION

Functions have been widely approximated by positive linear operators over the past decades. Sergei Natanovich Bernstein first used the known polynomials in the approximation theory to prove Weierstrass’ well-known theorem [6]. Bernstein polynomials of order n are given by n

    n i i z (1 − z)n−i h i n i=0

Bn (h; z) = ∑

(z ∈ [0, 1]),

(3.1)

for any continuous function h(z) defined on C[0, 1], the space of all real-valued continuous functions on [0, 1] endowed with the norm khkC[0,1] = supz∈[0,1] |h(z)|. The Bernstein basis polynomials play an important role in approximations of functions, computer-aided geometric design, differential equations, numerical analysis, ´ and constructing Bezier curves [12,13,15,21,22]. 77

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A shape parameter λ ∈ [−1, 1] was used to construct new Bernstein bases by Ye et al. in [30] since shape parameter λ provides more modeling flexibility. Shape parameter λ was also used by Cai et al. [7] to construct λ -Bernstein operators:   n i ˜ Bn,λ (h; z) = ∑ bn,i (λ ; z) h . (3.2) n i=0 Considering two nonnegative parameters α and β , which satisfy 0 ≤ α ≤ β , the α ,β λ -Stancu operators Sn (h; z; λ ) : C[0, 1] −→ C[0, 1] were defined by Srivastava et al. [23] as   n i+α ˜ Snα,β (h; z; λ ) = ∑ h bn,i (λ ; z), n+β i=0 for any n ∈ N. Many approximation properties of the bivariate variant of these operators were also investigated. The direct approximation theorem with the help of second-order modulus of smoothness was established, and the rate of convergence was calculated via Lipschitz-type function. Srivastava et al. also constructed the bivariate case of Stancu-type λ -Bernstein operators and studied their approximation behaviors. Kantorovich modified the well-known Bernstein polynomials (3.1) to approximate the Lebesgue integrable functions on the interval [0, 1] in [14]. Acu et al. [3] introduced the following integral modification of Bernstein operators: n

Kn,λ (h; z) = (n + 1) ∑ b˜ n,i (λ ; z)

Z

i=0

i+1 n+1 i n+1

(3.3)

h(t)dt.

They generalized both Bernstein (3.1) and λ -Bernstein (3.2) operators and obtained some asymptotic type results. Also, Cai proposed λ -Kantorovich operators (3.3) and ´ the following Bezier variant of Kantorovich-type λ -Bernstein operators [8]: n

(β )

Ln,λ ,β (h; z) = (n + 1) ∑ Qn,i (λ ; z) i=0

Z

i+1 n+1 i n+1

h(t)dt,

where h iβ h iβ n (β ) (β ) (β ) (β ) Qn,i (λ ; z) = Jn,i (λ ; z) − Jn,i+1 (λ ; z) and Jn,i (λ ; z) = ∑ b˜ n, j (λ ; z). j=i

´ Note that Bezier variants of Kantorovich-type λ -Bernstein operators reduce to the λ -Kantorovich operators (3.3) when β = 0. Approximation properties of λ -Kantorovich operators with shifted knots were introduced by Rahman et al. [24] as   Z i+α+1 n n+β n n+β +1 (α ,β ) (α ,β ) Kn,λ (h; z) = (n + β + 1) ∑ bˆ n,i (λ ; z) i+α h(t)dt. n i=0 n+β +1

Kantorovich Variant of λ −Schurer Operators

79

A family of GBS operators of bivariate tensor product of λ -Kantorovich type was constructed in [9]. An estimate for the rate of convergence of such operators was given for B-continuous and B-differentiable functions. Also, a Voronovskaja-type asymptotic formula was established for the bivariate case. Many researchers established some Kantorovich-type operators by modifying Bernstein-type operators to have better error estimation [16,24,26]. More recently, statistical approximation properties of univariate and bivari¨ ate λ -Kantorovich operators were considered by Ozger in [28]. The rate of weighted A-statistical convergence was estimated. A Voronovskaja-type approximation theorem by a family of linear operators was proved using the notion of weighted A-statistical convergence. Some estimates for differences of λ -Bernstein and λ -Durrmeyer, and λ -Bernstein and λ -Kantorovich operators were given. Finally, a Voronovskaja-type approximation theorem by weighted A-statistical convergence was established for the bivariate case of operators. Acu et al. extended λ -Bernstein operators to introduce a new type genuine Bernstein-Durrmeyer operators in [4]: ρ Un,λ (h; z) =

1" n−1 Z t iρ−1 (1 − t)(n−i)ρ−1

∑

i=1

0

B(iρ , (n − i)ρ )

# h(t)dt b˜ n,i (λ ; z)

+ h(0)b˜ n,0 (λ ; z) + h(1)b˜ n,n (λ ; z). Here, B(a, b) is Euler’s beta function and b˜ n,i (λ ; z) is the new Bernstein base, which was introduced by Ye et al. in [30]. ´ In [29], Qi et al. constructed λ -Szasz-Mirakian operators   n i Mn,λ (h; z) = ∑ m˜ n,i (λ ; z) h , z ∈ [0, ∞) , n i=0 where λ mn+1,1 (z), n+α +1 λ m˜ n,i (λ ; z) = mn,i (z) + 2 [(n − 2i + 1)mn+1,i (z) − (n − 2i − 1)mn+1,i+1 (z)] , n −1 λ m˜ n,n (λ ; z) = mn,n (z) − mn+1,n (z), n+1

m˜ n,0 (λ ; z) = mn,0 (z) −

and mn,,i (z) = e−nz

(nz)i . i!

The Schurer polynomials sn,i (z) were introduced by Frans Schurer in [20] as   n+α i sn,i (z) = z (1 − z)n+α −i (i = 0, 1, . . . , n + α), i

(3.4)

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where α is a nonnegative integer. The operators generated by these polynomials are called Schurer operators, which were introduced to extend the domain of function from C[0, 1] to C[0, 1 + α]. Some relevant works about the Schurer polynomials and Schurer operators may be found in [5,17–19]. The bases in [30] were modified by adding parameter α to introduce the following new bases in [27]: λ sn+1,1 (z), n+α +1 λ s˜n,i (λ ; z) = sn,i (z) + [(n + α − 2i + 1)sn+1,i (z) (n + α)2 − 1 −(n + α − 2i − 1)sn+1,i+1 (z)] (i = 1, 2 . . . , n + α − 1),

s˜n,0 (λ ; z) = sn,0 (z) −

s˜n,n+α (λ ; z) = sn,n+α (z) −

λ sn+1,n+α (z), n+α +1

(3.5)

where shape parameter λ ∈ [−1, 1]. In the same work, the λ -Schurer operators were introduced, and some approximation and statistical approximation properties were studied. Also, an estimate for the rate of weighted A-statistical convergence was obtained. Moreover, two Voronovskaja-type theorems, including a Voronovskaja-type approximation theorem using weighted A-statistical convergence, were proved. The rest of this chapter is organized as follows. In Section 3.2, we construct the λ -Schurer-Kantorovich operators, and find moments and central moments. Section 3.3 investigates the approximation behavior of defined λ -Schurer-Kantorovich operators (3.6) and obtains a global and a local direct estimate for the rate of convergence. Section 3.4 establishes three Voronovskaja-type theorems, including a quantitative type. Section 3.5 analyzes the approximation behavior of the defined operators by supporting theoretical results with numerical experiments and graphs. Finally, Section 3.6 summarizes our work and gives some suggestions for readers to carry out further after reading the work.

3.2

AUXILIARY RESULTS

Considering a given nonnegative integer α, we now introduce λ -Schurerλ : C[0, 1 + α] −→ C[0, 1] for any n ∈ N as Kantorovich operators Kn,α n+α λ Kn,α (h; z) = (n + α + 1)

∑

Z

s˜n,i (λ ; z)

i=0

i+1 n+α +1 i n+α+1

h(t)dt,

(3.6)

where new Schurer polynomials s˜n,i (λ ; z) are given in (3.5). Based on the above definition, we reveal the special cases of our new Schurer operators. Remark 3.1. We have the following results: a.

The λ -Schurer-Kantorovich operators (3.6) are reduced to the classical Schurer-Kantorovich operators if λ = 0.

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81

b.

The λ -Schurer-Kantorovich operators (3.6) are reduced to the classical Bernstein-Kantorovich operators if α = λ = 0. c. The λ -Schurer-Kantorovich operators (3.6) are reduced to the λ -BernsteinKantorovich operators if α = 0.

Now, we find moments and central moments of λ -Schurer-Kantorovich operators (3.6). First, we recall the following lemma. Lemma 3.1. [27] We have the following results for λ -Schurer operators: λ Sn,α (1; z) = 1, λ Sn,α (t; z) =

n+α 1 − 2z + zn+α+1 − (1 − z)n+α +1 z+ λ, n n(n + α − 1)

(n + α)2 2 n + α z + 2 (z − z2 ) n2 n 2(n + α )z − 1 − 4(n + α)z2 + (2(n + α) + 1)zn+α +1 + (1 − z)n+α +1 + λ. n2 (n + α − 1)

λ Sn,α (t 2 ; z) =

Lemma 3.2. Let λ ∈ [−1, 1] and α be a nonnegative integer, then the moments of λ -Schurer-Kantorovich operators are as follows: λ Kn,α (1; z) = 1, λ Kn,α (t; z) = λ Kn,α (t 2 ; z) =

1 + 2(n + α)z 1 − 2z + zn+α +1 − (1 − z)n+α+1 + λ, 2(n + α + 1) (n + α)2 − 1 (n + α )2 2 n+α 1 + 6zλ z + z(2 − z) + (n + α + 1)2 (n + α + 1)2 3(n + α + 1)2 +

2zn+α+1 4(n + α)z2 λ − λ. (n + α)2 − 1 (n + α + 1)((n + α)2 − 1)

Proof. Proof of the first part of the lemma is provided by the partition of unity property of Schurer and Kantorovich operators. Bearing in mind the definition of operators (3.6) and Lemma 3.1, we have n+α λ Kn,α (t; z) = (n + α + 1)

Z

∑ s˜n,i (λ ; z)

i=0 n+α

=

i+1 n+α+1 i n+α+1

t dt

2i + 1

∑ s˜n,i (λ ; z) 2(n + α + 1)

i=0

=

n 1 Sλ (t; z) + Sλ (1; z) n + α + 1 n,α 2(n + α + 1) n,α

=

1 + 2(n + α)z 1 − 2z + zn+α +1 − (1 − z)n+α +1 + λ, 2(n + α + 1) (n + α)2 − 1

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which completes the proof of the second part. Now, we prove the third part: n+α λ Kn,α (t 2 ; z) = (n + α + 1)

∑

i=0 n+α

=

i+1 n+α +1

Z

s˜n,i (λ ; z)

i n+α+1

t 2 dt

3i2 + 3i + 1

∑ s˜n,i (λ ; z) 3(n + α + 1)2

i=0

=

n2 n 1 Sλ (1; z) Sλ (t; z) + Sλ (t 2 ; z) + 3(n + α + 1)2 n,α (n + α + 1)2 n,α (n + α + 1)2 n,α

=

(n + α )2 2 n+α 1 + 6zλ z + z(2 − z) + (n + α + 1)2 (n + α + 1)2 3(n + α + 1)2 +

2zn+α+1 4(n + α )z2 λ. λ − (n + α)2 − 1 (n + α + 1)((n + α )2 − 1)

Thus, Lemma 3.2 is proved. Corollary 3.1. Let z ∈ [0, 1], α be a nonnegative integer, λ ∈ [−1, 1], and ψz = t − z, then we have the following central moments: λ Kn,α (ψz ; z) =

1 − 2z 1 − 2z + zn+α +1 − (1 − z)n+α +1 + λ, 2(n + α + 1) (n + α)2 − 1

λ Kn,α (ψz2 ; z) =

z − z2 1 + 6zλ + n + α + 1 3(n + α + 1)2 −

z(1 − 2z) + zn+α+1 (z − 1) − z(1 − z)n+α +1 2λ (n + α)2 − 1

−

4(n + α )z2 λ. (n + α + 1)((n + α)2 − 1)

λ (ψ ; z) and Corollary 3.2. The following relations hold for operators Kn,α z λ (ψ 2 ; z): Kn,α z

1 − 2z , 2 λ lim n Kn,α (ψz2 ; z) = z − z2 .

lim n Knλ,α (ψz ; z) =

n→∞

n→∞

3.3

APPROXIMATION BEHAVIOR OF λ -SCHURER-KANTOROVICH OPERATORS

The following theorem gives the uniform convergence property of λ -SchurerKantorovich operators (3.6) by the well-known Bohman-Korovkin-Popoviciu theorem: Theorem 3.1. Let h ∈ C[0, 1 + α], then λ lim Kn,α (h; z) = h(z)

n→∞

uniformly on [0, 1].

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83

Proof. As stated in the Bohman-Korovkin-Popoviciu theorem, it is sufficient to show that λ lim Kn,α (t i ; z) = zi , i = 0, 1, 2 n→∞

uniformly on [0, 1]. Using Lemma 3.2, we can easily get these three conditions. Hence, we have the result. We need the following notions and notation to achieve a local direct estimate for the rate of convergence of λ -Schurer-Kantorovich operators (3.6) and obtain a global approximation formula. Definition 3.1. Global approximation formulas via Ditzian-Totik uniform modulus of smoothness of first and second orders are defined by ωξ (h, δ ) := sup

sup

{|h(z + ρξ (z)) − h(z)|};

0 0, such that √ √ β β Ω−1 ω2 (h, δ ) ≤ K2,β (z) (h, δ ) ≤ Ωω2 (h, δ ).

(3.7)

First, we obtain the global approximation formula in terms of Ditzian-Totik uniform modulus of smoothness of first and second orders. Theorem 3.2. Let β (z) (β 6= 0) be an admissible step-weight function of modulus of smoothness in Definition 3.1 such that β 2 is concave and let ψz = t − z, h ∈ C[0, 1 + α], z ∈ [0, 1], and λ ∈ [−1, 1]. Then for Ω > 0, λ -SchurerKantorovich operators (3.6) verify λ |Kn,α (h; z) − h(z)| ≤

Ω

β ω2

 h,

[χn,α,λ (z) + χ¯ n,2 α ,λ (z)]1/2 2[(z − a)(b − z)]1/2



 + ωξ

 χ¯ n,α ,λ (z) h, , ξ (z)

λ (ψ ; z) and χ λ 2 where χ¯ n,α,λ (z) = Kn,α z n,α,λ (z) = Kn,α (ψz ; z) are given in Corollary 3.1.

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Proof. We define the following operators: λ λ K˘ n,α (h; z) = Kn,α (h; z) + h(z)  1 + 2(n + α )z 1 − 2z + zn+α+1 − (1 − z)n+α+1  −h + λ 2(n + α + 1) (n + α)2 − 1

(3.8)

where h ∈ C[0, 1 + α], z ∈ [0, 1], and λ ∈ [−1, 1]. We have the following relations λ λ (t; z) = x (1; z) = 1 and K˘ n,α K˘ n,α λ (ψ ; z) = 0. by Lemma 3.2. These relations imply K˘ n,α z Let u = ρz + (1 − ρ)t, ρ ∈ [0, 1]. Since β 2 is concave on [0, 1], it follows that β 2 (u) ≥ ρβ 2 (z) + (1 − ρ)β 2 (t) and

|t − u| ρ|z − t| |ψz | ≤ ≤ 2 . 2 2 2 β (u) ρβ (z) + (1 − ρ)β (t) β (z)

(3.9)

Hence, the following inequalities hold: λ λ λ (g; z) − g(z)| + |h(z) − g(z)| (h − g; z)| + |K˘ n,α (h; z) − h(z)| ≤ |K˘ n,α |K˘ n,α

(3.10)

λ (g; z) − g(z)|. ≤ 4kh − gkC[0,1+α] + |K˘ n,α

Applying Taylor’s formula, we obtain λ |K˘ n,α (g; z) − g(z)| (3.11)  Z z+χ¯  Z t n,α ,λ (z) λ |t − u| |g00 (u)|du ; z + z + χ¯ n,α ,λ (z) − u |g00 (u)| du ≤ Kn,α z z   Z t |t − u| λ 2 00 ≤ kβ 2 g00 kC[0,1+α ] Kn,α du z β 2 (u) ; z + kβ g kC[0,1+α] Z z+χ¯ ¯ n,α,λ (z) − u| n,α,λ (z) |z + χ × du β 2 (u) z

≤ β −2 (z)kβ 2 g00 kC[0,1+α] Knλ,α (ψz2 ; z) + β −2 (z)kβ 2 g00 kC[0,1+α] χ¯ n,2 α,λ (z). By definition of K-functional with relation (3.7) and inequalities (3.10)–(3.11), we have
λ (h; z) − h(z)| ≤ 4kh − gkC[0,1+α] + β −2 (z)kβ 2 g00 kC[0,1+α] χn,α,λ (z) + χ¯ n,2 α ,λ (z) |K˘ n,α   2 [χn,α,λ (z) + χ¯ n,α,λ (z)]1/2 β ≤ Ω ω2 h , . 2β (z) Also, by the help of uniform modulus of smoothness of first order in Definition 3.1, we have     χ¯ n,α,λ (z) χ¯ n,α,λ (z) |h(z + χ¯ n,α,λ (z)) − h(z)| = h z + ξ (z) − h(z) ≤ ωξ h, . ξ (z) ξ (z)

Kantorovich Variant of λ −Schurer Operators

85

Therefore, the following inequalities, which complete the proof, hold: λ |Knλ,α (h; z) − h(z)| ≤ |K˘ n,α (h; z) − h(z)| + h(z + χ¯ n,α,λ (z)) − h(z)    2 1/2  [χn,α,λ (z) + χ¯ n,α χ¯ n,α,λ (z) ,λ (z)] β . ≤ Ω ω2 h, + ωξ h, ξ (z) 2[(z − a)(b − z)]1/2

Theorem 3.3. Let h, h0 ∈ C[0, 1 + α] and z ∈ [0, 1], then the following inequality is satisfied: q q
λ |Kn,α (h; z) − h(z)| ≤ |χ¯ n,α,λ (z)| |h0 (z)| + 2 χn,α,λ (z)w h0 , χn,α,λ (z) , where χn,α,λ (z) and χ¯ n,α,λ (z) are given in Theorem 3.2. Proof. We have the following relation h(t) − h(z) = ψz h0 (z) +

Z t

(h0 (u) − h0 (z))du

(3.12)

z

λ (h; z) to both sides of (3.12), we have for any t, z ∈ [0, 1]. Applying operators Kn,α λ λ Knλ,α (h(t) − h(z); z) = h0 (z)Kn,α (ψz ; z) + Kn,α

Z

t

 (h (u) − h (z))du; z . 0

0

z

The following inequality holds for any δ > 0, u ∈ [0, 1], and h ∈ C[0, 1 + α]:   |u − z| |h(u) − h(z)| ≤ w(h, δ ) +1 , δ With above inequality, we get Z t   2 (h0 (u) − h0 (z))du ≤ w(h0 , δ ) ψz + |ψz | . z δ Hence, we have   1 λ λ λ λ (ψz ; z) . (h; z) − h(z)| ≤ |h0 (z)| |Kn,α (ψz ; z)| + w(h0 , δ ) Kn,α (ψz2 ; z) + Kn,α |Kn,α δ Applying Cauchy-Schwarz inequality on the right-hand side of above inequality, we have  q q 1 λ λ (ψ 2 ; z) + 1 λ (|ψ |; z). |Kn,α (h; z) − h(z)| ≤ h0 (z)|χ¯ n,α,λ (z)| + w(h0 , δ ) Kn,α Kn,α z z δ

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In the following theorem, we obtain a local direct estimate of the rate of converλ . gence via Lipschitz-type function involving two parameters for the operators Kn,α Before proceeding further, let us recall that (k ,k2 )

LipM1

n (η) := h ∈ C[0, 1] : |h(t) − h(z)| ≤ M

|ψz |η η

(k1 z2 + k2 z + t) 2

o ; z ∈ (0, 1],t ∈ [0, 1]

for k1 ≥ 0, k2 > 0, where η ∈ (0, 1] and M is a positive constant (see [25]). (k ,k2 )

Theorem 3.4. Let λ ∈ [−1, 1], z ∈ (0, 1], and η ∈ (0, 1] h ∈ LipM1 s λ (h; z) − h(z)| ≤ M |Kn,α

χnη,α ,λ (z) (k1 z2 + k2 z)η

(η), then

,

where χn,α,λ (z) is defined in Theorem 3.2. (k ,k )

Proof. Let h ∈ LipM1 2 (η) and η ∈ (0, 1]. First, we show that the statement holds for η = 1. Since we can express λ -Schurer-Kantorovich operators as n+α λ Kn,α (h; z) =

(n + α + 1)

Z 1 

∑ s˜n,i (λ ; z)

i=0

0

 i+t h dt, n+α +1

we have λ |Kn,α (h; z) − h(z)| ≤ |Kn,λ α (|h(t ) − h(z)|; z)| + h(z) |Knλ,α (1; z) − 1|  n+α  i+t − h(z) s˜n,i (λ ; z) ≤ ∑ h n+α +1 i=0 n+α

≤M

∑

i=0 (k ,k2 )

for h ∈ LipM1

i+t | n+α+1 − z| 1

(k1 z2 + k2 z + t ) 2

s˜n,i (λ ; z)

(1). By using

(k1 z2 + k2 z + t)−1/2 ≤ (k1 z2 + k2 z)−1/2 (k1 ≥ 0, k2 > 0) and applying Cauchy-Schwarz inequality, we obtain |Kn,λ α (h; z) − h(z)| ≤ M(k1 z2 + k2 z)−1/2

n+α

i+t

i=0 λ = M (k1 z + k2 z) |Kn,α (ψz ; z)| 1/2 2 ≤ M|χn,α,λ (z)| (k1 z + k2 z)−1/2 . 2

−1/2



∑ n + α + 1 − z s˜n,i (λ ; z)

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λ (h; z) and applyHence, the statement is true for η = 1. By the monotonicity of Kn,α ¨ ing Holder’ s inequality two times with a = 2/η and b = 2/(2 − η), we can see that the statement is true for η ∈ (0, 1], as follows: n+α  i + t  λ − h(z) s˜n,i (λ ; z) Kn,α (h; z) − h(z) ≤ ∑ h n+α +1 i=0 2  η  n+α  n+α    2−η 2 2 η i+t s˜n,i (λ ; z) − h(z) s˜n,i (λ ; z) ≤ ∑ h ∑ n+α +1 i=0 i=0
2 η  n+α i+t 2 α +1 − z s˜n,i (λ ; z) ≤ M ∑ n+i+t 2 i=0 n+α+1 + k1 z + k2 z  n+α  2 η 2 i+t ≤ M (k1 z2 + k2 z)−η /2 ∑ − z s˜n,i (λ ; z) n+α +1 i=0 h iη 2 λ ≤ M (k1 z2 + k2 z + t )−η/2 Kn,α (ψz2 ; z) s χn,η α ,λ (z) =M . (k1 z2 + k2 z)η

3.4

VORONOVSKAJA-TYPE APPROXIMATION THEOREMS

¨ In this section, we establish a quantitative Voronovskaja-type and a Grussλ (h; z) using modulus of smoothness, which is Voronovskaja type theorem for Kn,α given in Definition 3.1. This modulus is defined as       h z + ρβ (z) − h z − ρβ (z) , z ± ρ β (z) ∈ [0, 1] . ωβ (h, δ ) := sup 2 2 2 0 0 , g∈Wβ [0,1+α]

where Wβ [0, 1 + α] = {g : g ∈ ACloc [0, 1 + α], kβ g0 k < ∞} and ACloc [0, 1 + α] is the class of absolutely continuous functions defined on [a, b] ⊂ [0, 1 + α]. There is a constant Ω > 0 so that Kβ (h, δ ) ≤ Ω ωβ (h, δ ). Theorem 3.5. Let h ∈ C2 [0, 1 + α], then we have   λ Kn,α (h; z) − h(z) − χ¯ n,α,λ (z)h0 (z) − χn,α,λ (z) h00 (z) ≤ Ω β 2 (z)ωβ h00 , √1 n 2 n

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for every z ∈ [0, 1] and sufficiently large n, where Ω is a positive constant, and χn,α,λ (z) and χ¯ n,α,λ (z) are defined in Theorem 3.2. Proof. Consider the following equality: h(t) − h(z) − ψz h0 (z) =

Z t

(t − u)h00 (u)du

z

for h ∈ C[0, 1 + α]. It means we have h(t) − h(z) − ψz h0 (z) −

ψz2 00 h (z) = 2

Z t

(t − u)[h00 (u) − h00 (z)]du.

(3.13)

z

λ (h; z) to both sides of (3.13), we obtain Applying Kn,α

λ Knλ,α (ψz2 ; z) 00 λ 0 Kn,α (h; z) − h(z) − Kn,α (ψz ; z)h (z) − h (z) 2  Z t  λ 00 00 ≤ Kn,α |t − u| |h (u) − h (z)| du ; z .

(3.14)

z

The expression in the right-hand side of (3.14) can be estimated as Z t |t − u| |h00 (u) − h00 (z)| du ≤ 2kh00 − gkψz2 + 2kβ g0 kβ −1 (z)|ψz |3 ,

(3.15)

z

where g ∈ Wβ [0, 1 + α]. There exists Ω > 0 such that λ Kn,α (ψz2 ; z) ≤

Ω 2 β (z) 2n

and

λ Kn,α (ψz4 ; z) ≤

Ω 4 β (z) 2n2

(3.16)

for sufficiently large n. Using Cauchy-Schwarz inequality, we have λ (ψ 2 ; z) λ Kn,α z λ 0 00 Kn,α (h; z) − h(z) − Kn,α h (z) (ψz ; z)h (z) − 2 λ λ ≤ 2kh00 − gkKn,α (ψz2 ; z) + 2kβ g0 kβ −1 (z)Kn,α (|ψz |3 ; z) Ω λ λ ≤ (z − z2 )kh00 − gk + 2kβ g0 kβ −1 (z){Kn,α (ψz2 ; z)}1/2 {Kn,α (ψz4 ; z)}1/2 n n o Ω ≤ β 2 (z) kh00 − gk + n−1/2 kβ g0 k n

by (3.14)–(3.16). Taking infimum on the right-hand side over all g ∈ Wβ [0, 1 + α], we deduce   λ Kn,α (h; z) − h(z) − χ¯ n,α,λ (z)h0 (z) − χn,α ,λ (z) h00 (z) ≤ Ω β 2 (z)ωβ h00 , √1 . n 2 n The result follows immediately by applying Corollaries 3.1 and 3.2.

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As an immediate consequence of Theorem 3.5, we have the following result. Corollary 3.3. Let h ∈ C2 [0, 1 + α], then   χn,α ,λ (z) 00 λ 0 h (z) = 0, lim n Kn,α (h; z) − h(z) − χ¯ n,α,λ (z)h (z) − n→∞ 2 where χn,α,λ (z) and χ¯ n,α,λ (z) are defined in Theorem 3.2. ¨ inequality for the positive linear operators was first given by Acu et al. in A Gruss [2] by using the least concave majorant of the modulus of continuity. Then, Acar et al. ¨ ¨ [1] obtained a Gruss-type approximation theorem and a Gruss-V oronovskaja-type theorem for a class of sequences of linear positive operators. Now, we will obtain a λ : ¨ Gruss-V oronovskaja-type theorem for λ -Schurer-Kantorovich operators Kn,α Theorem 3.6. Let h, k ∈ C2 [0, 1 + α]. Then, for each z ∈ [0, 1], n o
λ λ lim n Kn,α ((hk) ; z) − Knλ,α (h; z) Kn,α (k; z) = h0 (z) k0 (z) z − z2 . n→∞

Proof. It can be easily seen that the following equality holds: λ λ λ Kn,α ((hk) ; z) − Kn,α (h; z) Kn,α (k; z)

χn,α ,λ (z) λ = Kn,α ((hk) ; z) − h (z) k (z) − (hk)0 (z) χ n,α,λ (z) − (hk)00 (z) 2   χ (z) n,α ,λ λ − k (z) Kn,α (h; z) − h (z) − h0 (z) χ n,α,λ (z) − h00 (z) 2   χn,α,λ (z) λ λ − Kn,α (h; z) Kn,α (k; z) − k (z) − k0 (z) χ n,α,λ (z) − k00 (z) 2 o χn,α,λ (z) n λ + h (z) k00 (z) + 2h0 (z) k0 (z) − k00 (z) Kn,α (h; z) 2 n o λ (h; z) . + χ n,α,λ (z) h (z) k0 (z) − k0 (z) Kn,α

Hence, by using Theorem 3.1 and Corollary 3.3, we have n o λ λ λ ((hk) ; z) − Kn,α (h; z) Kn,α (k; z) lim n Kn,α n→∞ n oχ χn,α,λ (z) n,α,λ (z) λ = lim nk00 (z) h (z) − Kn,α (h; z) + lim 2nh0 (z) k0 (z) n→∞ n→∞ 2 2 n o 0 λ + lim nk (z) h (z) − Kn,α (h; z) χ n,α,λ (z) n→∞
= h0 (z) k0 (z) z − z2 .

Finally, we obtain the following theorem by Taylor’s expansion theorem and Lemma 3.2, and Corollaries 3.1 and 3.2:

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Theorem 3.7. Let h ∈ C2 [0, 1 + α], then for each z ∈ [0, 1]  λ h0 (z) h00 (z) lim n Kn,α (h; z) − h(z) = (1 − 2z) + (z − z2 ) n→∞ 2 2 uniformly on [0, 1]. Proof. We first write the following equality by Taylor’s expansion theorem of function h(z) in C[0, 1]: 0 00 1 h(t) = h(z) + ψz h (z) + ψz2 h (z) + ψz2 rz (t), 2

(3.17)

where rz (t) is Peano form of the remainder, rz ∈ C[0, 1] and rz (t) → 0 as t → z. λ (·; z) to identity (3.17), we have Applying the operators Kn,α 00

0 λ λ Kn,α (h; z) − h(z) = h (z)Kn,α (ψz ; z) +

h (z) λ λ Kn,α (ψz2 ; z) + Kn,α (ψz2 rz (t); z). 2

Using Cauchy-Schwarz inequality, we have λ (ψz2 rz (t ); z) ≤ Kn,α

q

q λ (ψ 4 ; z) K λ (r 2 (t); z). Kn,α n,α z z

(3.18)

λ (r 2 (t); z) = 0 and hence We observe that limn Kn,α z λ lim n{Kn,α (ψz2 rz (t); z)} = 0.

n→∞

Thus, 00  h (z) λ λ Kn,α (ψz2 ; z) (h; z) − h(z)} = lim n Knλ,α (ψz ; z)h0 (z) + lim n{Kn,α n→∞ n→∞ 2 λ + Kn,α (ψz2 rz (t); z) .

3.5

GRAPHICAL AND NUMERICAL RESULTS

The aim of this section is to support the results given in the previous sections by graphs and numerical examples. We first study on the behavior of polynomials s˜n,i (λ ; z) for different values of shape parameter λ . In Figure 3.1, we demonstrate the classical Bernstein polynomials and λ -Schurer polynomials s˜n,i (λ ; z) to see the difference.

Kantorovich Variant of λ −Schurer Operators

91

(a)

(b)

(c)

(d)

Figure 3.1 Behavior of polynomials bn,i (z) and s˜n,i (λ ; z) for n = 3 and i = 0, 1, . . . , 3. (a) b3,i (z) for i = 0, 1, . . . , 3, (b) s˜3,i (λ ; z) with λ = 1 and α = 1, (c) s˜3,i (λ ; z) with λ = −1 and α = 1, and (d) s˜3,i (λ ; z) with λ = 0.5 and α = 1.

λ (h; z) = h (z) − K λ (h; z) Example 3.1. Let α = 2, λ = 0.5, h (z) = cos (πz), and En,α n,α be the error function of λ -Schurer-Kantorovich operators. The graphs of operators λ for n = 20, n = 50, n = 100 and the graph of the function h are illustrated in Kn,α λ are given in Figure 3.2b. Hence, we show the Figure 3.2a. The error functions En,α convergence and error of approximation of λ -Schurer-Kantorovich operators to the function h. Also, we see the error estimation from Table 3.1.

(a)

a=2 , l=0.5

0.2 Exact h(z) n=20 n=50 n=100

n=20 n=50 n=100

0.15 Error

t

0.5 0 −0.5 −1 0

a=2 , l=0.5

(b)

1

0.1 0.05

0.2

0.4

0.6 z

0.8

1

0 0

0.2

0.4

0.6

0.8

1

z

Figure 3.2 Convergence of our operators to h (z) = cos (πz). (a) Approximation process and (b) error of approximation.

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Table 3.1 The Errors of the Approximation n 20 50 100

α 2 2 2

Error 1.42e−01 6.05e−02 3.09e−02

λ 0.5 0.5 0.5




Example 3.2. Let n = 10, h (z) = z − 12 z − 83 z − 14 . In Figure 3.3a, we see 1 the graphs of λ -Schurer-Kantorovich operators Kn,1 2 , λ -Kantorovich operators Kn,0 0 that are given by Acu et al.[3], and K10,0 that are the classical Kantorovich operaλ are given in Figure 3.3b. tors and the graph of function h. The error functions En,α Also, we see the error estimation from Table 3.2. Hence, we show that the error for 1 is smaller than the ones for K 1 and K 0 . λ -Schurer-Kantorovich operators Kn,2 n,0 10,0

(a)

(b)

n=10 0.3 Exact h(z) a=0 , l=0 a=0 , l=1 a=2 , l=1

0.04 Error

t

0.2 0.1 0 −0.1 0

n=10 0.05 a=0 , l=0 a=0 , l=1 a=2 , l=1

0.03 0.02 0.01

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

z

Figure 3.3 Convergence of our operators to h (z) = z − 12 tion process and (b) error of approximation.




z − 83




0.8

1


z − 14 . (a) Approxima-

Table 3.2 Comparison of Operators via Errors n 10 10 10

0.6 z

α 0 0 2

λ 0 1 1

Error 4.76e−02 4.76e−02 4.09e−02

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93

When we compare the numerical results of this section and the results in [3,7,8,27], we see that our operators have less error of approximation.

3.6

CONCLUSION

One of the main contributions of our work is to approximate the Lebesgue integrable functions on the interval [0, 1] more precisely. This is achieved by considering new Schurer polynomials defined in [27] instead of the polynomials defined in [7]. We provide a comprehensive literature review about λ -Bernstein-type operators, Bernstein operators, Schurer operators, and Schurer and Bernstein polynomials. We ¨ obtain some Voronovskaja-type theorems, including a Gruss-V oronovskaja and a quantitative Voronovskaja-type theorem for λ -Schurer-Kantorovich operators. We explore the convergence of our operators, and we see that our new operators have several advantages. As it is theoretically and numerically shown, the new operators defined in this chapter generalize the results in [3,5–9,17,20,27]. We investigate some approximation properties and show the relevance of the results by examples, and hence, we get better error estimation for our newly defined operators. This study may be extended by considering q and (p, q) analogues of λ -Schurer, λ -Schurer-Kantorovich, λ -Schurer-Stancu, and λ -Schurer-Durrmeyer operators after introducing λ -Schurer-Durrmeyer and λ -Schurer-Stancu operators. All these operators may be considered for the functions of two variables, too.

REFERENCES [1] Acar T, Aral A, Rasa I. The new forms of Voronovskaja’s theorem in weighted spaces. Positivity 2016; 20(1):25–40. ¨ [2] Acu A M, Gonska H, Rasa I. Gruss-type and Ostrowski-type inequalities in approximation theory. Ukr. Math. J. 2011; 63(6):843–864. [3] Acu AM, Manav N, Sofonea S. Approximation properties of λ -Kantorovich operators. J Inequal Appl. 2018; 2018:202. ρ [4] Acu A M, Acar T, Radu V A. Approximation by modified Un,λ operators. RACSAM 2019; 113:2715–2729. ˘ ˘ [5] Barbosu D, Barbosu M. Some properties of the fundamental polynomials of BernsteinSchurer. Bul S¸tiint¸ Univ Baia Mare Ser B, Mat-Inf. 2002; 18(2):133–136. ´ ´ eme ` de Weierstrass fondee ´ sur le calcul des prob[6] Bernstein SN. Demonstration du theor ´ Communications of the Kharkov Mathematical Society 1912; 13(2):1–2. abilites. [7] Cai QB, Lian B-Y, Zhou G. Approximation properties of λ -Bernstein operators. J Ineq and App 2018; 2018:61. ´ [8] Cai QB. The Bezier variant of Kantorovich type λ -Bernstein operators. J Inequal Appl. 2018; 2018:90. [9] Cai QB, Zhou G. Blending type approximation by GBS operators of bivariate tensor product of λ -Bernstein-Kantorovich type. J Inequal Appl. 2018; 2018:268. [10] DeVore RA, Lorentz GG. Constructive Approximation. Berlin: Springer; 1993. [11] Ditzian Z, Totik V. Moduli of Smoothness. New York: Springer; 1987. [12] Farouki RT. The Bernstein polynomials basis: a centennial retrospective. Comput Aided Geom Des. 2012; 29:379–419.

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[13] Goldman R. Pyramid algorithms, a dynamic programming approach to curves and surfaces for geometric modeling. The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling San Francisco: Elsevier Science; 2002. [14] Kantorovich LV. Sur certains developements suivant les polynmes de la forme de S. Bernstein I, II. Dokl Akad Nauk SSSR 1930; (563)568:595–600. [15] Lorentz GG. Bernstein polynomials. Chelsea Pub. Comp: New York, 1986. ¨ [16] Mohiuddine SA, F. Ozger . Approximation of functions by Stancu variant of BernsteinKantorovich operators based on shape parameter α, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math. RACSAM 2020; 114:70. [17] Muraru C. On the monotonicity of Schurer type polynomials. Carpathian J Math. 2005; 21(1–2):89–94. [18] Mursaleen M, Ahasan M, Ansari KJ. Bivariate BernsteinSchurerStancu type GBS operators in (p, q)-analogue. Adv Differ Equ. 2020; 76(2020). [19] Mursaleen M, Al-Abeid AH, Ansari KJ. On approximation properties of Baskakov´ Schurer-Szasz-Stancu operators based on q-integers, Filomat 2018; 32(41):359-1378. [20] Schurer F. Linear positive operators in approximation theory. Math Inst Techn Univ Delft Report 1962. [21] Simsek Y. Construction a new generating function of Bernstein type polynomials. Appl Math Comput. 2011; 218:1072–1076. [22] Simsek Y. Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers. Math Method Appl Sci. 2015; 38:3007–3021. ¨ [23] Srivastava HM, Ozger F, Mohiuddine SA. Construction of Stancu-type Bernstein ´ operators based on Bezier bases with shape parameter λ . Symmetry 2019; 11(3):316. DOI:10.3390/symxx010005. [24] Rahman S, Mursaleen M, Acu AM. Approximation properties of λ -Bernstein Kantorovich operators with shifted knots. Math Meth Appl Sci. 2019; 42(11):042–4053. ¨ ˘ H. Local approximation for certain King type operators. Filo[25] Ozarslan MA, Aktuglu mat 2013; 27(1):173–181. ¨ [26] Ozarslan MA, Duman O. Smoothness properties of modified Bernstein-Kantorovich operators. Numer Func Anal Opt. 2016; 37(1):92–105. ¨ ´ [27] Ozger F. On new Bezier bases with Schurer polynomials and corresponding results in approximation theory. Commun Fac Sci Univ Ank Ser A1 Math Stat. 2019; 69(1): 376–393. ¨ [28] Ozger F. Weighted statistical approximation properties of univariate and bivariate λ -Kantorovich operators. Filomat 2019; 33(11):3473–3486. ´ [29] Qi Q, Guo D, Yang G. Approximation Properties of λ -Szasz-Mirakian Operators. International Journal of Engineering Research and Technology. 2019; 12(5):662–669. ´ [30] Ye Z, Long X, Zeng X-M. Adjustment algorithms for Bezier curve and surface. International Conference on Computer Science and Education 2010; 1712–1716. DOI: 10.1109/ICCSE.2010.5593563.

Characterizations of Rough 4 Fractional-Type Integral Operators on Variable Exponent Vanishing Morrey-Type Spaces Ferit Gurb ¨ uz ¨ Hakkari University

Shenghu Ding, Huili Han, and Pinhong Long Ningxia University

CONTENTS 4.1 Introduction ..................................................................................................... 95 4.2 Preliminaries and Main Results..................................................................... 100 4.2.1 Variable Exponent Lebesgue Spaces L p(·) ........................................ 100 4.2.2 Variable Exponent Morrey Spaces L p(·),λ (·) ..................................... 104 4.2.3 Variable Exponent Vanishing Generalized Morrey Spaces .............. 107 q(·),γ (·) 4.2.4 Variable Exponent-Generalized Campanato Spaces CΠ ........... 115 4.3 Conclusion..................................................................................................... 121 Funding .................................................................................................................. 122 References.............................................................................................................. 122

4.1

INTRODUCTION

In this work, we mainly focus on some operators and commutators on the variable exponent-generalized Morrey-type space. Precisely, our aim is to characterize the boundedness for the maximal operator, fractional integral operator, and fractional maximal operator with rough kernel as well as the corresponding commutators on the variable exponent vanishing generalized Morrey spaces. We first list a series of (somewhat standard) notation needed for later sections.

95

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1. Let x = (x1 , x2 , . . . , xn ), ξ = (ξ1 , ξ2 , . . . , ξn ) . . . . etc. be the points of the real n

n-dimensional space Rn . Let x.ξ = ∑ xi ξi stand for the usual dot product in i=1  1 2 n Rn and |x| = ∑ xi2 for the Euclidean norm of x. i=1

x By x0 , we always mean the unit vector corresponding to x, i.e., x0 = |x| for 6 0. any x = 3. Sn−1 = {x ∈ Rn :|x| = 1} represents the unit sphere in Euclidean n-dimensional space Rn (n ≥ 2), and dx0 is its surface measure. 4. B(x, r) = {y ∈ Rn : |x − y| < r} denotes x-centered Euclidean ball with radius r, BC (x, r) denotes its complement, and |B(x, r)| is the Lebesgue measure of n the ball B(x, r), |B(x, r)| = vn rn , where vn = |B(0, 1)| = nΓ2π 2n and B˜ (x, r) = (2) B(x, r) ∩ E, where E ⊂ Rn is an open set. Finally, we use the notation

2.

fB(x,r) =

1 |B (x, r)|

Z

f (y) dy. B˜ (x,r)

5. C stands for a positive constant that can change its value in each statement without explicit mention. 6. The exponents p0 (·) and s0 (·) always denote the conjugate index of any ex1 1 ponent 1 < p (x) < ∞ and 1 < s (x) < ∞, i.e., p01(x) := 1 − p(x) and s0 (x) := 1 1 − s(x) . 7. In the sequel, for any exponent 1 < p (x) < ∞ and bounded sets E ⊂ Rn , if we use −C 1 |p (x) − p (y)| ≤ |x − y| ≤ , x, y ∈ E, (4.1) log (|x − y|) 2

where C = C (p) > 0 does not depend on x, y, then we call that p (·) satisfies ¨ the local log-Holder continuity condition or Dini-Lipschitz condition. The ¨ important role of the local log-Holder continuity of p (x) is well known in variable analysis. On the other hand, the condition |p (x) − p (y)| ≤

C log (e + |x|)

|y| ≥ |x| ,

x, y ∈ E,

¨ introduced by Cruz-Uribe et al. in [4] is known as the log-Holder decay condition used for unbounded sets E. It is equivalent to the condition that there exists a number p∞ ∈ [1, ∞) such that 1 1 C∞ − for all x ∈ E, (4.2) p∞ p (x) ≤ log (e + |x|) where p∞ = lim p (x). |x|→∞

¨ If p (·) satisfies both (4.1) and (4.2), then we say that it is log-Holder continuous.

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8. Let F, G ≥ 0. Here and henceforth, the symbol F ≈ G means that F . G and G . F happen simultaneously, while F . G and G . F mean that there exists a constant C > 0 such that F ≤ CG. 9. Let Ω ∈ Ls (Sn−1 ) with 1 < s ≤ ∞ be the homogeneous function of degree 0 on Rn and satisfy the integral zero property over the unit sphere Sn−1 . !1 s

|Ω (z0 )|s d σ (z0 )

R

Moreover, note that kΩkLs (Sn−1 ) :=

and

Sn−1

1



s

Z

 kΩ (z − y)kLs (B˜(x,r)) = 

 |Ω ((z − y))|s dz

B˜ (x,r)

 Z

s

Ω (σ )

 .

1 s

Zr

B˜ (x,r)

ρ

n−1

 dρdσ 

0 n s

. kΩkLs (Sn−1 ) r , 10.

(4.3)

for z ∈ B(x, r). Suppose that 0 < α (x) < n, x ∈ E ⊂ Rn . Then, the rough Riesz-type potential operator with variable order IΩ,α(·) and the corresponding rough fractional maximal operator with variable order MΩ,α(·) are defined, respectively, by Ω(x − y) f (y)dy |x − y|n−α (x)

Z

IΩ,α(·) f (x) = E

and MΩ,α(·) f (x) = sup |B(x, r)|

α(x) n −1

r>0

Z

|Ω (x − y)| | f (y)|dy,

B˜ (x,r)

where E ⊂ Rn is an open set. On the other hand, if α (·) = 0, then the rough ´ Calderon-Zygmund-type singular integral operator TΩ in the sense of principal value Cauchy integral is defined by Z

TΩ f (x) = p.v.

Ω(x − y) f (y)dy, |x − y|n

E

and especially in the limiting case α (·) = 0, the rough fractional maximal operator with variable order MΩ,α reduces to the rough Hardy-Littlewood maximal operator MΩ , and MΩ is also defined by MΩ f (x) = sup r>0

1 |B(x, r)|

Z B˜ (x,r)

|Ω (y)| | f (x − y)| dy,

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where E ⊂ Rn is an open set. In fact, we can easily see that when Ω ≡ 1, M1,α(·) ≡ Mα(·) and I1,α(·) ≡ Iα(·) are the fractional maximal operator with variable order and the Riesz-type potential operator with variable order, and similarly, M and T are the Hardy-Littlewood maximal operator and the stan´ dard Calderon-Zygmund-type singular integral operator, respectively. 11. The boundedness properties of commutator operators is also an important aspect of harmonic analysis as these are useful in the study of characterization of function spaces and regularity theory of partial differential equations. The commutators of the operators TΩ , MΩ with rough kernel Ω with locally integrable function b are given by [b, TΩ ] f (x) = b (x) TΩ f (x) − TΩ (b f ) (x) Z

= p.v.

Ω(x − y) (b (x) − b (y)) f (y)dy |x − y|n

E

and [b, MΩ ] f (x) = b (x) MΩ f (x) − MΩ (b f ) (x) Z 1 |Ω (x − y)| (b (x) − |b (y)|) | f (y)| dy, = sup r>0 |B(x, r)| B˜ (x,r)

    similarly, define the rough commutators b, IΩ,α(·) , b, MΩ,α(·) generated by the function b and the operators IΩ,α(·) and MΩ,α(·) with rough kernel Ω and variable order α(·)(0 ≤ α (·) < n) by   b, IΩ,α(·) f (x) = b (x) IΩ,α(·) f (x) − IΩ,α(·) (b f ) (x) Z

= E

Ω(x − y) (b (x) − b (y)) f (y)dy |x − y|n−α(x)

and   b, MΩ,α(·) f (x) = b (x) MΩ,α(·) f (x) − MΩ,α(·) (b f ) (x) = sup |B(x, r)| r>0

α(x) n −1

Z

|Ω (x − y)| (b (x) − |b (y)|) | f (y)| dy.

B˜ (x,r)

Later, Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces can not handle. Morrey spaces that we have been handling are called classical Morrey spaces (see [16]). But classical Morrey spaces are not totally enough to describe the boundedness properties. To this end, we need to generalize parameters p and q, among others p, but this issue will exceed the scope of this chapter, so we pass this part. Though we do not consider the direct applications of

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Rough Fractional-Type Integral Operators

Morrey spaces to PDEs, Morrey spaces can be applied to PDEs. Applications to the second-order elliptic partial differential equations can be found in [9] and [21]. Recently, while we try out to resolve somewhat modern problems emerging inherently such that nonlinear elasticity theory, fluid mechanics, etc., it is well-known that classical function spaces are not anymore suitable spaces. It thus becomes essential to introduce and analyze the diverse function spaces from diverse viewpoints. One of such spaces is the variable exponent Lebesgue space L p(·) . This space is a generalization of the classical L p (Rn ) space, in which the constant exponent p is replaced by an exponent function p (·) : Rn → (0, ∞), and it consists of all functions R f such that | f (x)| p(x) dx < ∞. This theory got a boost in 1931 when Orlicz pubRn

lished his seminal paper [17]. The next major step in the investigation of variable ˇ and Rak ´ osn´ık in the early exponent spaces was the comprehensive paper by Kova´cik 1990s [14]. Since then, the theory of variable exponent spaces was applied to many fields: Refer to [3,23] for the image processing, [2] for thermorheological fluids, [19] for electrorheological fluids, and [11] for the differential equations with nonstandard growth. For the nonweighted and weighted variable exponent settings, refer to [5–8]. ˇ and Rak ´ osn´ık [14] established many of the basic properOn the other hand, Kova´cik ties of Lebesgue and Sobolev spaces. Moreover, since these authors clarified fundamental properties of the variable exponent Lebesgue and Sobolev spaces, there are many spaces studied, such as variable exponent Morrey, generalized Morrey, vanishing generalized Morrey, and Herz-Morrey spaces (see [1,10,12,13,15,20,22,24]). In the last decade, when the parameters that define the operator have changed from point to point, there has been a strong interest in fractional-type operators and the “variable setting” function spaces. The field called variable exponent analysis has become a fairly branched area with many interesting results obtained in the last decade, such as harmonic analysis, approximation theory, operator theory, and pseudo-differential operators. But the results in this paper lie in these spaces known as variable exponent Morrey-type spaces on the rough fractional-type operators with variable order of harmonic analysis, which has been extensively developed for the last ten years and continues to attract attention of researchers from various fields of mathematics. Many of problems about such spaces have been solved both in the classical setting and in the Euclidean setting, including fractional upper and lower dimensions. For example, in 2008, variable exponent Morrey spaces L p(·),λ (·) were introduced to study the boundedness of M and Iα(·) in the Euclidean setting by Almeida et al. [1]. In 2010, variable exponent-generalized Morrey spaces L p(·),w(·) (E) were introduced to consider the boundedness of M, Iα(·) , T for bounded sets E ⊂ Rn on L p(·),w(·) (E) in [10]. p(·),w(·)

In 2016, variable exponent vanishing generalized Morrey spaces V LΠ (E) were introduced to characterize the boundedness of M, Iα(·) , T for bounded or unbounded p(·),w(·)

sets E on V LΠ (E) in [15]. These results inspire us to ask whether the above operators (IΩ,α(·) , MΩ,α(·) , TΩ , and MΩ ) have the similar mapping properties on varip(·),w(·)

able exponent V LΠ (E), which includes variable exponent L p(·),λ (·) and L p(·) spaces. Our first results (see Theorem 4.3, Theorem 4.5, and Theorem 4.6 below) will give some affirmative answers to these questions. Another purpose of this chapter is

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    to prove the boundedness of above operators ( b, IΩ,α(·) , b, MΩ,α(·) , [b, TΩ ], and p(·),w(·)

[b, MΩ ]) on V LΠ

4.2

(E) spaces (see Theorem 4.7 below).

PRELIMINARIES AND MAIN RESULTS

In this section, we first recall the definitions and some properties of basic spaces that we need and then give the main results. 4.2.1

P(·)

VARIABLE EXPONENT LEBESGUE SPACES L

Unfortunately, the variable exponent Lebesgue spaces L p(·) and the classical cases have some undesired properties. For example, the variable L p(·) spaces are not translation invariant. As a consequence, the variable exponent Lebesgue spaces are not rearrangement-invariant Banach spaces, and so neither good-λ techniques nor rearrangement inequalities may be applied for a generalization of some standard results in classical Lebesgue spaces to the case of L p(·) . Now, we begin with a brief and necessarily incomplete review of the variable exponent Lebesgue spaces L p(·) . Definition 4.1. [18] Given an open set E ⊂ Rn and a measurable function p (·) : E → [1, ∞). We assume that 1 ≤ p− (E) ≤ p+ (E) < ∞, where p− (E) = essinfx∈E p (x) and p+ (E) = esssupx∈E p (x). The variable exponent Lebesgue space L p(·) (E) is the collection of all measurable functions f such that, for some λ > 0, ρ ( f /λ ) < ∞, where the modular is defined by Z

ρ ( f ) = ρ p(·) ( f ) =

| f (x)| p(x) dx.

E p(·)

Then, the spaces L p(·) (E) and Lloc (E ) are defined by 
L p(·) (E) = f is measurable : ρ p(·) λ −1 f < ∞ for some λ > 0 and p(·)

Lloc (E) =

n

o f is measurable : f ∈ L p(·) (K) for all compact K ⊂ E .

Moreover, we define k·kL p(·) (E) , which is called the variable Lebesgue norm or the Luxemburg norm as follows: 

k f kL p(·) (E) = inf λ ∈ (0, ∞) : ρ p(·) λ −1 f ≤ 1 ∪ {∞} f ∈ L p(·) (E) . (4.4)   Since p− (E) ≥ 1, k·kL p(·) (E ) is a norm and L p(·) (E ) , k·kL p(·) (E) is a Banach space.   However, if p− (E ) < 1, then k·kL p(·) (E ) is a quasinorm and L p(·) (E) , k·kL p(·) (E) is a quasi-Banach space. The variable exponent norm has the following property



λ λ

f p(·) = k f kLλ p(·) (E) , L

(E )

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for λ ≥

1 p− .

Moreover, these spaces are referred to as the variable L p spaces since

they generalize the standard L p spaces: If p (x) = p is constant, then L p(·) (E) is isometrically isomorphic to L p (E). As a result, using notations above (p− (E) and p+ (E)), we define a class of variable exponent as follows: Φ (E) = {p (·) : E → [1, ∞) , p− (E) ≥ 1, p+ (E) < ∞} . Now, we define two sets of exponents p (x) with 1 ≤ p− (E) ≤ p+ (E) < ∞. These will be denoted as follows: P log (E) =

and



p (·) : p− (E) ≥ 1, p+ (E) < ∞ and p (·) satisfy both conditions (4.1) and (4.2) (the latter required if E is unbounded)



n o B (E) = p (·) : p (·) ∈ P log (E) , M is bounded on L p(·) (E) ,

where M is the Hardy-Littlewood maximal operator. We recall that the generalized ¨ Holder inequality on Lebesgue spaces with the variable exponent Z Z 1 1 f (x) g (x) dx ≤ | f (x) g (x)| dx ≤ C p k f k p(·) kgk p0 (·) C p = 1+ − , L (E) L (E) p p − + E E 0

is known to hold for p (·) ∈ Φ (E ), f ∈ L p(·) (E), and g ∈ L p (·) (E) (see Theorem 2.1 in [14]). Now, we recall some recent results for the rough Riesz-type potential operator with the variable order IΩ,α(·) and the corresponding rough fractional maximal operator with variable order MΩ,α(·) on the variable exponent Lebesgue space L p(·) (E). The order α (x) of the potential is not assumed to be continuous. We assume that it is a measurable function on E satisfying the following assumptions:  α0 = essinf α (x) > 0  x∈E (4.5) esssup α (x) p (x) < n  . x∈E

First, the norm in the space L p(·) (E) seems to be complicated in a sense, to be calculated or estimated. So the following basic estimation of the boundedness of an operator B: kB f kL p(·) (E) . k f kL p(·) (E) (4.6) is not easy. However, in the case of linear operators, the above inequality between the norm and the modular and the homogeneity property kBkX→X = sup f ∈X

kB f kX = sup kB f kX k f kX k f kX =1

allow us to replace checking of (4.6) by a work with a modular: Z E

|B f (x)| p(x) dx, for all f with k f kL p(·) (E) ≤ 1,

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which is certainly easier. In that respect, the boundedness of the rough Riesz-type potential operator from the space L p(·) (Rn ) with the variable exponent p(x) into the space Lq(·) (Rn ) with the limiting Sobolev exponent 1 1 α (x) = − q (x) p (x) n

(4.7)

was an open problem for a long time. It was solved in the case of bounded domains. First, in [18], in the case of bounded domains E, there has the following conditional result. Theorem 4.1. [18] Let E be a bounded open set, Ω ∈ Ls (Sn−1 ) with 1 < s ≤ ∞, p (x) ∈ P log (E), α (x) satisfying assumptions (4.5) and (p0 )+ ≤ s. Define q (x) by (4.7). Then, the rough Riesz-type potential operator IΩ,α(·) is   L p(·) (E) → Lq(·) (E) -bounded, i.e., the Sobolev-type theorem

IΩ,α(·) f q(·) . k f k p(·) L (E) L (E)

(4.8)

is valid. Corollary 4.1. Let E be a bounded open set, Ω ∈ Ls (Sn−1 ) with 1 < s ≤ ∞ being a homogeneous function of degree 0 on Rn , sp0 ∈ B (E), and (p0 )+ ≤ s. Under  the conditions ofTheorem 4.1 (taking α (·) = 0 there), the operator TΩ is L p(·) (E) → L p(·) (E) -bounded, i.e., kTΩ f kL p(·) (E ) . k f kL p(·) (E)

(4.9)

is valid. On the other hand, the pointwise inequalities on variable exponent Lebesgue spaces are very useful. Indeed, we have | f (x)| ≤ |h (x)| implies that k f kL p(·) (E) . khkL p(·) (E) . Thus, if one operator is pointwise dominated by another one: |B f (x)| ≤ |D f (x)| , and we know that the operator D is bounded, then the boundedness of the operator B immediately follows. For example, by Theorem 4.1, we get the following: Theorem 4.2. Under the conditions of Theorem 4.1, the operator MΩ,α(·) is   L p(·) (E ) → Lq(·) (E) -bounded, i.e., the Sobolev-type theorem

MΩ,α(·) f q(·) . k f k p(·) L (E) L (E) is valid.

(4.10)

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Rough Fractional-Type Integral Operators

Corollary 4.2. Let E be a bounded open set, Ω ∈ Ls (Sn−1 ) with 1 < s ≤ ∞ being a homogeneous function of degree 0 on Rn , sp0 ∈ B (E), and (p0 )+ ≤ s. Under the conditions ofTheorem 4.2 (taking α (·) = 0 there), the operator MΩ is  L p(·) (E) → L p(·) (E) -bounded, i.e., kMΩ f kL p(·) (E) . k f kL p(·) (E)

(4.11)

is valid. We are now in a place of proving (4.10) in Theorem 4.2. Remark 4.1. The conclusion of (4.10) is a direct consequence of the following Lemma 4.1 and (4.8). In order to do this, we need to define an operator by Te|Ω|,α(·) (| f |) (x) =

Z E

|Ω(x − y)| | f (y)| dy |x − y|n−α (x)

0 < α (x) < n,

where Ω ∈ Ls (Sn−1 ) (s > 1) is homogeneous of degree zero on Rn . Using the idea of proving Corollary 3.1. in [22], we can obtain the following pointwise relation. Lemma 4.1. Let 0 < α (x) < n and Ω ∈ Ls (Sn−1 ) (s > 1). Then, we have MΩ,α(·) ( f ) (x) ≤ CTe|Ω|,α(·) (| f |) (x)

for x ∈ Rn ,

(4.12)

where C does not depend on f and x. Proof. To prove (4.12), we observe that for any x ∈ Rn , there exists an r = rx such that Z 2 |Ω (x − y)| | f (y)|dy, MΩ,α(·) ( f ) (x) ≤ |B (x, rx ) |n−α (x) B(x,rx )

and by the inequality above, we get Te|Ω|,α(·) (| f |) (x) ≥

Z

|Ω(x − y)| | f (y)| dy |x − y|n−α (x)

B(x,rx )

≥

C |B (x, rx ) |n−α (x)

Z

|Ω(x − y)| | f (y)| dy.

B(x,rx )

From the process proving (4.8) in [18], it is easy to see that the conclusions of (4.8) also hold for Te|Ω|,α(·) . Combining this with (4.12), we can immediately obtain (4.10), which completes the proof of Theorem 4.2.

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Remark 4.2. Taking α (·) = 0 in Lemma 4.1 and the inequality for x ∈ Rn ,

MΩ ( f ) (x) ≤ CTe|Ω| (| f |) (x) which follows from the definitions of the operators.

The above theorems (Theorem 4.1 and Theorem 4.2) allow us to use the known results for the boundedness of the operators MΩ,α(·) and IΩ,α(·) that are transferred to the various function spaces. The following fact is known (see Lemma 3.1. in [15]). Lemma 4.2. [15] Let E be a bounded open set, p (x) ∈ P log (E) and α (x) satisfying assumptions (4.5). Then,

n α (x)− p(x)

α(x)−n χB˜(x,r) .r .

|x − ·| L p(·) (E)

We will also make use of the estimate provided by the following fact (see [15]).



x ∈ E, p (x) ∈ P log (E) , (4.13)

χB˜(x,r) p(·) . rψ p (x,r) , L

(E)

where

( ψ p (x, r) =

4.2.2

n p(x) , n p(∞) ,

r≤1 r > 1. P(·),λ (·)

VARIABLE EXPONENT MORREY SPACES L

Here, we recall the variable exponent Morrey spaces and the integral inequalities. Definition 4.2. [1] Let E be a bounded open set and λ (x) be a measurable function on E with values in [0, n]. Then, the variable exponent Morrey space L p(·),λ (·) ≡ L p(·),λ (·) (E) is defined by

L p(·),λ (·) ≡ L p(·),λ (·) (E) =

   k f kL p(·),λ (·)

p(·)

f ∈ Lloc (E ) :

λ (x) −

= sup r p(x) f χB˜(x,r) x∈E ,r>0

L p(·) (E)

  n, then L p(·),λ (·) (E) = {0}. Lemma 4.3. Let E be a bounded open set, Ω ∈ Ls (Sn−1 ) with 1 < s < ∞, p (x) , q (x) ∈ P log (E), α (x) satisfying the following assumptions:   α0 = essinf α (x) > 0 x∈E (4.14) esssup [λ (x) + α (x) p (x)] < n  x∈E

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Rough Fractional-Type Integral Operators

α (·) 1 1 = p(·) − n−λ the rough Riesz-type potenand (p0 )+ ≤ s. Define q (x) by q(·) (·) . Then,   tial operator IΩ,α(·) is L p(·),λ (·) (E) → Lq(·),λ (·) (E) -bounded. Moreover,



IΩ,α(·) f q(·),λ (·) . k f k p(·),λ (·) . L (E) L (E ) Proof. By the embedding property in Lemma 7 in [1], we only need to prove that the operator IΩ,α(·) is bounded in L p(·),λ (·) (E). Hedberg’s trick: Ω(x − y) f (y)dy + |x − y|n−α (x)

Z

IΩ,α(·) f (x) =

Z

BC (x,2r)

B(x,2r)

Ω(x − y) f (y)dy |x − y|n−α (x)

= F (x, r) + G (x, r) .

(4.15)

We may assume that k f kL p(·),λ (·) (E) ≤ 1. For F (x, r), we first have to prove the following: Z Ω(x − y) 2n rα(x) F (x, r) := f (y)dy . M f (x) . (4.16) 2α (x) − 1 Ω |x − y|n−α (x) |x−y| 0

x∈Π

r > 0.

(4.20) p(·),w(·)

p(·),w(·)

Then, the variable exponent-generalized Morrey space LΠ ≡ LΠ (E) defined by  p(·)  f ∈ Ll oc (E) : p(·),w(·) p(·),w(·) − 1 LΠ ≡ LΠ (E) =  k f k p(·),w(·) = sup w(x, r) p(x) k f kL p(·) (B˜(x,r)) < ∞ LΠ

x∈Π,r>0

is   

(4.21) and one can also see that for bounded exponents p, there holds the following equivalence: Z p(y) p(·),w(·) f (y) dy < ∞. f ∈ LΠ if and only if sup w (x, r) x∈Π,r>0 B˜ (x,r)

On the other hand, the above definition recovers the definition of L p(·),λ (·) (E) if λ (x)

we choose w(x, r) = r p(x) and Π = E, i.e., p(·),w(·)

L p(·),λ (·) (E ) = LΠ

(E) |

λ (x)

w(x,r)=r p(x)

.

,

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Topics in Contemporary Mathematical Analysis and Applications p(·),w(·)

turns into the local generalized MorAlso, when Π = {x0 } and Π = E, LΠ p(·),w(·) p(·),w(·) rey space L{x } (E) and the global generalized Morrey space LE (E), respec0 tively. Moreover, we point out that w(x, r) is a measurable nonnegative function and no monotonicity-type condition is imposed on these spaces. Note that by the above definition of the norm in L p(·) (E) (see 4.4), we can also write that     p(y) Z   f (y) dy ≤ 1 . k f k p(·),w(·) = sup inf λ = λ (x, r) : λ w (x, r) LΠ   x∈Π,r>0   B˜ (x,r)

Then, recall that the concept of the variable exponent vanishing generalized Morrey p(·),w(·) space V LΠ (E ) has been introduced in [15] in the following form. Definition 4.4. [15] Let 1 ≤ p (x) ≤ p+ < ∞, Π ⊂ E ⊂ Rn , x ∈ Π, w(x, r): Π × (0, diam (E)) → R+ . Then, the variable exponent vanishing generalized Morrey p(·),w(·) p(·),w(·) space V LΠ ≡ V LΠ (E) is defined by   p(·),w(·) f ∈ LΠ (E) : lim sup M p(·),w(·) ( f ; x, r) = 0 , r→0 x∈Π

where r

n − p(x)

M p(·),w(·) ( f ; x, r) :=

k f kL p(·) (B˜(x,r)) 1

.

w(x, r) p(x) Naturally, it is suitable to impose on w(x,t) with the following conditions: t −ψ p (x,t)

lim sup

t→0 x∈Π

=0

1

(4.22)

w(x,t) p(x)

and inf sup w(x,t) > 0.

(4.23)

t>1 x∈Π

From (4.22) and (4.23), we easily know that the bounded functions with compact p(·),w(·) p(·),w(·) support belong to V LΠ (E ), which make the spaces V LΠ (E) nontrivial. p(·),w(·) The spaces V LΠ (E) are the Banach spaces with respect to the norm kfk

p(·),w(·)

V LΠ

≡ kfk

p(·),w(·)

LΠ

= sup M p(·),w(·) ( f ; x, r) . x∈Π,r>0

p(·),w(·)

The spaces V LΠ (E) are also the closed subspaces of the Banach spaces p(·),w(·) LΠ (E), which may be shown by standard means. Furthermore, we have the following embeddings: p(·),w(·)

V LΠ

p(·),w(·)

⊂ LΠ

,

kfk

p(·),w(·)

LΠ

≤ kfk

p(·),w(·)

V LΠ

.

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In 2016, for bounded or unbounded sets E, Long and Han [15] considered the p(·),w(·) Spanne-type boundedness of operators Mα(·) and Iα(·) on V LΠ (E). Now, in this section, we extend Theorem 4.3. in [15] to rough kernel versions. In other words, Theorem 4.3. in [15] allows us to use the known results for the boundedness of the operators Iα(·) and Mα(·) in generalized variable exponent Morrey spaces to transfer them to the operators IΩ,α(·) and MΩ,α(·) . We give two versions of such an extension, the one being a generalization of Spanne’s result for rough potential operators with variable order and the other extending the corresponding Adams’ result, respectively. In this context, we will give some answers to the above explanations in the following. Theorem 4.5. (Spanne-type result with variable α (x)) (our main result) Let E be a bounded open set, Ω ∈ Ls (Sn−1 ), 1 < s ≤ ∞, Ω(µx) = Ω(x) for any µ > 0, x ∈ Rn \ {0}, p (x) ∈ P log (E), α (x) satisfying assumption (4.5). Define q (x) by (4.7). Suppose that q (·) and α (·) satisfy (4.1). For s−s 1 < p− ≤ p (·) < αn(·) , the following pointwise estimate n

IΩ,α (·) f q(·) . r q(x) L (B˜ (x,r))

diam Z (E)

k f kL p(·) (B˜(x,t))

r

dt t

n +1 q(x)

(4.24)

p(·)

holds for any ball B˜ (x, r) and for all f ∈ Lloc (E). If the functions w1 (x, r) and w2 (x, r) satisfy (4.20) as well as the following Zygmund condition diam Z (E ) r

1 1 w1p(x) (x,t) q(x) dt . w 2 (x, r), t 1−α(x)

r ∈ (0, diam (E)]

(4.25)

and additionally these functions satisfy the conditions (4.22)-(4.23), 1

diam Z (E )

w p(x) (x,t ) sup 11−α(x) dt < ∞, x∈Π t

cδ :=

δ >0

(4.26)

δ

  p(·),w (·) q(·),w (·) then the operators IΩ,α(·) and MΩ,α(·) are V LΠ 1 (E) → V LΠ 2 (E) bounded. Moreover,

IΩ,α(·) f q(·),w (·) . k f k p(·),w (·) , (4.27) 2 1 V LΠ



MΩ,α(·) f

(E)

q(·),w2 (·)

V LΠ

(E)

V LΠ

. kfk

(E)

p(·),w1 (·)

V LΠ

(E)

.

Proof. Since inequality (4.24) is the key of the proof of (4.27), we first prove (4.24).

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For any x ∈ E, we write as f (y) = f1 (y) + f2 (y) ,

(4.28)

where f1 (y) = f (y) χB˜(x,2r) (y), r > 0 such that IΩ,α(·) f (y) = IΩ,α(·) f1 (y) + IΩ,α(·) f2 (y) . By using triangle inequality, we get





IΩ,α(·) f q(·)

IΩ,α(·) f1 q(·)

IΩ,α(·) f2 q(·) ≤ + . ˜ ˜ L (B(x,r)) L (B(x,r)) L (B˜ (x,r))



Now, let us estimate IΩ,α(·) f1 Lq(·) (B˜(x,r)) and IΩ,α (·) f2 Lq(·) (B˜(x,r)) , respectively. By Hardy-Littlewood-Sobolev-type inequality and Theorem 4.1, we obtain that



IΩ,α(·) f1 q(·) ≤ IΩ,α(·) f1 Lq(·) (E) . k f1 kL p(·) (E) = k f kL p(·) (B˜(x,2r)) L (B˜ (x,r)) n

≈ r q(x) k f kL p(·) (B˜(x,2r))

diam Z (E)

t

2r

≤r

n q(x)

diam Z (E)

k f kL p(·) (B˜(x,t ))

r

dt n +1 q(x)

dt t

n +1 q(x)

,

where in the last inequality, we have used the following fact: k f kL p(·) (B˜(x,2r)) ≤ k f kL p(·) (B(x,t) ˜ ) , for t > 2r.

Now, let us estimate the second part (= IΩ,α(·) f2 Lq(·) (B˜(x,r)) ). If |x − z| ≤ r and |z − y| ≥ r, then |x − y| ≤ |x − z| + |y − z| ≤ 2 |y − z|. By the generalized Minkowski’s inequality, we get



Z

Ω(z − y)

IΩ,α(·) f2 q(·) f (y)dy

˜ L (B(x,r) )=

|z − y|n−α(x)

E\B˜(x,2r)

q(·) ˜ L (B(x,r)) Z

|Ω(z − y)| | f (y)|

. dy

χ

q(·) . ˜ B (x,r) L (E) |x − y|n−α (x) E\B˜ (x,2r)

n Put γ > q(·) . Provided that 1 < s0 < p− ≤ p+ < ∞, sup (α (x) + γ − n) < ∞, x∈E   0  p(·) ¨ and inf n + (α (x) + γ − n) s0 < ∞, by generalized Holder’ s inequality for x∈E

L p(·) (E),Fubini’s theorem, and Lemma 4.2 and (4.3), we obtain

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Rough Fractional-Type Integral Operators

|Ω(z − y)| | f (y)| dy |x − y|n−α (x)

Z E\B˜ (x,2r)

|Ω(z − y)| | f (y)| dy |x − y|n−α(x)−γ

Z

. E\B˜ (x,2r) diam Z (E)

=

dt

Z

t γ+1 2r

{y∈E:2r≤|x−y|≤t}

diam Z (E)

dt t γ +1

|x−y|

|Ω(z − y)| | f (y)| dy |x − y|n−α (x)−γ

diam Z (E)



k f kL p(·) (B˜(x,t)) x − ·|α (x)+γ−n

.

˜ )) Lν(·) (B(x,t

2r diam Z (E)

k f kL p(·) (B˜(x,t))

. r

for

1 p(·)

1 s

+ +

1 ν(·)

kΩ (z − y)kLs (B˜(x,t))

dt t γ+1

dt t

(4.29)

n +1 q(x)

= 1. Thus, by (4.13), we get diam Z (E)

n

q(x)

IΩ,α (·) f2 q(·) ˜ L (B(x,r) ).r

k f kL p(·) (B˜(x,t))

r

dt t

n +1 q(x)

.



Combining all the estimates for IΩ,α(·) f1 Lq(·) (B˜(x,r)) and IΩ,α(·) f2 Lq(·) (B˜(x,r)) , we get (4.24). At last, by Definition 4.4, (4.24), and (4.25), we get r

IΩ,α(·) f

q(·),w2 (·)

V LΠ

(E )

n − q(x)

= sup



IΩ,α(·) f q(·) L (B˜ (x,r)) 1

x∈Π,r>0

w2 (x, r) q(x)

. sup x∈Π,r>0

. kfk

diam Z (E)

1 w2 (x, r)

p(·),w1 (·)

V LΠ

1 q(x)

k f kL p(·) (B˜(x,t ))

r

t

diam Z (E )

1

sup

(E) x∈Π,r>0

dt n +1 q(x)

1

w2 (x, r) q(x)

r

1

w1p(x) (x,t) dt t 1−α(x)

. kfk

p(·),w (·) V LΠ 1 (E)

and r lim sup

r→0 x∈Π

n − q(x)



IΩ,α(·) f q(·) L (B˜ (x,r)) w2 (x,t )

1 q(x)

r . lim sup

r→0 x∈Π

n − p(x)

k f kL p(·) (B˜(x,r)) 1

w1 (x,t) p(x)

= 0.

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Topics in Contemporary Mathematical Analysis and Applications

Thus, (4.27) holds. On the other hand, since MΩ,α(·) ( f ) . I|Ω|,α(·) (| f |) (see Lemma 4.1), we can also use the same method for MΩ,α(·) , so we omit the details. As a result, we complete the proof of Theorem 4.5. Definition 4.5. [13] (Rough (p, q)-admissible TΩ,α(·) -potential type operator with variable order) Let 1 ≤ p− (E) ≤ p (·) ≤ p+ (E) < ∞ oper . A rough sublinear ator with variable order T , i.e., T ( f + g) ≤ T ( f ) + T (g) and Ω,α (·) Ω,α(·) Ω,α(·) Ω,α (·) for ∀λ ∈ C TΩ,α(·) (λ f ) = |λ | TΩ,α(·) ( f ) , will be called rough (p, q)-admissible TΩ,α(·) -potential type operator with variable order if · TΩ,α(·) fulfills the following size condition: |Ω(x − y)| | f (y)| dy, |x − y|n−α (·)

Z


χB(z,r) (x) |TΩ,α(·) f χEB(z,2r) (x)| ≤ CχB(z,r) (x)

EB(z,2r)

(4.30)   · TΩ,α(·) is L p(·) (E) → Lq(·) (E) -bounded. Remark 4.3. Note that rough (p, q)-admissible potential type operators were introduced to study their boundedness on Morrey spaces with variable exponents in [13]. The operators MΩ,α(·) and IΩ,α(·) are also rough (p, q)-admissible potential type operators. Moreover, these operators satisfy (4.30). Corollary 4.3. Obviously, under the conditions  of Theorem 4.5, if the  rough (p, q)p(·) q(·) admissible TΩ,α(·) -potential type operator is L (E) → L (E) -bounded and satisfies (4.30), the result in Theorem 4.5 still holds. For α (x) = 0 in Theorem 4.5, we get the following new result. Corollary 4.4. Let E, Ω, p (x) be the same as in Theorem 4.5. Then, for p (·) ≤ p+ < ∞, the following pointwise estimate kTΩ f kL p(·) (B(x,r) ˜ ).r

n p(x)

diam Z (E)

t

n −1 − p(x)

s s−1

< p− ≤

k f kL p(·) (B˜(x,t)) dt

r p(·) holds for any ball B˜ (x, r) and for all f ∈ Lloc (E). If the function w (x, r) satisfies (4.20) as well as the following Zygmund condition diam Z (E) r

1 1 w p(x) (x,t) dt . w p(x) (x, r), t

r ∈ (0, diam (E)]

and additionally this function satisfies conditions (4.22) and (4.23), diam Z (E)

1

w p(x) (x,t) sup dt < ∞, t x∈Π

cδ := δ

δ >0

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Rough Fractional-Type Integral Operators p(·),w(·)

then the operators TΩ and MΩ are bounded on V LΠ kTΩ f k

p(·),w(·)

V LΠ

kMΩ f k

(E )

p(·),w(·)

V LΠ

(E )

. kfk

p(·),w(·)

V LΠ

. kfk

(E)

p(·),w(·)

V LΠ

(E). Moreover, , .

(E)

(4.31)

Theorem 4.6. (Adams-type result with variable α (x)) (our main result) Let E, Ω, n , p (x), q (x), α (x) be the same as in Theorem 4.5. Then, for s−s 1 < p− ≤ p (·) < α(·) the following pointwise estimate diam Z (E) n −1 α(x)− p(x) IΩ,α (·) f (x) . rα(x) MΩ f (x) + k f kL p (B˜(x,t)) dt t

(4.32)

r p(·) holds for any ball B˜ (x, r) and for all f ∈ Lloc (E). The function w (x,t) satisfies (4.20), (4.22)-(4.23) as well as the following conditions: diam Z (E) r

1 1 w p(x) (x,t) dt . w p(x) (x, r) , t

diam Z (E) r

1 α(x)p(x) w p(x) (x,t) − dt . r q(x)−p(x) , 1−α(x) t

(4.33)

where p (x) < q (x). Then, the operators IΩ,α(·) and MΩ,α(·) are ! 1 1 p(·),w p(·)

V LΠ

q(·),w q(·)

(E ) → V LΠ

(E) -bounded. Moreover,



IΩ,α(·) f

1 q(·),w q(·)

V LΠ



MΩ,α(·) f

. kfk (E)

1 q(·),w q(·)

V LΠ

1 p(·),w p(·)

V LΠ

. kfk (E)

, (E)

1 p(·),w p(·)

V LΠ

. (E)

Proof. As in the proof of Theorem 4.5, we represent the function f in the form (4.28) and have IΩ,α(·) f (x) = IΩ,α(·) f1 (x) + IΩ,α(·) f2 (x) . For IΩ,α(·) f1 (x), similar to the proof of (4.17), we obtain the following pointwise estimate: IΩ,α(·) f1 (x) . t α (x) MΩ f (x) . (4.34) ¨ For IΩ,α(·) f2 (x), similar to the proof of (4.29), applying Fubini’s theorem, Holder’ s inequality, and (4.3), we get IΩ,α(·) f2 (x) .

diam Z (E )

t r

n −1 α (x)− p(x)

k f kL p (B˜(x,t)) dt

(4.35)

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Topics in Contemporary Mathematical Analysis and Applications

and by (4.34) and (4.35) complete the proof of (4.32). Since MΩ,α(·) ( f ) . I|Ω|,α(·) (| f |) (see Lemma 4.1), it suffices to treat only the case of the operator IΩ,α(·) . In this sense, by (4.32) and (4.33), we obtain α(x) p(x) IΩ,α(·) f (x) . rα (x) MΩ f (x) + r− q(x)−p(x) k f k

p(·),w(·)

V LΠ

kfk

p(·),w(·) V LΠ (E)

Then, choosing r =

(E)

.

! q(x)−p(x) α (x)p(x)

for every x ∈ E supposing that f is not

MΩ f (x)

equal 0, thus we have p(x) p(x) 1− IΩ,α(·) f (x) . (MΩ f (x)) q(x) k f k q(x) p(·),w(·)

V LΠ

(E)

.

(4.36)

Finally, by Definition 4.4, (4.36), and (4.31), we get r

IΩ,α(·) f

1 q(·),w q(·)

V LΠ

n − q(x )

= sup (E)



IΩ,α(·) f p(·) L (B˜ (x,r)) 1

x∈Π,r>0

w(x, r) q(x)

p(x)

. kfk

1− q(x)

p(·),w(·)

V LΠ

r

sup

(E ) x∈Π,r>0

n − q(x ) 1

. kfk

p(·),w(·)

V LΠ

x∈Π,r>0

p(x)

. kfk

p(·),w(·)

(B˜(x,r)) ! p(x)

n − p(x)

q(x)

1

w(x, r) p(x)

kMΩ f kL p(·) (B˜(x,r))

p(x)

1− q(x)

V LΠ

r

sup

(E)

L

w(x, r) q(x)

p(x)

1− q(x)

p(x)

kMΩ f k q(px(·))

(E)

kMΩ f k q(x)

1 p(·),w p(·)

V LΠ

(E)

. kfk

1 p(·),w p(·) V LΠ

(E)

if p (x) < q (x) and r lim sup

r→0 x∈Π

n − q(x)



IΩ,α (·) f q(·) L (B˜ (x,r)) w2 (x,t)

1 q(x)

r . lim sup

r→0 x∈Π

n − p(x)

k f kL p(·) (B˜(x,r)) 1

= 0,

w1 (x,t) p(x)

which completes the proof of Theorem 4.6. Corollary 4.5. Obviously, under the conditions  of Theorem 4.6, if the  rough (p, q)admissible TΩ,α(·) -potential type operator is L p(·) (E) → Lq(·) (E) -bounded and satisfies (4.30), the result in Theorem 4.6 still holds. Remark 4.4. Let E be a bounded open set and λ (x) be a measurable function on E with values in [0, n]. Then, the variable exponent vanishing Morrey space

115

Rough Fractional-Type Integral Operators p(·),λ (·)

p(·),λ (·)

≡ V LΠ

V LΠ

p(·),λ (·)

V LΠ

p(·),λ (·)

≡ V LΠ

(E) is defined by

(E) =

    k f kV LΠp(·),λ (·) 

f ∈ L p(·),λ (·) (E)

:

− λ (x) = lim sup t p(x) f χB˜ (x,t)

L p(·) (E)

r→0 x∈E 0 0,

δ

      p (·),w1 (·) q(·),w (·) then the operators b, IΩ,α(·) and b, MΩ,α(·) are V LΠ1 (E ) → V LΠ 2 (E) bounded. Moreover,

 

b, IΩ,α(·) f q(·),w (·) . kbk p (·),γ(·) k f k p (·),w (·) , 2 2 1 1 V LΠ

 

b, MΩ,α(·) f

(E )

q(·),w2 (·)

V LΠ

(E )

CΠ

. kbk

p (·),γ(·)

CΠ2

V LΠ

kfk

(E)

p (·),w1 (·) (E)

V LΠ1

.

    Proof. Since b, MΩ,α(·) ( f ) . b, I|Ω|,α(·) (| f |), it suffices to treat only the case of   the operator b, IΩ,α(·) . As in the proof of Theorem 4.5, we represent the function f in the form (4.28) and have  


b, IΩ,α(·) f (x) = b (x) − bB(x,r) IΩ,α(·) f1 (x) − IΩ,α(·) b (·) − bB(x,r) f1 (x)

117

Rough Fractional-Type Integral Operators


+ b (x) − bB(x,r) IΩ,α(·) f2 (x) − IΩ,α(·)



b (·) − bB(x,r) f2 (x)

≡ F1 + F2 + F3 + F4 . Hence, we get   k b, IΩ,α(·) f kLq(·) (B(x,r)) ≤ kF1 kLq(·) (B(x,r) ˜ ˜ ) + kF2 kLq(·) (B˜(x,r)) + kF3 kLq(·) (B˜(x,r)) + kF4 kLq(·) (B˜(x,r)) . ¨ First, we use the Holder’ s inequality such that

(4.40)

1 q(·)

=

1 p2 (·)

+ q 1(·) , the boundedness of 1

IΩ,α(·) from L p(·) into Lq(·) (see Theorem 4.1) and (4.37) to estimate kF1 kLq(·) (B˜(x,r)) , and we obtain



kF1 kLq(·) (B(x,r) ˜ ) = (b (·) − bB ) IΩ,α(·) f1 (·) Lq(·) (B˜(x,r))

. k(b (·) − bB )kL p2 (·) (B˜(x,r)) IΩ,α(·) f1 (·) Lq1 (·) (B˜(x,r)) n

. r p2 (x)

+nγ(x)

kbk

= kbk

p (·),γ (·) CΠ2

r

p (·),γ(·) CΠ2

k f1 kL p1 (·)

n + n +nγ(x) p2 (x) q1 (x)

(B˜(x,r))

k f kL p1 (·)

diam Z (E)

t

(B˜(x,2r))

−1− q n(x) 1 dt

2r n

. kbk

p (·),γ(·)

CΠ2

diam Z (E)

 t nγ (x)− q n(x) −1 1 1 + ln k f kL p1 (·) B˜(x,t) t dt . ( ) r

r q(x) 2r

Second, for kF2 kLq(·) (B˜(x,r)) , applying the boundedness of IΩ,α(·) from L p(·) into Lq(·) ¨ (see Theorem 4.1), generalized Holder’ s inequality such that 1 q(·)

=

1 p2 (·)

1 p(·)

=

1 p1 (·)

+

1 p2 (·) ,

+ q 1(·) and (4.37), we know that 1




kF2 kLq(·) (B(x,r) ˜ ) = IΩ,α(·) b (·) − bB(x,r) f1 Lq(·) (B˜(x,r)) . k(b (·) − bB ) f1 kL p(·) (B˜(x,r)) . k(b (·) − bB )kL p2 (·) (B˜(x,r)) k f1 kL p1 (·) (B˜(x,r)) . kfk

p (·),γ(·) CΠ2

. kbk

p (·),γ(·) CΠ2

r

n + n +nγ (x) p2 (x) q1 (x)

k f kL p1 (·)

diam Z (E)

t

(B˜(x,2r))

−1− q n(x) 1 dt

2r n

diam Z (E)

 t nγ (x)− q n(x) −1 1 dt. 1 + ln k f kL p1 (·) B˜(x,t) t ( ) r

r q(x) 2r

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Topics in Contemporary Mathematical Analysis and Applications

s Third, for kF3 kLq(·) (B(x,r) ˜ ) , similar to the proof of (4.29), when s−1 ≤ p1 (·), by ¨ Fubini’s theorem, generalized Holder’ s inequality, and (4.3), we have

Z

IΩ,α(·) f2 (x) .

E\B˜ (x,2r)

|Ω(z − y)| | f (y)| dy |x − y|n−α (x)

diam Z (E)

k f kL p1 (·) (B˜(x,t)) t

.

−1− q n(x) 1

(4.41)

dt.

2r 1 ¨ Thus, by generalized Holder’ s inequality such that q(·) = p 1(·) + q 1(·) , (4.37), and 2 1 (4.41), we obtain




b (·) − bB(x,r) IΩ,α(·) f2 (·) q(·) kF3 kLq(·) (B(x ˜ ,r)) = L (B˜ (x,r))




. b (·) − bB(x,r) L p2 (·) (B˜(x,r)) IΩ,α (·) f2 (·) Lq1 (·) (B˜(x,r))

.r

n +nγ (x) p2 (x)

. kbk

p (·),γ(·) CΠ2

kbk

p (·),γ (·) CΠ2

n

r

n q1 (x)

diam Z (E)

k f kL p1 (·)

(B˜(x,t))

t

−1− q n(x) 1 dt

2r

diam Z (E)

 t nγ (x)− q n(x) −1 1 1 + ln k f kL p1 (·) B˜(x,t) t dt. ( ) r

r q(x) 2r

Finally, we consider the term kF4 kLq(·) (B˜(x,r)) = IΩ,α(·)



b (·) − bB(x,r) f2 (·) Lq(·) (B˜(x,r)) .

s For z ∈ B (x, r), when s−1 ≤ p (·), by the Fubini’s theorem, applying the generalized ¨ Holder’ s inequality, and from (4.3) and (4.37), we have

IΩ,α(·) b (·) − bB(x,r) f2 (z)

. ≈ . + . t

b (y) − bB(x,r) |Ω (z − y)|

2r diam R (E)

R

2r 2r0

n − q(x)

 t  nγ (x)− q n(x) −1 1 1 + ln t r

1 q(x)

2r

× k f kL p1 (·) (B˜(x,t )) dt . kbk

p (·),γ(·) CΠ2

kfk

p (·),w1 (·) (E) x∈Π,r>0 V LΠ1

1 p1 (x)

× t α(x)+nγ(x) w1 . kbk

p (·),γ(·)

CΠ2

sup

(x, t )

kfk

w2 (x, r)

1 q(x)

r

 t 1 + ln r

dt t

p(·),w1 (·)

V LΠ

diam Z (E)

1

(E)

and r lim sup

r→0 x∈Π

n − q(x)

 

b, IΩ,α (·) f q(·) L (B˜ (x,r)) 1

w2 (x,t) q(x)

which completes the proof of Theorem 4.7.

r . lim sup

r→0 x∈Π

− p(nx)

k f kL p(·) (B˜(x,r)) 1

w1 (x,t) p(x)

= 0,

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Topics in Contemporary Mathematical Analysis and Applications

Corollary 4.8. Let E, Ω, p (x), q (x), α (x) be the same as in Theorem 4.5. Suppose that q (·) and α (·) satisfy (4.1). Then, for s−s 1 < p− ≤ p (·) < αn(·) and b ∈ BMO (E ), the following pointwise estimate n   k b, IΩ,α(·) f kLq(·) (B(x,r)) . kbkBMO r q(x) ˜

diam Z (E )

 t  n −1 1 + ln t q(x) k f kL p(·) (B˜(x,t)) dt r

2r p(·) holds for any ball B˜ (x, r) and for all f ∈ Lloc (E). If the functions w1 (x, r) and w2 (x, r) satisfy (4.20) as well as the following Zygmund condition 1

diam Z (E )

p(x)  1 t  w1 (x,t) q(x) 1 + ln dt . w 2 (x, r), r t 1−α(x)

r

r ∈ (0, diam (E)]

and additionally these functions satisfy conditions (4.22) and (4.23), 1

diam Z (E)

p(x)  t  w1 (x,t) sup 1 + ln dt < ∞, r t 1−α(x) x∈Π

dδ :=

δ > 0,

δ

      p(·),w (·) q(·),w (·) then the operators b, IΩ,α(·) and b, MΩ,α(·) are V LΠ 1 (E ) → V LΠ 2 (E) bounded. Moreover,

 

b, IΩ,α(·) f q(·),w (·) . kbkBMO k f k p (·),w (·) , V LΠ

 

b, MΩ,α(·) f

2

q(·),w2 (·)

V LΠ

V LΠ1

(E )

(E )

. kbkBMO k f k

1

(E)

p (·),w1 (·) (E)

V LΠ1

.

For α (x) = 0 in Theorem 4.7, we get the following new result. Corollary 4.9. Let E, Ω, p (x) be the same as in Theorem 4.5. Let also 1 p2 (·) and s − s−1 < p

p (·),γ(·) b ∈ CΠ2 (E). Suppose that + ≤ p (·) ≤ p < ∞, the following

k [b, TΩ ] f kL p(·) (B(x,r)) . kbk ˜

p (·),γ(·) CΠ2

r

n p(x)

1 p(·)

=

1 p1 (·)

+

p1 (·) and p2 (·) satisfy (4.1). Then, for pointwise estimate

diam Z (E)

 t  nγ(x)− p n(x) −1 1 k f kL p1 (·) (B(x,t) 1 + ln t ˜ ) dt r

2r

p1 (·) holds for any ball B˜ (x, r) and for all f ∈ Lloc (E ). If the function w (x, r) satisfies (4.20) as well as the following Zygmund condition diam Z (E) r

1  1 t  w p1 (x) (x,t) p(x) (x, r), 1 + ln dt . w r t 1−nγ(x)

r ∈ (0, diam (E)]

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Rough Fractional-Type Integral Operators

and additionally this function satisfies conditions (4.22) and (4.23), diam Z (E)

1  t  w p1 (x) (x,t) sup 1 + ln dt < ∞, r t 1−nγ(x) x∈Π

dδ :=

δ > 0,

δ

then the operators [b, TΩ ] and [b, MΩ ] are bounded. Moreover, k[b, TΩ ] f k

p(·),w(·)

V LΠ

k[b, MΩ ] f k

(E )

p(·),w(·)

V LΠ

(E )

. kbk

  p (·),w(·) p(·),w(·) V LΠ1 (E) → V LΠ (E) -

p (·),γ(·)

kfk

p (·),γ (·)

kfk

CΠ2

. kbk

CΠ2

p (·),w(·)

V LΠ1

(E)

p (·),w(·)

V LΠ1

,

(E)

.

From Corollary 4.9, we get the following. Corollary 4.10. Let E, Ω, p (x) be the same as in Theorem 4.5. Then, for p− ≤ p (·) ≤ p+ < ∞ and b ∈ BMO (E), the following pointwise estimate k [b, TΩ ] f kL p(·) (B(x,r)) . kbkBMO r ˜

n p(x)

s s−1


0,

δ p(·),w(·)

then the operators [b, TΩ ] and [b, MΩ ] are bounded on V LΠ k[b, TΩ ] f k

p(·),w(·)

V LΠ

k[b, MΩ ] f k

p(·),w(·)

V LΠ

4.3

(E) (E)

. kbkBMO k f k

p(·),w(·)

V LΠ

. kbkBMO k f k

(E)

p(·),w(·)

V LΠ

(E). Moreover, ,

(E)

.

CONCLUSION

It is well known that, for the purpose of researching non-smoothness partial differential equation, mathematicians pay more attention to the singular integrals with rough kernel. Moreover, the fractional-type operators and their weighted boundedness

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theory play important roles in harmonic analysis and other fields, and the multilinear operators arise in numerous situations involving product-like operations. Moreover, the multilinear operators are natural generalizations of linear case. In recent years, these topics gain the attention from a vast number of researchers in different function spaces. Especially, more and more researches focus on function spaces based on variable exponent Morrey spaces to fill in some gaps in the theory of Morrey-type spaces. Moreover, these spaces are useful in harmonic analysis and PDEs. But this topic exceeds the scope of this paper. Thus, we omit the details here. In this paper, we have shown that various classical operators (such as rough maximal, potential, and singular integral operators) and their commutators are bounded in different proper closed subspaces of variable exponent-generalized Morrey spaces. On the other hand, all these subspaces together provide an explicit description of the closure of nice functions in variable exponent Morrey norm in terms of vanishing properties of Morrey functions. Therefore, our results give a contribution for the development of harmonic analysis on variable exponent-generalized Morrey spaces. As is well known, variable exponent Morrey-type spaces attract a lot of attention from the applications side. By this reason, we are convinced that the results exhibited in this paper can also be useful for regularity results in the theory of certain partial differential equations. Finally, the results presented here are sure to be new and potentially useful. Since the research subject here and its related ones are so popular, the content of this paper may attract interested readers who have been interested in this and related research subjects. Therefore, the results in this paper are worthwhile to record.

FUNDING This work is funded by Hakkari University Scientific Research Project (Grant No. FM18BAP1) under the research project “Some estimates for rough Riesz-type potential operator with variable order and rough fractional maximal operator with variable order both on generalized variable exponent Morrey spaces and on vanishing generalized variable exponent Morrey spaces,” Institution of Higher Education Scientific Research Project in Ningxia (Grant No. NGY2017011) and Natural Science Foundation of China (Grant Nos. 11461053 and 11762016).

REFERENCES [1] A. Almeida, J.J. Hasanov and S.G. Samko, Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J., 2008, 15(2): 195–208. [2] S.N. Antontsev and J.F. Rodrigues, On stationary thermorheological viscous flows, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 2006, 52(1): 19–36. [3] Y. Chen, S.E. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 2006, 66(4): 1383–1406. [4] D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable L p spaces, Ann. Acad. Sci Fenn. Math., 2003, 28(1): 223–238. ¨ o¨ and M. Ru¨zi ˇ cka, ˇ [5] L. Diening, P. Harjulehto, P. Hast Lebesgue and Sobolev Space with Variable Exponents, Springer Lecture Notes, Vol. 2017, Berlin: Springer-Verlag, 2011.

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¨ o¨ and A. Nekvinda, Open problems in variable exponent Lebesgue [6] L. Diening, P. Hast and Sobolev spaces, in Proceedings of the Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA’04), pp. 38–58, Milovy, Czech, 2004. ˇ cka, ˇ ´ [7] L. Diening and M. Ru¨zi Calderon-Zygmund operators on generalized Lebesgue Spaces L p(x) (Ω) and problems related to fluid dynamics, J. Reine Angew. Math., 2003, 563: 197–220. [8] X.L. Fan and D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 2001, 263(2): 424–446. [9] M. Giaquinta, Multiple integrals in the calculus of variations and non-linear elliptic systems. Princeton, New Jersey: Princeton Univ. Press, 1983. [10] V.S. Guliyev, J.J. Hasanov and S.G. Samko, Boundedness of the maximal, potential and singular integral operators in generalized variable exponent Morrey spaces, Math. Scand., 2010, 107(2): 285–304. ´ V. Leˆ and M. Nuortio, Overview of differential equations with ¨ o, ¨ Ut [11] P. Harjulehto, P. Hast nonstandard growth, Nonlinear. Anal., 2010, 72(12): 4551–4574. [12] K. Ho, The fractional integral operators on Morrey spaces with variable exponent on unbounded domains, Math. Inequal. Appl., 2013, 16(2): 363–373. [13] K. Ho, Fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents, J. Math. Soc. Japan, 2017, 69(3): 1059–1077. ˇ and J. Rak ´ osn´ık, On spaces L p(x) and W k,p(x) , Czechoslovak Math. J., 1991, [14] O. Kova´cik 41(11): 592–618. [15] P. Long and H. Han, Characterizations of some operators on the vanishing generalized Morrey spaces with variable exponent, J. Math. Anal. Appl., 2016, 437(1): 419–430. [16] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 1938, 43: 126–166. ¨ [17] W. Orlicz, Uber konjugierte Exponentenfolgen, Studia Math., 1931, 3: 200–211. [18] H. Rafeiro and S.G. Samko, On maximal and potential operators with rough kernels in variable exponent spaces, Rend. Lincei Mat. Appl., 2016,27(3): 309–325. ˇ cka, ˇ [19] M. Ruzi Electrorheological fluids: modeling and mathematical theory, Berlin: Springer, 2013. [20] Y. Sawano, K. Ho, D.C. Yang and S. Yang, Hardy spaces for ball quasi-Banach function spaces, Dissertationes Math., 2017, 525: 102 pp. [21] M.E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Volume 81 of Math. Surveys and Monogr. AMS, Providence, R.I., 2000. [22] J. Wu, Boundedness for Riesz-type potential operators on Herz-Morrey spaces with variable exponent, Math. Inequal. Appl., 2015, 18(2): 471–484. [23] T. Wunderli, On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudosolutions, J. Math. Anal. Appl., 2010, 364(2): 591–598. [24] L.J. Wang and Sh. P. Tao, Boundedness of Littlewood-Paley operators and their commutators on Herz-Morrey spaces with variable exponent, J. Inequal. Appl., 2014, 2014: 227.

Operators in 5 Compact-Like Vector Lattices Normed by Locally Solid Lattices Abdullah Aydın Mus¸ Alparslan University

CONTENTS 5.1 Introduction ................................................................................................... 125 5.2 Preliminaries.................................................................................................. 126 5.3 pτ -Continuous and pτ -Bounded Operators ................................................... 128 5.4 upτ -Continuous Operators............................................................................. 132 5.5 The Compact-Like Operators ........................................................................ 134 Bibliography .......................................................................................................... 140

5.1

INTRODUCTION

Lattice-valued norms on vector lattices play an important and effective role in the functional analysis. It is well known that order convergence in vector lattice is not topological unless dimension is finite. However, thanks to the order convergence, some properties like order continuous in vector lattice can be defined. We refer the reader for an exposition on basic notions and properties of vector lattice (or, Riesz space) to [1–3,13,16,25–27,31,32] and on the theory of lattice-normed vector lattices to [9–11,16,23,24] and to locally solid vector lattices [2–6,20,32]. In this chapter, we introduce and study continuous and bounded operators with respect to the pτ -convergence, which was introduced and investigated by Aydın in [5], and we also introduce the compact operators in vector lattice normed by locally solid lattices. The concept of the unbounded convergence is crucial for the concept of pτ -convergence. The uo-convergence was introduced in [28] under the name individual convergence and the name “unbounded order convergence” was first proposed by DeMarr in [14]. The relation between weak and uo-convergence in Banach lattices was studied by Wickstead in [30], and Kaplan established two characterizations of uo-convergence on Dedekind complete Riesz spaces with the weak unit in [22], and the un-convergence was introduced in [29] under the name d-convergence. We refer the reader for an exposition on uo-convergence to [15,17–19], on un-convergence to

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[15,21,29], on the unbounded p-convergence to [10–12], and on the pτ -convergence to [5,6]. For applications of uo-convergence, we refer to [4,7,8,18,19]. The structure of this chapter is as follows. In Section 5.2, we give some basic notions and properties of vector lattices, lattice-normed spaces, locally solid Riesz spaces, unbounded convergence, and unbounded pτ -convergence. In Section 5.3, we introduce the definitions of the pτ -continuous and the pτ bounded operators between lattice-normed locally solid vector lattices. We show that p-boundedness coincides with pτ -boundedness, and also, we prove that a dominated operator is pτ -bounded; see Propositions 5.1 and 5.2, respectively. Also, we get a relation between the order continuity and pτ -continuity; see Proposition 5.3. We show that a sequentially pτ -continuous operator is norm continuous; see Proposition 5.4. In Section 5.4, we introduce the notions of upτ -continuous and sequentially upτ -continuous operator between lattice-normed locally solid vector lattices. We prove that a dominated surjective lattice homomorphism operator is sequentially upτ -continuous; see Theorem 5.1. We give some kind of upτ -continuous operator; see Propositions 5.6 and 5.7. In the last section, we introduce the notions of sequentially pτ -compact and pτ -compact operators, and we give some basic properties of them. We show that a sequence of order-bounded sequentially pτ -compact operators is sequentially pτ -compact (see Theorem 5.2). Also, it holds for the equicontinuously and uniformly convergence (see Theorem 5.3). We prove that a sequentially pτ -compact operator is compact (see Proposition 5.9). Also, a GAM-compact operator is sequentially pτ -compact (see Proposition 5.12).

5.2

PRELIMINARIES

First of all, let us recall some notations and terminologies used in this paper. In this chapter, all vector spaces are supposed to be real. Let E be a vector space. Then, E is called ordered vector space if it has an order relation “≤” (i.e, it is reflexive, antisymmetric, and transitive) that is compatible with the algebraic structure of E, it means that y ≤ x implies y + z ≤ x + z for all z ∈ E and λ y ≤ λ x for each positive scalar λ ≥ 0. An ordered vector E is said to be vector lattice (or, Riesz space) if, for each pair of vectors x, y ∈ E, the supremum x ∨ y = sup{x, y} and the infimum x ∧ y = inf{x, y} both exist in E. Then, x+ := x ∨ 0, x− := (−x) ∨ 0, and |x| := x ∨ (−x) are called the positive part, the negative part, and the absolute value of x ∈ E, respectively. Also, two vectors x and y in a vector lattice are said to be disjoint whenever |x| ∧ |y| = 0. A vector lattice is called order complete if every nonempty bounded above subset has a supremum (or, equivalently, whenever every nonempty bounded below subset has an infimum). A partially ordered set I is called directed if, for each a1 , a2 ∈ I, there is another a ∈ I such that a ≥ a1 and a ≥ a2 (or, a ≤ a1 and a ≤ a2 ). A function from a directed set I into a set E is called a net in E. A vector lattice is order complete if 0 ≤ xα ↑≤ x implies the existence of sup xα . A net (xα )α∈A in a vector lattice X is called order convergent (or shortly, o-convergent) to x ∈ X if there exists another net (yβ )β ∈B

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127

satisfying yβ ↓ 0, and for any β ∈ B, there exists αβ ∈ A such that |xα − x| ≤ yβ o for all α ≥ αβ . In this case, we write xα → − x; for more details, see, for example, [3,31,32]. In a vector lattice X, a net (xα ) is unbounded order convergent to x ∈ X if o |xα − x| ∧ u → − 0 for every u ∈ X+ ; see, for example, [15,17–19,28]. Recall that every linear topology τ on a vector space E has a base N for the zero neighborhoods satisfying the following four properties: For each V ∈ N , we have λV ⊆ V for all scalar |λ | ≤ 1; for any V1 ,V2 ∈ N , there is another V ∈ N such that V ⊆ V1 ∩ V2 ; for each V ∈ N , there exists another U ∈ N with U + U ⊆ V ; and for any scalar λ and each V ∈ N , the set λV is also in N ; for much more detail, see [2,3]. A subset A of a vector lattice is called solid whenever |x| ≤ |y| and y ∈ A imply x ∈ A. A solid vector subspace is referred to as an order ideal. An order closed ideal is referred to as a band. A sublattice Y of a vector lattice is majorizing E if, for every x ∈ E, there exists y ∈ Y with x ≤ y; see, for example, [1–3,32]. Let E be a vector lattice and τ be a linear topology on E that has a base at zero consisting of solid sets. Then, the pair (E, τ) is said to be a locally solid vector lattice (or, locally solid Riesz space). It should be noted that all topologies considered throughout this chapter are assumed to be Hausdorff. It follows from [2, Thm. 2.28.] that a linear topology τ on a vector lattice E is a locally solid if it is generated by a family of Riesz pseudonorms {ρ j } j∈J . Moreover, if a family of Riesz pseudonorms generates a locally solid topolτ ogy τ on a vector lattice E, then xα → − x if ρ j (xα − x) → 0 in R for each j ∈ J. Since E is Hausdorff, the family {ρ j } j∈J of Riesz pseudonorms is separating; i.e., if ρ j (x) = 0 for all j ∈ J, then x = 0. A locally solid vector lattice is said to have o τ Lebesgue property if xα → − 0 in E implies xα → − 0. In this chapter, unless otherwise, the pair (E, τ) refers to as a locally solid vector lattice, and the topologies in locally solid vector lattices are generated by a family of Riesz pseudonorms {ρ j } j∈J . Let X be a vector space, E be a vector lattice, and p : X → E+ be a vector norm (i.e., p(x) = 0 ⇔ x = 0, p(λ x) = |λ |p(x) for all λ ∈ R, x ∈ X, and p(x + y) ≤ p(x) + p(y) for all x, y ∈ X), then the triple (X, p, E) is called a latticenormed space, abbreviated as LNS. The lattice norm p in an LNS (X, p, E) is said to be decomposable if, for all x ∈ X and e1 , e2 ∈ E+ , it follows from p(x) = e1 + e2 that there exists x1 , x2 ∈ X such that x = x1 + x2 and p(xk ) = ek for k = 1, 2. We refer the reader for more information on LNSs to [13,23,24] and [10–12]. A linear operator T between two LNSs (X, p, E) and (Y, m, F) is said to be dominated if there is a positive operator S : E → F satisfying m(T x) ≤ S(p(x)) for all x ∈ X. In an LNS (X, p, E), a subset A of X is called p-bounded if there exists e ∈ E such that p(a) ≤ e for all a ∈ A; see [11, Def. 2.]. If X is a vector lattice and the vector norm p is monotone (i.e., |x| ≤ |y| ⇒ p(x) ≤ p(y)), then the triple (X, p, E) is called a lattice-normed vector lattice, abbreviated as LNV L; see [10–12]. Let (X, p, E) be an LNVL with (E, τ), which is a locally solid vector lattice, then (X, p, Eτ ) is said to be a lattice-normed locally solid Riesz space (or lattice-normed locally solid vector lattice), abbreviated as LSNV L in [5]. Throughout this chapter, we use X instead of (X, p, Eτ ) and Y instead of (Y, m, Fτ´ ).

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Note also that L(X,Y ) denotes the space of all linear operators between vector spaces X and Y . If X is a normed space, then X ∗ denotes the topological dual of X and BX denotes the closed unit ball of X. pτ τ We abbreviate the convergence p(xα − x) → − 0 as xα −→ x, and say in this case that (xα ) pτ -converges to x. A net (xα )α∈A in an LSNV L (X, p, Eτ ) is said to be pτ -Cauchy if the net (xα − xα 0 )(α,α 0 )∈A×A pτ -converges to 0. An LSNV L (X, p, Eτ ) is called (sequentially) pτ -complete if every pτ -Cauchy (sequence) net in X is pτ convergent. In an LSNV L (X, p, Eτ ), a subset A of X is called pτ -bounded if p(A) is o τ -bounded in E. An LSNV L (X, p, Eτ ) is called opτ -continuous if xα → − 0 implies that τ p(xα ) → − 0. A net (xα ) in an LSNV L (X, p, Eτ ) is said to be unbounded pτ -convergent upτ τ to x ∈ X (shortly, (xα ) upτ -converges to x or xα −−→ x) if p(|xα − x| ∧ u) → − 0 for all u ∈ X+ ; see for much more information [5]. Let (X, p, E) be an LNS and (E, k·kE ) be a normed vector lattice. The mixed-norm on X is defined by p-kxkE = kp(x)kE for all x ∈ X. In this case, the normed space (X, p-k·kE ) is called a mixed-normed space; see [23, 7.1.1, p. 292]. Lastly, it should be noticed that the theory of lattice-normed spaces is well developed in the case of decomposable lattice norms. In this chapter, we usually do not assume lattice norms to be decomposable. On the other hand, throughout this chapter, all vector lattices are assumed to be Archimedean, and also, we frequently use the following lemmas and so we shall keep in mind them; see [5, Lem. 1.1.] and [11, Lem. 1.], respectively. Lemma 5.1. If (xα )α∈A and (yα )α∈A be two nets in a locally solid vector lattice τ τ (E, τ ) such that |xα | ≤ |yα | for all α ∈ A and yα → − 0, then xα → − 0. Lemma 5.2. Let (X, p, E) be an LNS such that (E, k·kE ) is a Banach space. If (X, p, E) is sequentially p-complete, then (X, p-k·kE ) is the Banach space.

5.3

pτ -CONTINUOUS AND pτ -BOUNDED OPERATORS

In this section, we define the notions of pτ -continuous and pτ -bounded operators between LSNVLs. Recall that an operator T between two LNVLs X and Y is called p p p-continuous if xα − → 0 in X implies T xα − → 0 in Y , and if the condition holds only for sequences, then T is called sequentially p-continuous. Moreover, T is also called p-bounded if it maps p-bounded sets in X to p-bounded sets in Y ; see [11]. Motivated by these definitions, we give the following notions. Definition 5.1. Let X and Y be two LSNV Ls and T ∈ L(X,Y ). Then, T is called pτ

pτ

pτ -continuous if xα −→ 0 in X implies T xα −→ 0 in Y , and if the condition holds only for sequences, then T is called sequentially pτ -continuous, 2. pτ -bounded if it maps pτ -bounded sets to pτ -bounded sets. 1.

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Remark 5.1. Let T be an operator between LSNV Ls (X, p, Eτ ) and (Y, m, Fτ´ ) with (E, τ) and (F, τ´) having order-bounded neighborhoods of zero. Then, by applying [2, Thm. 2.19(i)] and [20, Thm. 2.2.], one can see that T is p-bounded if it is pτ -bounded. Moreover, T : (E, |·|, Eτ ) → (F, |·|, Fτ´ ) is pτ -bounded if T : X → Y is order-bounded. ii. Let X be a vector lattice and (Y, k·kY ) be a normed space. Then, T ∈ L(X,Y ) i.

k·kY

o

is called order-to-norm continuous if xα → − 0 in X implies T xα −−→ 0; see [27, Sec. 4., p. 468]. For a locally solid lattice (X, τ) with the Lebesgue property, the pτ -continuity of T : (X, |·|, Xτ ) → (Y, k·kY , R) implies order-tonorm continuity of it. iii. Let X be a vector lattice and (Y, m, Fτ´ ) be an LSNV L, and T : X → Y be a strictly positive operator. Define p : X → F+ , by p(x) = (m ◦ T )(|x|). Then, (X, p, Fτ´ ) is an LSNV L, and also, the map T : (X, p, Fτ´ ) → (Y, m, Fτ´ ) is pτ continuous. In the following work, we show that the collection of all pτ -continuous operators between LSNV Ls is the vector space. Lemma 5.3. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be two LSNV Ls. If T, S : (X, p, Eτ ) → (Y, m, Fτ´ ) are pτ -continuous operators, then λ S + µT is pτ -continuous for any real numbers λ and µ. In particular, if H = T − S, then H is pτ -continuous. ii. If −T1 ≤ T ≤ T2 with T1 and T2 positive and pτ -continuous operators, then T is pτ -continuous. i.

pτ

τ´

− 0 in F. To do this, we Proof. (i) We show xα −→ 0 in X implies m(Sxα + T xα ) → consider the following inequality: τ´

m(Sxα + T xα ) ≤ m(Sxα ) + m(T xα ) → − 0. So S + T is also pτ -continuous. Now, for arbitrary real numbers λ and µ, we have m(λ Sxα + µT xα ) ≤ |λ |m(Sxα ) + |µ|m(T xα )
τ´ ≤ (|λ | + |µ |) m(Sxα ) + m(T xα ) → − 0. Therefore, λ S + µT is pτ -continuous. (ii) If −T1 ≤ T ≤ T2 with T1 and T2 positive and pτ -continuous, then we have 0 ≤ T + T1 ≤ T2 + T1 where T + T1 is positive and pτ -continuous. Thus, it follows from (i) that T = (T + T1 ) − T1 is pτ -continuous.

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By the following result, we prove that there is a relation between p- and pτ bounded notions. Proposition 5.1. Let T be an operator between LSNV Ls (X, p, Eτ ) and (Y, m, Fτ´ ) with (E, τ) and (F, τ´) having order-bounded neighborhoods of zero. Then, T is pbounded if it is pτ -bounded. Proof. We show only one direction. Assume T is p-bounded. Suppose A is pτ bounded subset in X, and so p(A) is τ-bounded in E. Thus, p(A) is also orderbounded in E; see
[20, Thm. 2.2.]. Since T is p-bounded, T
(A) is p-bounded in Y , and so m T (A) is order-bounded in F. Therefore, m T (A) is τ´-bounded in F; see [2, Thm. 2.19(i)]. Hence, T (A) is pτ -bounded in Y . Proposition 5.2. Any dominated operator T from an LSNV L (X, p, Eτ ) with (E, τ) having order-bounded τ-neighborhood of zero to an LSNV L (Y, m, Fτ´ ) is pτ bounded. Proof. Consider a pτ -bounded subset A in X. That is, p(A) is τ-bounded in E. So p(A) is order-bounded in E; see
[20, Thm. 2.2.]. Let S be the dominant of T . Since
S is a positive operator, S p(A) is order-bounded in F. Also, we know that m T (a ) ≤

S p(a) for all a ∈ A, and
so m T (A) is order-bounded in F. Hence, by applying [2, Thm. 2.19(i)], m T (A) is τ´-bounded in F. Therefore, T is pτ -bounded.  The converse of Proposition 5.2 is not true in general. For instance, consider `∞ with the norm topology and R with the usual topology, and the identity operator I : (`∞ , |·|, `∞ ) → (`∞ , k·k, R). It is pτ -bounded. Indeed, for any pτ -bounded set A

in `∞ , |A| is τ-bounded in `∞ . Thus, |A| = kAk is bounded in R. But it is not dominated; see [13, Rem., p. 388]. The next proposition gives a relation between the pτ - and order continuity. Proposition 5.3. Let (Y, m, Fτ´ ) be arbitrary and (X, p, Eτ ) be opτ -continuous LSNV Ls and T : (X, p, Eτ ) → (Y, m, Fτ´ ) be a (sequentially) pτ -continuous positive operator. Then, T : X → Y is (σ -) order continuous operator. τ

Proof. Assume xα ↓ 0 in X. Since X is opτ -continuous, we have p(xα ) → − 0, and so pτ

τ´

xα −→ 0 in X. By the pτ -continuity of T , m(T xα ) → − 0 in F. It can be seen that T xα ↓ because T is positive. Then, applying [5, Prop. 2.4.], we get T xα ↓ 0. Thus, T is order continuous. Corollary 5.1. Let (X, p, E) be an opτ -continuous LSNV L and (Y, m, Fτ´ ) be an LSNV L with Y being order complete. If T : (X, p, E) → (Y, m, F) is pτ -continuous and T ∈ L∼ (X,Y ), then T : X → Y is order continuous. Proof. Since Y is order-complete and T is order-bounded, by Riesz-Kantorovich formula, we have T = T + − T − . Now, Proposition 5.3 implies that T + and T − are both order continuous, and so T is order continuous.

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In the following result, which is pτ -version of [11, Prop. 3.], we give norm continuity of sequentially pτ -continuous operator on the mixed-norms. Proposition 5.4. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be two LSNV Ls with (E, k·kE ) and (F, k·kF ) being normed vector lattices, and where τ and τ´ are generated by the norms. If T : (X, p, Eτ ) → (Y, m, Fτ´ ) is sequentially pτ -continuous, then T : (X, p-k·kE ) → (Y, m-k·kF ) is norm continuous. Proof. (⇒) Assume T : (X, |·|, X) → (X, k·kX , R) to be sequentially pτ -continuous, k·k

τ

− 0 in X, and since T is sequentially and a sequence xn −→ 0 in X. Thus, xn → k·k

pτ -continuous, T xn −→ 0. pτ (⇐) Assume that T is norm continuous, and let xn −→ 0 in (X, |·|, X). Then, k·k

τ

k·k

− 0 or xn −→ 0 in X. Hence, T xn −→ 0. Therefore, T : (X, |·|, X) → (X, k·kX , R) |xn | → is sequentially p-continuous.  Remark 5.2. By applying [2, Thm. 2.19(i)] and [11, Prop. 4.], one can see that every pτ -continuous operator is pτ -bounded. But a pτ -continuous operator T : (X, p, Eτ ) → (Y, m, Fτ´ ) need not to be order-bounded from X to Y . Indeed, consider the classical “Fourier coefficients” operator T : L1 [0, 1] → c0 defined by the formula Z T(f) =

1

Z 1

f (x)sinx dx, 0

 f (x)sin2x dx, ... .

0

Then, T : L1 [0, 1] → c0 is norm-bounded, but it is not order-bounded; see [2, Exer. 10., p. 289]. So T : (L1 [0, 1], k·kL1 , R) → (c0 , k·k∞ , R) is pτ -continuous and is not order-bounded. Using [20, Thm. 2.2.] in Remark 5.2, it can be seen that pτ -continuity implies order boundedness if (F, τ´) has order-bounded τ´-neighborhood of zero. Recall that an operator T ∈ L(X,Y ), where X and Y are the normed spaces, is called Dunfordw

k·k

Pettis if xn − → 0 in X implies T xn −→ 0 in Y . We show the following result, which is pτ -version of [11, Prop. 5.], so we omit its proof. Proposition 5.5. Let (X, k·kX ) be a normed vector lattice and (Y, k·kY ) be a normed ∗ space. Put E := RX and define p : X → E+ , by p(x)[ f ] = | f |(|x|) for f ∈ X ∗ . It is easy to see that (X, p, Eτ ), where τ is the topology generated by the norm k·kX ∗ , is an LSNV L. Then, the followings hold: If T ∈ L(X,Y ) is a Dunford-Pettis operator then T : (X, p, Eτ ) → (Y, k·kY , R) is sequentially pτ -continuous, ii. The converse holds if the lattice operations of X are weakly sequentially continuous. i.

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upτ -CONTINUOUS OPERATORS

Recall that a net (xα ) in an LSNV L (X, p, Eτ ) is said to be unbounded pτ -convergent τ to x if p(|xα − x| ∧ u) → − 0 for all u ∈ X+ ; see [5]. Motivated by up-continuous operators in [11] and un-continuous functionals in [21, p. 16], and by using the upτ convergence, we introduce the following notion. Definition 5.2. An operator T between two LSNV Ls X and Y is called upτ continuous if it maps the upτ -convergent net to upτ -convergent nets. If it holds only for sequence, then T is called sequentially upτ -continuous. It is clear that if T is (sequentially) pτ -continuous operator, then T is (sequentially) upτ -continuous. For an LSNV L (X, p, Eτ ), a sublattice Y of X is called upτ upτ upτ -regular if, for any net (yα ) in Y , the convergence yα −−→ 0 in Y implies yα −−→ 0 in X. The following is a more general extension of [21, Prop. 9.4.]. Theorem 5.1. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be LSNV Ls with (E, k·kE ) being a Banach lattice and (F, k·kF ) being a normed vector lattice, and also τ and τ´ are being generated by the norms. Then, the followings hold: A dominated surjective lattice homomorphism operator T ∈ L(X,Y ) is sequentially upτ -continuous; ii. If T ∈ L(X,Y ) is a dominated lattice homomorphism operator and T (X) is upτ -regular in Y , then it is sequentially upτ -continuous; iii. If T ∈ L(X,Y ) is a dominated lattice homomorphism operator and IT (X) (the ideal generated by T (X)) is upτ -regular in Y , then it is sequentially upτ continuous. i.

upτ

Proof. (i) Let’s fix a sequence xn −−→ 0 in X and u ∈ Y+ . Since T is a surjective lattice τ homomorphism, we have some v ∈ X+ such that T v = u. So we have p(|xn | ∧ v) → −0 in E. Since T is dominated, there is a positive operator S : E → F such that

m T (|xn | ∧ v) ≤ S p(|xn | ∧ v) . Taking into
account that T
is a lattice homomorphism and T v = u, we get m |T xn | ∧ u ≤ S p(|xn | ∧ v) . By using [2, Thm. 4.3.], we know that every positive operator from a Banach lattice to the normed vector lattice is continuous, and so
τ´
τ´ S is continuous. Hence, we get S p(|xn | ∧ v) → − 0 in F. That is, m |T xn | ∧ u → − 0, and we get the desired result. (ii) Since
T is a lattice homomorphism, T (X) is a vector sublattice of Y . So
T (X), m, Fτ´ is an LSNV L. Thus, by (i), we have T : (X, p, Eτ ) → T (X), m, Fτ´ is sequentially upτ -continuous. Next, we show that T : (X, p, Eτ ) → (Y, m, Fτ´ ) is sequentially upτ -continuous. upτ Consider an upτ -convergent to zero sequence (xn ) in X. That is, T xn −−→ 0 in T (X).

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Since T (X) is upτ -regular in Y , T (xn ) −−→ 0 in Y . Therefore, T is sequentially upτ continuous. upτ

τ

(iii) Let (xn ) −−→ 0 sequence in X. Thus, p(|xn | ∧ u) → − 0 in E for all u ∈ X+ . Fix 0 ≤ w ∈ IT (X) . Then, there is x ∈ X+ such that 0 ≤ w ≤ T x. For a dominant S, we

have m T (|xn | ∧ x) ≤ S p(|xn | ∧ x) , and so, by taking lattice homomorphism of T , we have

m (|T xn | ∧ T x) ≤ S p(|xn | ∧ x) .

It follows from 0 ≤ w ≤ T x that m (|T xn | ∧ w) ≤ S p(|T xn | ∧ x) . Now, the argument given in the proof of
(i) can be repeated here as well. Thus, we see that T : (X, p, Eτ ) → IT (X) , m, Fτ´ is sequentially upτ -continuous. Since IT (X) is upτ regular in Y , it can be easily seen by (ii) that T : X → Y is sequentially upτ continuous. It should be mentioned, by using Theorem 5.1, that an operator, surjective lattice homomorphism with order continuous dominant, is upτ -continuous. Proposition 5.6. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be two LSNV Ls with Y being ordercomplete vector lattice. For a positive upτ -continuous operator T : (X, p, Eτ ) → (Y, m, Fτ´ ), consider the operator S : (X+ , p, Eτ ) → (Y+ , m, Fτ´ ) defined by S(x) = upτ sup{T (xα ∧ x) : xα ∈ X+ , xα −−→ 0} for each x ∈ X+ . Then, we have the followings: i. S is the upτ -continuous operator; ii. The Kantorovich extension of S is the upτ -continuous operator.
τ´ upτ upτ Proof. (i) Suppose X+ 3 yβ −−→ 0. Then, Tyβ −−→ 0 in Y , and so m Tyβ ∧ w → − 0 in F for all w ∈ Y+ . For any net xα ∈ X+ and fixed w ∈ Y+ , we have τ´

T (xα ∧ yβ ) ∧ w ≤ T (yβ ) ∧ w → − 0. Therefore, we get the desired result. (ii) We show first that S has the Kantorovich extension. To show this, let’s see additivity of it. By using [3, Lem. 1.4.], for any upτ -null net (xα ) in X+ , we have
T (x + y) ∧ xα ≤ T (x ∧ xα ) + T (y ∧ xα ) ≤ S(x) + S(y). So by taking supremum, we get S(x + y) ≤ S(x) + S(y). On the other hand, for any two upτ -null nets (xα ) and (yβ ) in X+ , using the formula in the proof of [3, Thm. 1.28.], we get
T (x ∧ xα ) + T (y ∧ yβ ) = T (x ∧ xα + y ∧ yβ ) ≤ T (x + y) ∧ (xα + yβ ) ≤ S(x + y). So S(x) + S(y) ≤ S(x + y). By [3, Thm. 1.10.], S extends to a positive operator, denoted by Sˆ : (X, p, Eτ ) → (Y, m, Fτ´ ). That is, Sˆx = S(x+ ) − S(x− ) for all x ∈ X. Now, upτ upτ upτ ˆ Fix a net wβ − we show upτ -continuity of S. −→ 0 in X. Then, w+ −→ 0 and w− −→ 0 β − β − − upτ − upτ + upτ + ˆ in X, and so S(w ) −−→ 0 and S(w ) −−→ 0 in Y . Hence, Swβ = S(w ) − S(w ) −−→ 0 β

in Y .

β

β

β

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We complete this section with the following technical work. Proposition 5.7. Consider a positive upτ -continuous operator T between LSNV Ls X and Y , and an ideal A in X. Then, an operator S : (X, p, Eτ ) → (Y, m, Fτ´ ) defined by S(x) = sup T (|x| ∧ a) for each x ∈ X is upτ -continuous operator. a∈A upτ

upτ

upτ

Proof. Let xα −−→ 0 be a net in X. Then, |xα | −−→ 0, and so T (|xα |) −−→ 0 in Y . Thus, for each u ∈ Y+ , we have upτ |S(xα )| ∧ u = sup T (|xα | ∧ a) ∧ u ≤ T (|xα |) ∧ u ≤ T (|xα |) ∧ u −−→ 0. a∈A

upτ

Therefore, S(xα ) −−→ 0 in Y .

5.5

THE COMPACT-LIKE OPERATORS

In this section, we define the notions of pτ -compact and sequentially pτ -compact operators in LSNV Ls and study their properties. A linear operator T between normed spaces is called compact if T (BX ) is relatively compact, or equivalently, T is compact if, for any norm-bounded sequence (xn ), there is a subsequence (xnk ) such that the sequence (T xnk ) is convergent. Similarly, we introduce the following notions. Definition 5.3. Let X and Y be two LSNV Ls and T ∈ L(X,Y ). Then, T is called pτ -compact if, for any pτ -bounded net (xα ) in X, there is a subnet (xαβ ) such that pτ

T xαβ −→ y in Y for some y ∈ Y . If it holds only for sequence, then T is called sequentially pτ -compact. Example 5.1. Let (X, k·kX ) and (Y, k·kY ) be the normed spaces. Then, T : (X, k·kX , R) → (Y, k·kY , R) is (sequentially) pτ -compact if T : X → Y is compact. ii. Let X be a vector lattice and Y be a normed space. An operator T ∈ L(X,Y ) is said to be AM-compact if T [−x, x] is relatively compact for every x ∈ X+ ; see [26, Def. 3.7.1.]. Let (X, τ) be a locally solid vector lattice with orderbounded τ-neighborhood and (Y, k·kY ) be a normed vector lattice. Then, T ∈ L(X,Y ) is the AM-compact operator if T : (X, |·|, Xτ ) → (Y, k·kY , R) is pτ -compact; apply [20, Thm. 2.2.] and [2, Thm. 2.19(i)]. iii. Consider a topology τ in `2 with the norm k·k2 , and so (`2 , |·|, `2τ ) and (`2 , k·k2 , R) are two LSNV Ls. Thus, a linear operator Tn : (`2 , |·|, `2τ ) → (`2 , k·k2 , R) defined by Tn x = (x1 , x2 , · · · , xn , 0, 0 · · · ) for all x ∈ `2 is pτ compact. Indeed, let (xn ) be a pτ -bounded, or |(xn )| is k·k2 -bounded sequence in `2 . Hence, (T xn ) is k·k2 -bounded in `2 . Since every bounded sequence in Rn has a convergent subsequence, it follows that Tn is pτ -compact. i.

Lemma 5.4. If S and T are (sequentially) pτ -compact operators between LSNV Ls, then T + S and λ T , for any real number λ , are also (sequentially) pτ -compact operators.

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Proof. We show that only T + S is pτ -compact, and the other cases are analogous. Let (xα ) be a pτ -bounded net in X. Since T is pτ -compact, there is a subnet (xαβ ) τ´

such that m(T xαβ − y) → − 0 in F for some y ∈ Y . On the other hand, (xαβ ) is also pτ -bounded in X and S is pτ -compact, and then there is a subnet (xαβγ ) such that τ´

τ´

m(Sxαβγ − z) → − 0 in F for some z ∈ Y . Also, we have m(T xαβγ − y) → − 0 in F. Thus, τ´

m[(S + T )(xαβγ ) − y − z)] → − 0. Hence, S + T is pτ -compact. Proposition 5.8. Let (X, p, Eτ ) be an LSNV L and R, T, S ∈ L(X). If T is a (sequentially) pτ -compact and S is a (sequentially) pτ -continuous operator, then S ◦ T is (sequentially) pτ -compact. ii. If T is a (sequentially) pτ -compact and R is a pτ -bounded operator, then T ◦ R is (sequentially) pτ -compact. i.

Proof. We give the only pτ -compactness, and the sequential case is analogous. i. Consider a pτ -bounded net (xα ) in X. Since T is pτ -compact, there are a pτ subnet (xαβ ) and x ∈ X such that T xαβ −→ x. By pτ -continuity of S, we have pτ

S(T xαβ ) −→ Sx in E. Therefore, S ◦ T is pτ -compact. ii. Suppose (xα ) is a pτ -bounded net in X. Since R is pτ -bounded, (Rxα ) is pτ -bounded. Then, by pτ -compact of T , there are a subnet (xαβ ) and y ∈ X pτ

such that T (Rxαβ ) −→ y. Therefore, T ◦ R is pτ -compact.

Remark 5.3. Let X be an LSNV L and (Y, τ´) be a locally solid vector lattice with Y being compact. Then, each operator T : (X, p, Eτ ) → (Y, |·|,Yτ´ ) is (sequentially) pτ -compact. ii. Let X be an LSNV L and (Y, k·kY ) be a finite dimensional normed space, and τ´ be the topology generated by this norm. If T : (X, p, Eτ ) → (Y, |·|,Yτ´ ) is pτ -bounded operator, then it is sequentially pτ -compact. iii. Let (X, τ) be a locally solid vector lattice with an order-bounded τ-neighborhood of zero and (Y, m, Fτ´ ) be an opτ -continuous LSNV L with Y being an atomic KB-space. If T : X → Y is an order-bounded operator, then T : (X, |·|, Xτ ) → (Y, m, Fτ´ ) is pτ -compact; see [20, Thm. 2.2.] and [11, Rem. 6.]. i.

Remark 5.4. Let (Tm ) be a sequence of sequentially pτ -compact operators from X to Y . For a given pτ -bounded sequence (xn ) in X, by a standard diagonal argument, pτ there exists a subsequence (xnk ) such that, for any m ∈ N, Tm xnk −→ ym for some ym ∈ Y .

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Theorem 5.2. Let (Tm ) be a sequence of order-bounded sequentially pτ -compact operators from (X, p, Eτ ) to a sequentially pτ -complete opτ -continuous (Y, q, Fτ´ ) with o Y being order complete. If Tm → − T in Lb (X,Y ), then T is sequentially pτ -compact. Proof. Let (xn ) be a pτ -bounded sequence in X. By Remark 5.4, there exists a subpτ sequence (xnk ) such that, for any m ∈ N, Tm xnk −→ ym for some ym ∈ Y . We show that (ym ) is a pτ -Cauchy sequence. Consider the following formula q(ym − y j ) ≤ q(ym − Tm xnk ) + q(Tm xnk − T j xnk ) + q(T j xnk − y j ).

(5.1)

The first and the third terms in the last inequality both τ´-converge to zero as m → ∞ o o and j → ∞, respectively. Since Tm → − T , we have Tm xnk → − T xnk for all xnk ; see [31, Thm. VIII.2.3.]. Then, for a fixed index k, we have o

|Tm xnk − T j xnk | ≤ |Tm xnk − T xnk | + |T xnk − T j xnk | → −0 o

as m, j → ∞, and so (Tm − T j )xnk → − 0 in Y . Hence, by opτ -continuity of (Y, q, Fτ´ ), τ´

we get q(Tm xnk − T j xnk ) → − 0 in F. By formula (1), (ym ) is pτ -Cauchy. Since Y is τ´

sequentially pτ -complete, there is y ∈ Y such that q(ym − y) → − 0 in F as m → ∞. So, for arbitrary m, if we take τ´-limit with k in the following formula q(T xnk − y) ≤ q(T xnk − Tm xnk ) + q(Tm xnk − ym ) + q(ym − y), we get τ´ − lim q(T xnk − y) ≤ q(T xnk − Tm xnk ) + q(ym − y) because of q(Tm xnk − τ´

τ´

ym ) → − 0. Since m is arbitrary, τ´ − lim q(T xnk − y) → − 0. Thus, T is sequentially pτ compact. Similar to Theorem 5.2, we give the following theorem by using equicontinuously and uniformly convergence. Theorem 5.3. Let (Tm ) be a sequence of sequentially pτ -compact operators from (X, |·|, Xτ ) to a sequentially pτ -complete LSNV L (Y, |·|,Yτ ). Then, the followings hold: i. If (Tm ) converges equicontinuously to an operator T : (X, |·|, Xτ ) → (Y, |·|,Yτ´ ), then T is sequentially pτ -compact, ii. If (Tm ) uniformly converges on zero neighborhoods to an operator T : (X, |·|, Xτ ) → (Y, |·|,Yτ´ ), then T is sequentially pτ -compact. Let (X, E) be a decomposable LNS and (Y, F) be an LNS with F being order complete. Then, each dominated operator T : X → Y has the exact dominant |T | : E → F; see [23, 4.1.2., p. 142]. For a sequence (Tn ) in the set of dominated operators τ´

M(X,Y ), we call Tn → T in M(X,Y ) whenever |Tn − T |(e) → − 0 in F for each e ∈ E. Theorem 5.4. Let (X, p, Eτ ) be a decomposable and (Y, q, Fτ´ ) be a sequentially pτ complete LSNV Ls with F being order complete. If (Tm ) is a sequence of sequentially pτ -compact operators and Tm → T in M(X,Y ) then T is sequentially pτ -compact.

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Proof. Let (xn ) be a pτ -bounded sequence in X. By Remark 5.4, there exists a subpτ sequence (xnk ) and a sequence (ym ) in Y such that, for any m ∈ N, Tm xnk −→ ym . We show that (ym ) is pτ -Cauchy sequence in Y . Consider formula (1) of Theorem 5.2. Similarly, the first and the third terms in the last inequality of (1) both τ´-converge to zero as m → ∞ and j → ∞, respectively. Since Tm ∈ M(X,Y ) for all m ∈ N,


τ´ q(Tm xnk − T j xnk ) ≤ |Tm − T j | p(xnk ) ≤ |Tm − T | p(xnk ) + |T − T j | p(xnk ) → −0 τ´

as m, j → ∞. Thus, q(ym − y j ) → − 0 in F as m, j → ∞. Therefore, (ym ) is pτ -Cauchy. τ´

Since Y is sequentially pτ -complete, there is y ∈ Y such that q(ym − y) → − 0 in F as m → ∞. By the following formula q(T xnk − y) ≤ q(T xnk − Tm xnk ) + q(Tm xnk − ym ) + q(ym − y)
≤ |Tm − T|| p(xnk ) + q(Tm xnk − ym ) + q(ym − y) τ´

and by repeating the same of last part of Theorem 5.2, we get q(T xnk −y) → − 0. Therefore, T is sequentially pτ -compact. Proposition 5.9. Let (X, p, Eτ ) be an LSNV L, where (E, k·kE ) is an AM-space with a strong unit, and (Y, m, Fτ´ ) be an LSNV L, where (F, k·kF ) is the normed vector lattice and τ´ is generated by the norm k·kF . If T : (X, p, Eτ ) → (Y, m, Fτ´ ) is sequentially pτ -compact, then T : (X, p-k·kE ) → (Y, m-k·kF ) is compact. Proof. Let (xn ) be a normed bounded sequence in (X, p-k·kE ). That is, p-kxn kE = kp(xn )kE < ∞ for all n ∈ N. Since (E, k·kE ) is an AM-space with a strong unit, p(xn ) is order-bounded in E. Thus, p(xn ) is τ-bounded in E; see [2, Thm. 2.19(i)]. So (xn ) is a pτ -bounded sequence in (X, p, Eτ ). Since T is sequentially τ´

pτ -compact, there are a subsequence xnk and y ∈ Y such that m(T xnk − y) → −0 in F. Then, km(T xnk − y)kF → 0 or m-kT xnk − ykF → 0 in F. Thus, the operator T : (X, p-k·kE ) → (Y, m-k·kF ) is compact. It is known that a finite rank operator is compact. Similarly, we give the following result. Proposition 5.10. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be LSNV Ls with (F, τ´) having the Lebesgue property. Consider an operator T : (X, p, Eτ ) → (Y, m, Fτ´ ) defined by T x = f (x)y0 , where y0 ∈ Y and f is a linear functional on X. If f : (X, p, Eτ ) → (R, |·|, R) is pτ -bounded, then T is (sequentially) pτ -compact. Proof. Suppose (xα ) is a pτ -bounded net in X. Since f is pτ -bounded, f (xα ) is bounded in R. Then, there is a subnet (xαβ ) such that f (xαβ ) → λ for some λ ∈ R. For y0 ∈ Y , we have the following formula:

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o m(T xαβ −λ y0 ) = m( f (xαβ )y0 −λ y0 ) = m ( f (xαβ )−λ )y0 = | f (xαβ )−λ |m(y0 ) → − 0. τ

By the Lebesgue property of F, we get m(T xαβ − λ y) → − 0 in E. Thus, T is pτ compact. Proposition 5.11. Let (X, p, Eτ ) be an LSNV L with (E, τ) having an order-bounded τ-neighborhood and (Y, m, Fτ´ ) be an LSNV L, where (Y, k·kY ) is an order continuous atomic KB-space and τ´ is generated by k·kY . If T : (X, p, Eτ ) → (Y, |·|,Yτ´ ) is p-bounded or dominated operator, then it is pτ -compact. Recall that a linear operator T from an LNS (X, E) to a Banach space (Y, k·kY ) is called the generalized AM-compact or GAM-compact if, for any p-bounded set A in X, T (A) is relatively compact in (Y, k·kY ). Proposition 5.12. Let (X, p, Eτ ) be an LSNV L with (E, τ) having an order-bounded τ-neighborhood and (Y, m, Fτ´ ) be an opτ -continuous LSNV L with a Banach lattice (Y, k·kY ). If T : (X, p, Eτ ) → (Y, k·kY ) is GAM-compact, then T : (X, p, Eτ ) → (Y, m, Fτ´ ) is sequentially pτ -compact. Proof. Let (xn ) be a pτ -bounded sequence in X. By [20, Thm. 2.2.], (xn ) is p-bounded in (X, p, Eτ ). Since T is GAM-compact, there are a subsequence (xnk ) and some y ∈ Y such that kT xnk − ykY → 0. Since (Y, k·kY ) is Banach lattice then, o by [31, Thm. VII.2.1.], there is a further subsequence (xnk j ) such that T xnk j → − y in Y . pτ

Then, by opτ -continuity of (Y , m, Fτ´ ), we get T xnk j −→ y in Y . Hence, T is sequentially pτ -compact. Proposition 5.13. Let (X, k·kX ) be a normed lattice and (Y, k·kY ) be a Banach lattice. If T : (X, k·kX , R) → (Y, |·|,Yτ´ ) is sequentially pτ -compact and p-bounded, and f : Y → R is σ -order continuous, then ( f ◦ T ) : X → R is compact. Proof. Assume (xn ) be a norm-bounded sequence in X. Since T is sequentially pτ pτ compact, there are a subsequence (xnk ) and y ∈ Y such that T xnk −→ y or |T xnk − k·kY

τ´

y| → − 0 or T xnk −−→ y in Y . Since (Y, k·kY ) be a Banach lattice, there is a further o subsequence (xnk j ) such that T xnk j → − y in Y ; see [31, Thm. VII.2.1.]. By σ -order continuity of f , we have ( f ◦ T )xnk j → f (y) in R. We now turn our attention to the upτ -compact operators. Definition 5.4. Let X and Y be two LSNV Ls and T ∈ L(X,Y ). Then, T is called upτ -compact if, for any pτ -bounded net (xα ) in X, there is a subnet (xαβ ) such that upτ

T xαβ −−→ y in Y for some y ∈ Y . If the condition holds only for sequences, then T is called sequentially-upτ -compact. It is clear that a pτ -compact operator is upτ -compact, and similar to Lemma 5.4, linear properties hold for upτ -compact operators. Moreover, an operator T ∈ L(X,Y )

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is (sequentially) un-compact if T : (X, k·kX , R) → (Y, k·kY , R) is (sequentially) upτ compact; see [21, Sec. 9., p. 28]. Similar to Proposition 5.8, we give the following results. Proposition 5.14. Let (X, p, Eτ ) be an LSNV L and R, T, S, H ∈ L(X). i. ii.

If T is an (sequentially) upτ -compact and S is a (sequentially) pτ continuous, then S ◦ T is (sequentially) upτ -compact. If T is an (sequentially) upτ -compact and R is a pτ -bounded, then T ◦ R is (sequentially) upτ -compact.

Now, we investigate a relation between sequentially upτ -compact operators and dominated lattice homomorphisms. The following is a more general extension of [21, Prop. 9.4.] and [11, Thm. 8.], and its proof is similar to Theorem 5.1. Theorem 5.5. Let (X, p, Eτ ), (Y, m, Fτ´ ), and (Z, q, Gτˆ ) be LSNV Ls with (F, k·kF ) being Banach lattice and (G, k·kG ) normed lattice, and τ´ and τˆ are being generated by the norms. Then, the followings hold: If T ∈ L(X,Y ) is a sequentially upτ -compact operator and S ∈ L(Y, Z) is a dominated surjective lattice homomorphism, then S ◦ T is sequentially upτ compact; ii. If T ∈ L(X,Y ) is a sequentially upτ -compact, S ∈ L(Y, Z) is a dominated lattice homomorphism, and S(Y ) is upτ -regular in Z, then S ◦ T is sequentially upτ -compact; iii. If T ∈ L(X,Y ) is a sequentially upτ -compact, S ∈ L(Y, Z) is a dominated lattice homomorphism operator, and IS(Y ) (the ideal generated by S(Y )) is upτ -regular in Z, then S ◦ T is sequentially upτ -compact. i.

Proposition 5.15. Let (X, p, Eτ ) be an LSNV L and (Y, m, Fτ´ ) be an upτ -complete LSNV L, and S, T : (X, p, Eτ ) → (Y, m, Fτ´ ) be operators with 0 ≤ S ≤ T . If T is a lattice homomorphism and (sequentially) upτ -compact, then S is (sequentially) upτ compact. Proof. We will prove the sequential case; the other case is similar. Let (xn ) be a pτ bounded sequence in X. So there are a subsequence (xnk ) and some y ∈ Y such that upτ T xnk −−→ y in Y . In particular, it is upτ -Cauchy. Fix u ∈ Y+ , and note that τ´

−0 |Sxnk − Sxn j | ∧ u ≤ (S|xnk − xn j |) ∧ u ≤ (T |xnk − xn j |) ∧ u = |T xnk − T xn j | ∧ u → as k, j → ∞. Thus, we get (Sxnk ), which is a upτ -Cauchy sequence in Y . Therefore, it follows from upτ -complete of Y . Lemma 5.5. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be two LSNV Ls with Y being ordercomplete vector lattice. If T : (X, p, Eτ ) → (Y, m, Fτ´ ) is a positive upτ -compact operator, then the operator S : (X+ , p, Eτ ) → (Y+ , m, Fτ´ ) defined by S(x) = sup{T (u ∧ x) : u ∈ X+ } for each x ∈ X+ is also upτ -compact operator.

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Proof. Suppose (yβ ) is a pτ -bounded net in X+ . Then, there is a subnet (yβγ ) such
τ´ upτ − 0 in F for all w ∈ Y+ . that Tyβγ −−→ y for some y ∈ Y , and so m |Tyβγ − y| ∧ w → For u ∈ X+ and fixed w ∈ Y+ , we have 0 ≤ T (u ∧ yβγ ) ≤ T (yβγ ), and so |T (u ∧ yβγ ) − y| ∧ w ≤ |T (yβγ ) − y| ∧ w. By taking supremum over u ∈ X+ , we get |Syβγ − y| ∧ w ≤ τ´

|T (yβγ ) − y| ∧ w → − 0, and so we get the desired result. Remark 5.5. The sum of two pτ -bounded subsets is also pτ -bounded since the sum of two solid subsets is solid. Moreover, for a pτ -bounded net (xα ) in an LSNV L (X, p, Eτ ), the nets (xα+ ) and (xα− ) are pτ -bounded. The following theorem is upτ -compact version of Proposition 5.6, so we omit its proof. Theorem 5.6. Let (X, p, Eτ ) and (Y, m, Fτ´ ) be two LSNV Ls with Y being ordercomplete vector lattice. If T : (X, p, Eτ ) → (Y, m, Fτ´ ) is a positive upτ -compact operator, then the Kantorovich extension of S : (X+ , p, Eτ ) → (Y+ , m, Fτ´ ) defined by S(x) = sup{T (xα ∧x) : xα ∈ X+ is pτ -bounded} for each x ∈ X+ is also upτ -compact.

BIBLIOGRAPHY [1] Y.A. Abramovich and C.D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, Vol. 50, American Mathematical Society, Rhoda Island, (2002). [2] C.D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces with applications to economics, Pure and Applied Mathematics, Vol. 105, American Mathematical Society, Indianapolish, (2003). [3] C.D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Vol. 119, Dordrecht, (2006). [4] A. Aydın, Topological algebras of bounded operators with locally solid Riesz spaces, Journal of Science and Technology of Erzincan University, Vol. 11, 543–549, (2018). [5] A. Aydın, Unbounded pτ -convergence in vector lattices normed by locally solid lattices, Academic studies in mathematic and natural sciences, IVPE, Cetinje-Montenegro, 118–134, (2019). [6] A. Aydın, Convergence via filter in locally solid Riesz spaces, International Journal of Science and Research, Vol. 8, 351–353, (2019). [7] A. Aydın, Multiplicative order convergence in f -algebras, Hacettepe Journal of Mathematics and Statistics, Vol. 49, 998–1005, (2020). [8] A. Aydın, Multiplicative norm convergence in Banach lattice f -algebras, Hacettepe Journal of Mathematics and Statistics, in press, (2020). [9] A. Aydın and M. C ¸ ınar, Multiplicative norm compact operators on Banach lattice f algebras, International Journal on Mathematics, Engineering and Natural Science, Vol. 9, 8–13, (2019). ¨ [10] A. Aydın, E.Y. Emel’yanov, N.E. Ozcan and M.A.A. Marabeh, Compact-like operators in lattice-normed spaces, Indagationes Mathematicae, Vol. 2, 633–656, (2018). ¨ [11] A. Aydın, E.Y. Emel’yanov, N.E. Ozcan and M.A.A. Marabeh, Unbounded pconvergence in lattice-normed vector lattices, Siberian Advances in Mathematics, Vol. 29, 164–182, (2019).

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¨ Nonstandard hulls of lattice-normed ordered [12] A. Aydın, S.G. Gorokhova and H. Gul, vector spaces, Turkish Journal of Mathematics, Vol. 42, 155–163, (2018). [13] A.V. Bukhvalov, A.E. Gutman, V.B. Korotkov, A.G. Kusraev, S.S. Kutateladze and B.M. Makarov, Vector lattices and integral operators, Mathematics and its Applications, Vol. 358, Kluwer Academic Publishers Group, Dordrecht, (1996). [14] R. DeMarr, Partially ordered linear spaces and locally convex linear topological spaces, Illinois Journal of Mathematics, Vol. 8, 601–606, (1964). [15] Y. Deng, M. O’Brien and V.G. Troitsky, Unbounded norm convergence in Banach lattices, Positivity, Vol. 21, 963–974, (2017). [16] E.Y. Emel’yanov, Infinitesimal analysis and vector lattices, Siberian Advances in Mathematics, Vol. 6, 19–70, (1996). [17] N. Gao, Unbounded order convergence in dual spaces, Journal of Mathematical Analysis and Applications, Vol. 419, 347–354, (2014). [18] N. Gao and F. Xanthos, Unbounded order convergence and application to martingales without probability, Journal of Mathematical Analysis and Applications, Vol. 415, 931–947, (2014). ´ [19] N. Gao, V.G. Troitsky and F. Xanthos, Uo-convergence and its applications to Cesaro means in Banach lattices, Israel Journal of Mathematics, Vol. 220, 649–689, (2017). [20] L. Hong, On order bounded subsets of locally solid Riesz spaces, Quaestiones Mathematicae, Vol. 39, 381–389, (2016). ´ M.A.A. Marabeh and V.G. Troitsky, Unbounded norm topology in Banach [21] M. Kandic, lattices, Mathematical Analysis and Applications, Vol. 451, 259–279, (2017). [22] S. Kaplan, On unbounded order convergence, Real Analysis Exchange, Vol. 23, 175– 184, (1998). [23] A.G. Kusraev, Dominated operators, Mathematics and its Applications, (2000). [24] A.G. Kusraev and S.S. Kutateladze, Boolean valued analysis, Mathematics and its Applications, (1999). [25] W.A.J. Luxemburg and A.C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, (1971). [26] P. Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, (1991). [27] O.V. Maslyuchenko, V.V. Mykhaylyuk and M.M. Popov, A lattice approach to narrow operators, Positivity, Vol. 13, 459–495, (2009). [28] H. Nakano, Ergodic theorems in semi-ordered linear spaces, Annals of Mathematics, Vol. 49, 538–556, (1948). [29] V.G. Troitsky, Measures of non-compactness of operators on Banach lattices, Positivity, Vol. 8, 165–178, (2004). [30] A.W. Wickstead, Weak and unbounded order convergence in Banach lattices, Journal of the Australian Mathematical Society, Vol. 24, 312–319, (1977). [31] B.Z. Vulikh, Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, (1967). [32] A.C. Zaanen, Riesz spaces II, North-Holland Mathematical Library, Vol. 30, NorthHolland Publishing Co., Amsterdam, (1983).

Indexed Product 6 On Summability of an Infinite Series B. P. Padhy KIIT Deemed to be University

P. Baliarsingh Gangadhar Meher University

CONTENTS 6.1 Introduction ................................................................................................... 143 6.1.1 Historical Background....................................................................... 143 6.1.2 Notations and Definitions .................................................................. 143 6.2 Known Results .............................................................................................. 145 6.3 Main Results.................................................................................................. 146 6.4 Proof of Main Results.................................................................................... 149 6.5 Conclusion..................................................................................................... 162 References.............................................................................................................. 162

6.1 6.1.1

INTRODUCTION HISTORICAL BACKGROUND

Initially, in 1952, Szasz [10] published some results on products of summability methods. Subsequently, Rajgopal [7] in 1954, Parameswaran [6] in 1957, Ramanujan [8] in 1958, etc. published some more results on the products of summability methods. Later on, Das [2] in 1969 proved some related results on the absolute product summability. In 2008, Sulaiman [9] provided a result on the indexed product summability of an infinite series. The result found by Sulaiman was then extended by Paikray et al. [5] in 2010. Also, we can see some more developments on this idea in Ref. [1,3,4] in the recent past. 6.1.2

NOTATIONS AND DEFINITIONS

Let ∑ an be an infinite series with the sequence of partial sums {sn }. Let {pn } be a sequence of positive real constants such that 143

144

Topics in Contemporary Mathematical Analysis and Applications

Pn = p0 + p1 + · · · + pn → ∞ as n → ∞ (P−i = p−1 = 0) .

(6.1)

The sequence-to-sequence transformation tn =

1 n ∑ pν sv Pn v=0

(6.2)

defines (R, pn ) transform of {sn } generated by {pn }. The series ∑ an is said to be summable |R, pn |k , k ≥ 1, if ∞

∑ nk−1 |tn − tn−1 |k < ∞.

(6.3)

n=1

Similarly, the sequence-to-sequence transformation Tn =

1 n ∑ pn−v sv Pn v=0

defines the (N, pn ) transform of {sn } generated by {pn }. Let {τn } be the sequence of (N, qn ) transform of the (N, pn ) transform of {sn }, generated by the sequences {qn } and {pn }, respectively. Then, the series ∑ an is said to be summable |(N, qn ) (N, pn )|k , k ≥ 1, if ∞

∑ nk−1 |τn − τn−1 |k < ∞,

(6.4)

n=1

and the series ∑ an is said to be summable |(N, qn )(N, pn ), δ |k , k ≥ 1, 1 ≥ δ k ≥ 0, if ∞

∑ n(δ k+k−1) |τn − τn−1 |k < ∞.

(6.5)

n=1

Similarly, if {αn } is any sequence of positive numbers, then the series ∑ an is said to be summable |(N, qn ) (N, pn ) , αn |k , k ≥ 1, if ∞

∑ αnk−1 |τn − τn−1 |k < ∞,

(6.6)

n=1

and the series ∑ an is said to be summable |(N, qn )(N, pn ), αn ; δ |k , k ≥ 1, 1 ≥ kδ ≥ 0, if ∞

∑ αnδ k+k−1 |τn − τn−1 |k < ∞.

(6.7)

n=1

Let f be a function of αn , if ∞

∑ { f (αn )}k (αn )k−1 |τn − τn−1 |k < ∞,

(6.8)

n=1

then the series ∑ an is said to be |(N, qn ) (N, pn ) , αn ; f |k , k ≥ 1 summable. Clearly for f (αn ) = αnδ , δ ≥ 0, |(N, qn ) (N, pn ) , αn ; f |k = |(N, qn ) (N, pn ) , αn ; δ |k ., and for δ = 0, |(N, qn ) (N, pn ) , αn ; f |k = |(N, qn ) (N, pn ) , αn |k . We assume throughout this chapter that Qn = q0 + · · · + qn → ∞ as n → ∞ and Pn = p0 + · · · + pn → ∞ as n → ∞

145

On Indexed Product Summability

6.2

KNOWN RESULTS

In 2008, Sulaiman [9] has proved the following theorem. Theorem 6.1. Let k ≥ 1 and (λm ) be a sequence of constants. Let us define n

fv =

n

qr

∑ Pr , Fv = ∑ pr

r=v

(6.9)

fr

r=v

Let pn Qn = O (Pn ) such that nk−1 qk ∑ Qk Qn−n1 = O n=v+1 n ∞

vqk−1 v Qkv


! .

(6.10)

Then, the sufficient conditions for the implication that ∑ an is summable |R, rn |k ⇒ ∑ an λn is summable |(R, qn ) (R, pn )|k are |λv | Fv = O (Qv ) ,

(6.11)

|λv | = O (Qν ) ,

(6.12)

pv Rv |λv | = O (Qv ) ,

(6.13)

pv qv Rv |λv | = O (Qv Qv−1 rv ) ,

(6.14)

pn qn Rn |λn | = O (Pn Qn rn ) ,

(6.15)

Rv−1 |∆ λv | Fv+1 = O (Qv rv ) ,

(6.16)

Rv−1 |∆ λv | = O (Qv rv ) .

(6.17)

and Subsequently, Paikray et al. [5] generalized the above theorem by replacing the (R, pn ) summability by A-summability and stated the results as follows. Theorem 6.2. Let k ≥ 1, {λn } be a sequence of constants. Let us define n

fv =

n

∑ qr arν ,

r=v

Fv =

∑ fr .

(6.18)

r=v

Then, the sufficient conditions for the implication that ∑ an is summable |R, rn |k ⇒ ∑ an λn is summable |(R, qn ) (A)|k are   nk−1 qkn 1 ∑ Qk Qn−1 = O λ k , v n=v+1 n m+1



n

k k−1

∑ qr

r=v

(6.19)

 = O (qv ) ,

(6.20)

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Topics in Contemporary Mathematical Analysis and Applications n



∑

akr,v



  = O vk−1 ,

(6.21)

r=v

Rv = O(rv ), qn = O(1), Qn qn λn an,n = O(1), Qn−1   (∆λv )k k−1 = O v , 1 qk− v ∆λv = O(1), λv

(6.22) (6.23) (6.24) (6.25) (6.26)

and

  λvk = O vk−1 . (6.27) k−1 qv In this chapter, we established the following theorems on the product summability of the infinite series ∑ an λn .

6.3

MAIN RESULTS

Theorem 6.3. For the sequences of real constants {pn } and {qn }, define n

n qn−i pi−v and Fv = ∑ fi . Pi i=v i=v

fv = ∑

(6.28)

Let Qn = O (qn Pn ) and nδ k+k−1 qk ∑ Qk Qn−1 n = O n n=v+1 m+1

(vqv )k−1 Qkv

(6.29) ! as m → ∞.

(6.30)

Then for any sequences {rn } and {λn }, the sufficient conditions for the implication that ∑ an is summable |R, rn |k ⇒ ∑ an λn is summable |(N, qn ) (N, pn ) ; δ |k , k ≥ 1, 1 ≥ k δ ≥ 0, are |λn | Fv = O (Qv ) , (6.31) |λn | = O (Qn ) ,

(6.32)

Rv Fv |λv | = O (Qv rv ) ,

(6.33)

qn Rn Fn |λn | = O (Qn Qn−1 rn ) ,

(6.34)

Rv−1 Fv+1 |∆λv | = O (Qv rv ) ,

(6.35)

Rv−1 |∆λv | = O (Qv rv ) ,

(6.36)

qn Rn |λn | = O (Qn Qn−1 rn ) ,

(6.37)

where Rn = r1 + r2 + ...... + rn .

147

On Indexed Product Summability

Theorem 6.4. For the sequences of real constants {pn } and {qn } and the sequence of positive numbers {αn }, define n

n qn−i pi−v and Fv = ∑ fi . Pi i=v i=v

fv = ∑

(6.38)

Let Qn = O (qn Pn ) and m+1

α k−1 qk ∑ QknQn−1n = O n=v+1 n

(vqv )k−1 Qvk

(6.39) ! as m → ∞.

(6.40)

Then, for any sequences {rn } and {λn }, the sufficient conditions for the implication that ∑ an is summable |R, rn |k ⇒ ∑ an λn is summable |(N, qn ) (N, pn ) , αn |k , k ≥ 1, are |λν | Fv = O (Qv ) , (6.41) |λn | = O (Qn ) ,

(6.42)

Rv Fv |λv | = O (Qv rv ) ,

(6.43)

qn Rn Fn |λn | = O (Qn Qn−1 rn ) ,

(6.44)

Rv−1 Fv+1 |∆λv | = O (Qv rv ) ,

(6.45)

Rv−1 |∆λv | = O (Qv rv ) ,

(6.46)

qn Rn |λn | = O (Qn Qn−1 rn ) ,

(6.47)

∞

∑ nk−1 |tn |k = O(1),

(6.48)

n=1

and

∞

∑ αnk−1 |tn |k = O(1),

(6.49)

n=2

0

0

where Rn = r1 + r2 + ...... + rn tn = ∆ tn and tn is the n-th (R, rn ) transform of ∑ an . Theorem 6.5. For the sequences of real constants {pn } and {qn } and the sequence of positive numbers {αn }, define n

n qn−i pi−v and Fv = ∑ fi Pi i=v i=v

fv = ∑

(6.50)

Let Qn = O (qn Pn ) and m+1

α δ k+k−1 qk ∑ Qn k Qn−1 n = O n n=v+1

(vqv )k−1 Qkv

(6.51) ! as m → ∞.

(6.52)

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Topics in Contemporary Mathematical Analysis and Applications

Then for any sequences {rn } and {λn }, the sufficient conditions for the implication ∑ an is summable |R, rn |k ⇒ ∑ an λn is summable |(N, qn )(N, pn ), αn ; δ |k , k ≥ 1, 1 ≥ kδ ≥ 0, are |λn | Fv = O (Qv ) , (6.53) |λn | = O (Qn ) ,

(6.54)

Rv Fv |λv | = O (Qv rv ) ,

(6.55)

qn Rn Fn |λn | αnδ = O (Qn Qn−1 rn ) ,

(6.56)

Rv−1 Fv+1 |∆λv | = O (Qv rv ) ,

(6.57)

Rv−1 |∆λv | = O (Qv rv ) ,

(6.58)

qn Rn |λn | αnδ = O (Qn Qn−1 rn ) ,

(6.59)

∞

∑ nk−1 |tn |k = O(1),

(6.60)

n=1

and

∞

∑ αnk−1 |tn |k = O(1),

(6.61)

n=2

where Rn = r1 + r2 + ...... + rn . Theorem 6.6. For the sequences of real constants {pn } and {qn } and the sequence of positive numbers {αn }, we define n

n qn−i pi−v and Fv = ∑ fi . Pi i=v i=v

fv = ∑

(6.62)

Let Qn = O (qn Pn ) and { f (αn )}k (αn )k−1 qkn =O ∑ Qnk Qn−1 n=v+1 m+1

(vqv )k−1 Qkv

(6.63) ! as m → ∞.

(6.64)

Then for any sequences {rn } and {λn }, the sufficient conditions for the implication ∑ an is summable |R, rn |k ⇒ ∑ an λn is |(N, qn ) (N, pn ) , αn ; f |k , k ≥ 1 is summable, are |λn | Fv = O (Qv ) , (6.65) |λn | = O (Qn ) ,

(6.66)

Rv Fv |λv | = O (Qv rv ) ,

(6.67)

qn Rn Fn |λn | = O (Qn Qn−1 rn ) ,

(6.68)

Rv−1 Fv+1 |∆λv | = O (Qv rv ) ,

(6.69)

Rv−1 |∆λv | = O (Qv rv ) ,

(6.70)

149

On Indexed Product Summability

qn Rn |λn | = O (Qn Qn−1 rn ) ,

(6.71)

∞

∑ nk−1 |tn |k = O(1),

(6.72)

∑ { f (αn )}k (αn )k−1 |tn |k = O(1),

(6.73)

n=1

and

∞

n=2

where Rn = r1 + r2 + ...... + rn .

6.4

PROOF OF MAIN RESULTS

In this section, we provide the detailed proof of all theorems as stated above. Proof of Theorem 6.3. Let {t 0 n } be the (R, rn ) transform of the series ∑ an . Then, 0

tn =

1 n ∑ rν sν . R ν=0

Then tn = t 0 n − t 0 n−1 =

n rn Rv−1 av . ∑ Rn Rn−1 v=1

Let {sn } be the sequence of partial sums of the series ∑ an λn and {τn } the sequence of (N, qn ) (N, pn ) transform of the series ∑ an λn . Then, τn =

1 n 1 r q n−r ∑ ∑ pr=v sv , Qn r=0 Pr v=0 1 n ∑ sv Qn v=0

=

n

qn−v pr−v , Pr r=v

∑

1 n ∑ f v sv . Qn v=0

(6.74)

Hence, Tn = τn − τn−1 =

1 n 1 n−1 fv sv − ∑ ∑ fv sv Qn v=0 Qn−1 v=0

=−

n qn fn sn fv sv + ∑ Qn Qn−1 v=0 Qn−1

=−

n qn fr ∑ Qn Qn−1 r=0 n

=−

r

∑

av λv +

v=0

qn ∑ av λv Qn Qn−1 r=0

v=0

n

a v λv Qn−1 ∑ v=0 n

r

∑

fn

fr +

fn ∑ a v λv Qn−1 v=0

(6.75)

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Topics in Contemporary Mathematical Analysis and Applications

    n qn λv n q0 p0 n λv =− Rv−1 av ∑ fr + Pn Qn−1 ∑ Rv−1 av Rv−1 Qn Qn−1 v∑ Rv−1 r=v =1 v=1 " !  ! #  n−1 v n n λv qn λn =− ∑ Rr−1 ar ∆ Rv−1 ∑ fr + ∑ Rv−1 av Rn−1 fn Qn Qn−1 v∑ r=v =1 r=1 v=1 ! " !  #  n λn λv p0 q0 n−1 v + ∑ Rr−1 av ∆ Rv+1 + ∑ Rv−1 av Rn−1 Pn Qn−1 v∑ =1 r=1 v=1 "  #  n−1 qn Rv−1 Rv−1 Rn =− ∑ λv Fvtv + rv fv λv tv + rv (∆λv ) Fv+1tv + rn λn Fn tn Qn Qn−1 v=1 "  #  Rv−1 Rn p0 q0 n−1 + ∑ λv tv + rv (∆λv )tν + rn λn tn Pn Qn−1 v= 1 7

= ∑ Tn,i ,

say.

(6.76)

i=1

In order to prove the theorem, using Minkowski’s inequality, it is sufficient to show that ∞

∑ nδ k+k−1 |Tn, i |k < ∞,

for i = 1, 2, 3, 4, 5, 6, 7.

n=1

Now, on applying Holder’s inequality, we have k q m+1 m+1 n−1 n k δ k+k−1 δ k+k−1 | | = n λ F t n T v v v n,1 ∑ ∑ ∑ Q Q n n−1 n=2 n=2 v=1 m+1

≤

δ k+k−1

∑n

n=2

m

qkn n−1 |λv |k Fvk |tv |k ∑ qk−1 Qkn Qn−1 v=1 v

1

= O(1) ∑

k−1 v=1 qv m

1

= O(1) ∑

k−1 v−1 qv m

k−1

= O(1) ∑ v

|λv |k Fvk |tv |k



|tv |

v=1 m

= O(1) ∑ vk−1 |tv |k ,

|λv | Fv Qv

∑n

n=2

|Tn2 | =

m+1

using (6.31)

∑n

n=2

k q n−1 Rv−1 n f λ t v v v ∑ rv Qn Qn−1 v=1

k−1

∑ qv

(vqv )k−1 , using (6.30) Qvk k

Next, k

v=1

nδ k+k−1 qkn k n=v+1 Qn Qn−1

= O(1), on m → ∞.

δ k+k−1

Qn−1

∑

v=1

m+1

n−1

m+1

|λv |k Fvk |tv |k k

1

!k−1

151

On Indexed Product Summability

qk n−1 Rkν Fvk |λv |k ≤ ∑ nδ k+k−1 k n ∑ qk−1 rk Qn Qn−1 v=1 v n=2 v m+1

1

n−1

Qn−1

v=1

!k−1

∑ qv

Rkv Fvk |λv |k |tv |k m+1 nδ k+k−1 qnk ∑ k k qk−1 v rv v=1 n=v+1 Qn Qn−1   m Rv Fv |λv | k = O(1) ∑ vk−1 |tv |k rv Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk−1 |tv |k , using (6.33) v=1

= O(1), as m → ∞. k q n−1 R n v−1 ∑ nδ k+k−1 |Tn,3 |k = ∑ nδ k+k−1 Qn Qn−1 ∑ rv (∆λv ) Fv+1tv n=2 n=2 v=1

m+1

m+1

qk n−1 Rkv−1 ≤ ∑ nδ k+k−1 k n ∑ rk qk−1 |∆λv |k Fvk+1 |tv |k Qn Qn−1 v=1 n=2 v v m+1

m

k |∆λ |k Rv− r 1

1

n−1

Qn−1

v=1

!k−1

∑ qv

m+1

nδ k+k−1 qnk , by (6.30) k v=1 n=v+1 Qn Qn−1   m Rv−1 Fv+1 |∆λv | k = O(1) ∑ vk−1 |tv |k rv Qv v=1

= O(1) ∑

rvk qk−1 v

k |tv |k Fv+1

∑

m

= O(1) ∑ vk−1 |tv |k , using (6.35) v=1

= O(1), as m → ∞. m+1 δ k+k−1

∑n

n=2

qn Rn λn fntn k |Tn,4 | = ∑ n Qn Qn−1 r n n=2   m+1 qn Rn Fn |λn | k ≤ ∑ nk−1 |tn |k Qn Qn−1 rn n=2 m+1

k

δ k+k−1

m+1

= O(1)

∑ nk−1 |tn |k , using

(6.34)

n=2

= O(1), as m → ∞. m+1

k k m+1 δ k+k−1 p0 q0 n−1 Tn,5 = ∑ n ∑ λv tv P Q n n− 1 n=2 v=1

δ k+k−1

∑n

n=2

m+1

≤ O(1)

δ k+k−1

∑n

n=2

n−1 |λv |k k 1 ∑ qk−1 |tv | Pnk Qn−1 v=1 v

1

n−1

Qn−1

v=1

∑ qv

!k−1

152

Topics in Contemporary Mathematical Analysis and Applications

|λv |k |tv |k m+1 δ k+k−1 1 n · k ∑ k−1 Pn Qn−1 v=1 qv n=v+1 m

= O(1) ∑

|λv |k |tv |k m+1 nk−1 qkn , using (6.29) ∑ k k−1 v=1 qv n=v+1 Qn Qn−1 !k m |λn |k k k = O(1) ∑ v |tv | Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk |tv |k , using (6.34) v=1

= O(1), as m → ∞. p q n−1 R m+1 k 0 0 v−1 ∑ nδ k+k−1 Tn,6 = ∑ nδ k+k−1 Pn Qn−1 ∑ rv (∆λv ) n=2 n=2 v=1

m+1

n−1 Rk 1 v−1 |∆λv |k |tv |k ≤ O(1) ∑ nδ k+k−1 · k ∑ k−1 k P Q n n−1 v=1 rv qv n=2 m+1

m

= O(1) ∑

v=1

Rkv−1 |∆λv |k |tv |k rvk qk−1 v

m

= O(1) ∑ vk−1 |tv |k v=1 m



m+1

∑

nδ k+k−1 ·

n=v+1

Rv−1 |∆λv | rv Qv

k tv 1

n−1

Qn−1

v=1

∑ qv

1 Pnk Qn−1

k

= O(1) ∑ vk−1 |tv |k , using (6.36) v=1

= O(1), as m → ∞. Finally, m+1

∑

n=2

k δ k+k−1 p0 q0 Rn λn tn n ∑ Pn Qn−1 rn n=2   m+1 Rn |λn | k = O(1) ∑ nk−1 |tn |k Pn Qn−1 rn n=2   m+1 qn Rn |λn | k = O(1) ∑ nk−1 |tn |k Qn Qn−1 rn n=2

nδ k+k−1 |Tn,7 |k =

m+1

m+1

= O(1)

∑ nk−1 |tn |k , using

n=2

= O(1), as m → ∞. This completes the Proof of Theorem 6.3.

(6.37)

!n−1

153

On Indexed Product Summability

Proof of Theorem 6.4. In order to prove this theorem, using (6.76) and Minkowski’s inequality, it is sufficient to show that ∞

∑ αnk−1 |Tn, i |k < ∞, for

i = 1, 2, 3, 4, 5, 6, 7.

n=1

Now, on applying Holder’s inequality, we have k q n−1 n ∑ αnk−1 |Tn,1 |k = ∑ αnk−1 Qn Qn−1 ∑ λv Fv tv n=2 n=2 v=1

m+1

m+1

m+1

≤

∑

αnk−1

n=2

qnk n−1 |λv |k Fvk |tv |k Qkn Qn−1 v∑ qk−1 v =1

m

1

|λ |k Fvk |tv |k k−1 v q v v=1

= O(1) ∑

1

n−1

Qn−1

v=1

!k−1

∑ qv

m+1

αnk−1 qnk ∑ k n=v+1 Qn Qn−1

k−1 k k k (vqv ) | |λ | | t , using (6.40) F v v v k−1 Qkv v−1 qv   m |λv | Fv k = O(1) ∑ vk−1 |tv |k Qv v=1 m

1

= O(1) ∑

m

= O(1) ∑ vk−1 |tv |k , using (6.41) v=1

= O(1), as m → ∞. Next, m+1

∑

αnk−1 |Tn2 |k

m+1

=

n=2

∑

n=2

k qn n−1 Rv−1 ∑ rv fv λv tv Qn Qn−1 v=1



αnk−1

m+1

≤

∑

αnk−1

n=2

qkn n−1 Rkν Fvk |λv |k ∑ qk−1 rk k Qn Qn−1 v= v 1 v

1

n−1

Qn−1

v=1

Rvk Fvk |λv |k |tv |k m+1 αnk−1 qnk ∑ k qvk−1 rvk v=1 n=v+1 Qn Qn−1   m Rv Fv |λv | k k k−1 = O(1) ∑ v |tv | rv Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk−1 |tv |k , using (6.43) v=1

= O(1), as m → ∞.

∑ qv

!k−1

154

Topics in Contemporary Mathematical Analysis and Applications

Further, m+1

∑

αnk−1 |Tn,3 |k

m+1

=

n=2

∑

n=2

k qn n−1 Rv−1 (∆λv ) Fv+1tv ∑ Qn Qn−1 v=1 rv



αnk−1

m+1

≤

∑

αnk−1

n=2

k qkn n−1 Rv−1 |∆λv |k Fvk+1 |tv |k ∑ Qnk Qn−1 v=1 rvk qvk−1

Rkv−1 |∆λr |k

1

n−1

Qn−1

v=1

∑ qv

αnk−1 qkn k v=1 n=v+1 Qn Qn−1   m Rv−1 Fv+1 |∆λv | k = O(1) ∑ vk−1 |tv |k , by (6.40) rv Qv v=1 m

= O(1) ∑

rvk qk−1 v

k |tv |k Fv+1

m+1

∑

m

= O(1) ∑ vk−1 |tv |k , using (6.45) v=1

= O(1), as m → ∞. Again, m+1

∑

αnk−1 |Tn,4 |k

n=2

qn Rn λn fntn k = ∑ Qn Qn−1 rn n=2   m+1 qn Rn Fn |λn | k ≤ ∑ αnk−1 |tn |k Qn Qn−1 rn n=2 m+1



αnk−1

m+1

= O(1)

∑ αnk−1 |tn |k , using

(6.44)

n=2

= O(1), as m → ∞. Next, k p q n−1 m+1 k 0 0 ∑ αnk−1 Tn,5 = ∑ αnk−1 Pn Qn−1 ∑ λv tv v=1 n=2 n=2

m+1

m+1

≤ O(1)

∑

n=2

αnk−1

n−1 |λv |k k 1 |tv | ∑ Pnk Qn−1 v=1 qvk−1

1

n−1

Qn−1

v=1

∑ qv

|λv |k |tv |k m+1 k−1 1 αn · k ∑ k−1 Pn Qn−1 v=1 qv n=v+1 m

= O(1) ∑

|λv |k |tv |k m+1 αnk−1 qkn , using (6.39) ∑ k k−1 v=1 qv n=v+1 Qn Qn−1 !k m |λn |k k k = O(1) ∑ v |tv | Qv v=1 m

= O(1) ∑

!k−1

!k−1

155

On Indexed Product Summability m

= O(1) ∑ vk |tv |k , using (6.44) v=1

= O(1), as m → ∞. Again, p q n−1 R m+1 k 0 0 v−1 ∑ αnk−1 Tn,6 = ∑ αnk−1 Pn Qn−1 ∑ rv (∆λv ) n=2 n=2 v=1

m+1

m+1

≤ O(1)

n−1 Rk 1 v−1 |∆λv |k |tv |k k k−1 Pnk Qn−1 v∑ =1 rv qv

∑

αnk−1

m

Rkv−1 |∆λv |k |tv |k

m+1

rvk qk−1 v

n=v+1

n=2

= O(1) ∑

v=1

k tv

m

= O(1) ∑ vk−1 |tv |k



v=1 m

∑

αnk−1

Rv−1 |∆λv | rv Qv

1

n−1

Qn−1

v=1

!n−1

∑ qv

1 Pnk Qn−1

k

= O(1) ∑ vk−1 |tv |k , using (6.46) v=1

= O(1), as m → ∞. Finally, m+1

∑

k k−1 p0 q0 Rn λn tn α ∑ n Pn Qn−1 rn n=2   m+1 Rn |λn | k = O(1) ∑ αnk−1 |tn |k Pn Qn−1 rn n=2   m+1 qn Rn |λn | k = O(1) ∑ αnk−1 |tn |k Qn Qn−1 rn n=2

αnk−1 |Tn,7 |k =

n=2

m+1

m+1

= O(1)

∑ αnk−1 |tn |k , using

(6.47)

n=2

= O(1), as m → ∞. This completes the Proof of Theorem 6.4. Proof of Theorem 6.5. In order to prove this theorem, using (6.76) and Minkowski’s inequality, it is sufficient to show that ∞

∑ αnδ k+k−1 |Tn, i |k < ∞, for

n=1

i = 1, 2, 3, 4, 5, 6, 7.

156

Topics in Contemporary Mathematical Analysis and Applications

On applying Holder’s inequality, we have k q m+1 m+1 n−1 n k δ k+k−1 δ k+k−1 |Tn,1 | = ∑ αn λv Fv tv ∑ αn ∑ Qn Qn−1 v=1 n=2 n=2 m+1

≤

∑

αnδ k+k−1

n=2 m

= O(1) ∑

qnk n−1 |λv |k Fvk |tv |k ∑ qk−1 Qnk Qn−1 v=1 v

1

k−1 v=1 qv

|λv |k Fvk |tv |k

m+1

∑

1

n−1

Qn−1

v=1

!k−1

∑ qv

αnδ k+k−1 qkn Qkn Qn−1

n=v+1 k−1 1 k k k (vqv ) |λ | | | = O(1) t , F v v k−1 v Qvk v−1 qv   m |λv | Fv k = O(1) vk−1 |tv |k Qv v=1 m

∑

using (6.51)

∑ m

= O(1) ∑ vk−1 |tv |k , using (6.53) v=1

= O(1), as m → ∞. Next, m+1

∑

αnδ k+k−1 |Tn,2 |k

k qn n−1 Rv−1 f λ t v v v ∑ rv Qn Qn−1 v=1



m+1

=

n=2

∑

n=2 m+1

≤

∑

αnδ k+k−1 αnδ k+k−1

n=2

qkn n−1 Rkν Fvk |λv |k ∑ qk−1 rk Qkn Qn−1 v=1 v v

1

n−1

Qn−1

v=1

!k−1

∑ qv

Rkv Fvk |λv |k |tv |k m+1 αnδ k+k−1 qkn ∑ k k qk−1 v rv n=v+1 Qn Qn−1 v=1   m Rv Fv |λv | k = O(1) ∑ vk−1 |tv |k rv Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk−1 |tv |k , using (6.55) v=1

= O(1), as m → ∞. Further, m+1

∑ αn

δ k+k−1

k

|Tn,3 | =

n=2

m+1

∑

n=2 m+1

≤

∑

n=2

αnδ k+k−1

k qn n−1 Rv−1 (∆λ ) F t v v v+1 ∑ rv Qn Qn−1 v=1



αnδ k+k−1

qkn n−1 Rkv−1 k k |∆λv |k Fv+ 1 |tv | k qk−1 Qkn Qn−1 v∑ r v =1 v

1

n−1

Qn−1

v=1

∑ qv

!k−1

157

On Indexed Product Summability m

= O(1) ∑

Rkv−1 |∆λr |k 1 rvk qk− v

v=1 m

= O(1) ∑ vk−1 |tv |k

m+1

k |tv |k Fv+1

∑

n=v+1



v=1 m

Rv−1 Fv+1 |∆λv | rv Qv

αnδ k+k−1 qkn , by (6.52) Qkn Qn−1

k

= O(1) ∑ vk−1 |tv |k , using (6.56) v=1

= O(1), as m → ∞. Again, m+1

∑

qn Rn λn fntn k ∑ Qn Qn−1 rn n=2   m+1 qn Rn Fn |λn | k ≤ ∑ αnδ k+k−1 |tn |k Qn Qn−1 rn n=2 !k m+1 qn Rn Fn |λn | αnδ k k−1 = ∑ αn |tn | Qn Qn−1 rn n=2

αnδ k+k−1 |Tn,4 |k =

n=2

m+1

αnδ k+k−1

m+1

= O(1)

∑ αnk−1 |tn |k , using

(6.56)

n=2

= O(1), as m → ∞. Next, m+1

∑

n=2

k αnδ k+k−1 Tn,5

k p q n−1 0 0 λ t ∑ v v Pn Qn−1 v=1

m+1

=

∑

n=2

αnδ k+k−1 m+1

≤ O(1)

∑

αnδ k+k−1

n=2

n−1 |λv |k k 1 ∑ qk−1 |tv | Pnk Qn−1 v=1 v

1

n−1

Qn−1

v=1

∑ qv

|λv |k |tv |k m+1 δ k+k−1 1 ∑ αn k−1 Pnk Qn−1 v=1 qv n=v+1 m

= O(1) ∑

|λv |k |tv |k m+1 αnδ k+k−1 qkn , using (6.51) ∑ k k−1 v=1 qv n=v+1 Qn Qn−1 ! k k m | | λ n k = O(1) ∑ vk |tv | Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk |tv |k , using (6.55) v=1

= O(1), as m → ∞.

!k−1

158

Topics in Contemporary Mathematical Analysis and Applications

Again, k m+1 δ k+k−1 p0 q0 n−1 Rv−1 Tn,6 = ∑ αn ∑ rv (∆λv ) Pn Qn−1 v=1 n=2

m+1

∑ αn

δ k+k−1

n=2

m+1

≤ O(1)

n−1 Rk 1 |∆λv |k |tv |k ∑ rk qv−1 k−1 Pnk Qn−1 v=1 v v

∑

αnδ k+k−1

m

Rkv−1 |∆λv |k |tv |k

m+1

rvk qk−1 v

n=v+1

n=2

= O(1) ∑

v=1 m k−1

= O(1) ∑ v

k

|tv |

v=1 m



∑

αnδ k+k−1

Rv−1 |∆λv | rv Qv

k tv 1

n−1

Qn−1

v=1

!n−1

∑ qv

1 Pnk Qn−1

k

= O(1) ∑ vk−1 |tv |k , using (6.58) v=1

= O(1), as m → ∞. Finally, m+1

∑

αnδ k+k−1 |Tn,7 |k

n=2

p0 q0 Rn λntn k = ∑ Pn Qn−1 rn n=2   m+1 Rn |λn | k k δ k+k−1 |tn | = O(1) ∑ αn Pn Qn−1 rn n=2   m+1 qn Rn |λn | k k δ k+k−1 |tn | = O(1) ∑ αn Qn Qn−1 rn n=2 !k m+1 δ |λ | q R α n n n k n = O(1) ∑ αnk−1 |tn | Qn Qn−1 rn n=2 m+1

αnδ k+k−1

m+1

= O(1)

∑ αnk−1 |tn |k , using

(6.59)

n=2

= O(1), as m → ∞. This completes the Proof of the Theorem 6.5. Proof of Theorem 6.6. In order to prove this theorem, using (6.76) and Minkowski’s inequality, it is sufficient to show that ∞

∑ { f (αn )}k (αn )k−1 |Tn, i |k < ∞, for

n=1

i = 1, 2, 3, 4, 5, 6, 7.

159

On Indexed Product Summability

Now, on applying Holder’s inequality, we have m+1

∑ { f (αn )}k (αn )k−1 |Tn,1 |k

n=2

k q n−1 n λv Fv tv ∑ Qn Qn−1 v=1

m+1 k

k−1

k

k−1

∑ { f (αn )} (αn )

=

n=2

m+1

≤

∑ { f (αn )} (αn )

n=2

m

qkn n−1 |λv |k Fvk |tv |k ∑ qk−1 k Qn Qn−1 v= v 1

1

|λ |k Fvk |tv |k k−1 v

= O(1) ∑

v=1 qv m

1

|λ |k Fvk |tv |k k−1 v

= O(1) ∑

v−1 qv m

k−1

= O(1) ∑ v



k

|tv |

v=1 m

|λv | Fv Qv

1

n−1

Qn−1

v=1

!k−1

∑ qv

{ f (αn )}k (αn )k−1 qnk Qkn Qn−1 n=v+1 m+1

∑

(vqv )k−1 , using (6.64) Qkv k

= O(1) ∑ vk−1 |tv |k , using (6.65) v=1

= O(1), as m → ∞. Next, m+1

∑ { f (αn )}k (αn )k−1 |Tn,2 |k

n=2

m+1

=

k−1

k

k−1

∑ { f (αn )} (αn )

n=2

m+1

≤

k q n−1 Rv−1 n fv λv tv ∑ Qn Qn−1 v=1 rv

k

∑ { f (αn )} (αn )

n=2

qkn n−1 Rνk Fvk |λv |k k k−1 k Qn Qn−1 v∑ =1 qv rv

1

n−1

Qn−1

v=1

Rkv Fvk |λv |k |tv |k m+1 { f (αn )}k (αn )k−1 qkn ∑ k Qkn Qn−1 qk−1 v rv v=1 n=v+1   m Rv Fv |λv | k = O(1) ∑ vk−1 |tv |k rv Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk−1 |tv |k , using (6.67) v=1

= O(1), as m → ∞.

∑ qv

!k−1

160

Topics in Contemporary Mathematical Analysis and Applications

Further, m+1

∑ { f (αn )}k (αn )k−1 |Tn,3 |k

n=2

m+1

=

k q n−1 Rv−1 n (∆λ ) F t v v+1 v ∑ rv Qn Qn−1 v= 1

k

k−1

k

k−1

∑ { f (αn )} (αn )

n=2

m+1

≤

∑ { f (αn )} (αn )

n=2

Rkv−1 |∆λr |k

qkn n−1 Rkv−1 k |tv |k ∑ rk qk−1 |∆λv |k Fv+1 k Qn Qn−1 v=1 v v

1

n−1

Qn−1

v=1

∑ qv

{ f (αn )}k (αn )k−1 qnk , by (6.64) Qkn Qn−1 rvk qvk−1 v=1 n=v+1   m Rv−1 Fv+1 |∆λv | k k k−1 = O(1) ∑ v |tv | rv Qv v=1 m

= O(1) ∑

k k Fv+ 1 |tv |

m+1

∑

m

= O(1) ∑ vk−1 |tv |k , using (6.69) v=1

= O(1), as m → ∞. Again, m+1

∑ { f (αn )}k (αn )k−1 |Tn,4 |k

n=2

qn Rn λn fntn k = ∑ { f (αn )} (αn ) Q Q r n n n − 1 n=2   m+1 qn Rn Fn |λn | k k k k−1 ≤ ∑ { f (αn )} (αn ) |tn | Qn Qn−1 rn n=2   m+1 qn Rn Fn |λn | k k k k−1 = ∑ { f (αn )} (αn ) |tn | Qn Qn−1 rn n=2 m+1

k

k−1

m+1

= O(1)

∑ { f (αn )}k (αn )k−1 |tn |k , using (6.68)

n=2

= O(1), as m → ∞. Next, m+1

k

∑ { f (αn )}k (αn )k−1 Tn,5

n=2

m+1

=

k

∑ { f (αn )} (αn )

n=2

p q n−1 k 0 0 ∑ λv tv Pn Qn−1 v=1

k−1

!k−1

161

On Indexed Product Summability n−1 |λv |k k 1 |tv | ≤ O(1) ∑ { f (αn )}k (αn )k−1 k ∑ Pn Qn−1 v=1 qkv−1 n=2

1

n−1

Qn−1

v=1

m+1

!k−1

∑ qv

|λv |k |tv |k m+1 { f (αn )}k (αn )k−1 ∑ k−1 Pnk Qn−1 v=1 qv n=v+1 m

= O(1) ∑

|λv |k |tv |k m+1 { f (αn )}k (αn )k−1 qkn , using (6.63) ∑ k−1 Qkn Qn−1 v=1 qv n=v+1 !k m |λn |k k k = O(1) ∑ v |tv | Qv v=1 m

= O(1) ∑

m

= O(1) ∑ vk |tv |k , using (6.68) v=1

= O(1), as m → ∞. Again, m+1

k

∑ { f (αn )}k (αn )k−1 Tn,6

n=2

m+1

=

k

∑ { f (αn )} (αn )

n=2

m+1

≤ O(1)

p q n−1 R 0 0 v−1 ∑ rv (∆λv ) Pn Qn−1 v=1

k−1

k

k−1

∑ { f (αn )} (αn )

n=2

Rkv−1 |∆λv |k |tv |k

k tv

n−1 Rk 1 |∆λv |k |tv |k ∑ rk qv−1 k k−1 Pn Qn−1 v=1 v v

1

n−1

Qn−1

v=1

{ f (αn )}k (αn )k−1 Pnk Qn−1 rvk qk−1 v v=1 n=v+1   m Rv−1 |∆λv | k = O(1) ∑ vk−1 |tv |k rv Qv v=1 m

= O(1) ∑

m+1

∑

m

= O(1) ∑ vk−1 |tv |k , using (6.70) v=1

= O(1), as m → ∞. Finally, m+1

∑ { f (αn )}k (αn )k−1 |Tn,7 |k

n=2

k k k−1 p0 q0 Rn λn tn { f ( α )} ( α ) n n ∑ Pn Qn−1 rn n=2   m+1 Rn |λn | k = O(1) ∑ { f (αn )}k (αn )k−1 |tn |k Pn Qn−1 rn n=2 m+1

=

∑ qv

!n−1

162

Topics in Contemporary Mathematical Analysis and Applications m+1

= O(1)

k

k−1

k

k−1

∑ { f (αn )} (αn )

k



qn Rn |λn | Qn Qn−1 rn

k



qn Rn |λn | Qn Qn−1 rn

k

|tn |

n=2

m+1

= O(1)

∑ { f (αn )} (αn )

n=2 m+1

= O(1)

k

|tn |

∑ { f (αn )}k (αn )k−1 |tn |k , using

(6.71)

n=2

= O(1), as m → ∞. This completes the Proof of the Theorem 6.6.

6.5

CONCLUSION

From the above results and discussions, we are in a conclusion that our results are more generalized and in particular, these generalize the results of Sulaiman [9], Paikray et al. [5], and Das [2]. We have also observed that the sufficient conditions for ∑ an λn using the absolute indexed product summability |(N, qn )(N, pn ), αn ; δ |k with k ≥ 1, 1 ≥ kδ ≥ 0 and |(N, qn ) (N, pn ) , αn ; f |k , k ≥ 1 generalize the sufficient conditions for ∑ an λn using the absolute indexed product summability |(N, qn ) (N, pn ) , αn |k , k ≥ 1. Similarly, the sufficient conditions for ∑ an λn using the absolute indexed product summabilities |(N, qn ) (N, pn ) ; δ |k , k ≥ 1, 1 ≥ kδ ≥ 0, and |(N, qn ) (N, pn ) , αn |k , k ≥ 1 generalize the sufficient conditions for ∑ an λn using the absolute indexed product summability |(N, qn ) (N, pn )|k . As a future scope of this work, one may approach in the similar way to find the sufficient conditions for • ∑ an λn is summable |(N, qn ) (N, pn ) , αn , δ , µ|k if ∑ an is |R, rn |k -summable, where µ is a real number. • ∑ an λn is summable |(R, qn )(R, pn ), δ |k , |(R, qn )(R, pn ), αn |k , |(R, qn )(R, pn ), αn ; δ |k and |(R, qn )(R, pn ), αn , f |k if ∑ an is |R, rn |k -summable. • ∑ an λn is summable |(R, qn )(C, 1), δ |k , |(C, 1)(R, pn ), δ |k , |(R, qn )(C, 1), αn |k , |(C, 1)(R, pn ), αn |k , |(C, 1)(R, pn ), αn ; δ |k , |(R, qn )(C, 1), αn ; δ |k if ∑ an is |C, 1|k -summable.

REFERENCES [1] Aasma, A., Dutta, H. and Natarajan, P. N., An Introductory Course in Summability Theory, First Edition, John Wiley & Sons, Inc., USA, (2017). [2] Das, G., Tauberian theorems for absolute Norlund summability, Proceedings of the London Mathematical Society, Vol. 19(2), (1969), 357−384. [3] Dutta, H. and Rhoades, B.E. (Eds.), Current Topics in Summability Theory and Applications, First Edition, Springer, Singapore, (2016). [4] Padhy, B.P., Misra, U.K. and Misra, M., Summability Methods and its Applications, Lap Lambart Academic Publications, Germany, (2012).

On Indexed Product Summability

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[5] Paikray, S.K., Misra, U.K. and Sahoo, N.C., Product Summability of an Infinite Series, International Journal of Computer and Mathematical Sciences, Vol-1(7), (2010), 853−863. [6] Parameswaran, M.R., Some product theorems in summability, Mathematische Zeitscher, Vol. 68, (1957), 19−26. [7] Rajgopal, C.T., Theorems on product of two summability methods, The Journal of Indian Mathematical Society, Vol. 18(1), (1954) . [8] Ramanujan, M.S., On products of summability methods, Mathematische Zeitscher, Vol. 69(1), (1958), 423−428. [9] Sulaiman, W.T., A Note on product summability of an infinite series, International Journal of Mathematical Sciences, Hindawi publishing corporation, (2008), Article ID 372604. [10] Szasz, O., On products of summability methods, Proceedings of American Mathematical Society, Vol. 3(2), (1952).

Some Important 7 On Inequalities Zlatko Pavic´ University of Osijek

CONTENTS 7.1 Concepts of Affinity and Convexity .............................................................. 165 7.1.1 Affine and Convex Sets and Functions ............................................. 165 7.1.2 Effect of Affine and Convex Combinations in Rn ............................ 166 7.1.3 Coefficients of Affine and Convex Combinations ............................ 167 7.1.4 Support and Secant Hyperplanes ...................................................... 169 7.2 The Jensen Inequality ................................................................................... 170 7.2.1 Discrete and Integral Forms of the Jensen Inequality....................... 170 7.2.2 Generalizations of the Jensen Inequality .......................................... 173 7.3 The Hermite-Hadamard Inequality ............................................................... 175 7.3.1 The Classic Form of the Hermite-Hadamard Inequality ..................175 7.3.2 Generalizations of the Hermite-Hadamard Inequality...................... 177 ¨ 7.4 The Rogers-Holder Inequality ...................................................................... 184 ¨ 7.4.1 Integral and Discrete Forms of the Rogers-Holder Inequality ......... 185 ¨ 7.4.2 Generalizations of the Rogers-Holder Inequality ............................. 187 7.5 The Minkowski Inequality ............................................................................ 188 7.5.1 Integral and Discrete Forms of the Minkowski Inequality ............... 189 7.5.2 Generalizations of the Minkowski Inequality................................... 190 Bibliography .......................................................................................................... 192

7.1 7.1.1

CONCEPTS OF AFFINITY AND CONVEXITY AFFINE AND CONVEX SETS AND FUNCTIONS

Throughout this chapter, we will use a vector space X over the field of real numbers R. We think about a binomial linear combination of points x1 , x2 ∈ X and coefficients λ1 , λ2 ∈ R as the sum λ1 x1 + λ2 x2 .

(7.1)

We briefly say that the above sum is a linear combination of points x1 and x2 . A linear combination in formula (7.1) is said to be affine if λ1 + λ2 = 1. A set A ⊆ X is said to be affine if it contains each affine combination of each pair of its 165

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points. A function h : A → R is said to be affine if the equality h(λ1 x1 + λ2 x2 ) = λ1 h(x1 ) + λ2 h(x2 )

(7.2)

holds for each affine combination λ1 x1 + λ2 x2 of each pair of points x1 , x2 ∈ A. A linear combination in formula (7.1) is said to be convex if λ1 + λ2 = 1 and λ1 , λ2 ≥ 0. Really, we have that λ1 , λ2 ∈ [0, 1]. A set C ⊆ X is said to be convex if it contains each convex combination of each pair of its points. A function f : C → R is said to be convex if the inequality f (λ1 x1 + λ2 x2 ) ≤ λ1 f (x1 ) + λ2 f (x2 )

(7.3)

holds for each convex combination λ1 x1 + λ2 x2 of each pair of points x1 , x2 ∈ C. In relation to this, a function f is said to be concave if − f is convex. A convex combination is affine. An affine set is convex. The empty set and singleton are affine and convex. Let S ⊆ X be a set. The affine (convex) hull affS (convS) of the set S is defined as the set containing each binomial affine (convex) combination of each pair of points from S. The set affS (convS) is the smallest affine (convex) set containing S. Let n ≥ 2 be an integer. By using the method of mathematical induction, it can be demonstrated that the set affinity (convexity), function affinity (convexity), and affine (convex) hull apply to n-member affine (convex) combinations. Comprehensive presentation of convex sets, convex functions, and their inequalities can be found in books [12,18]. 7.1.2

EFFECT OF AFFINE AND CONVEX COMBINATIONS IN Rn

Let a and b be the distinct points in the line R. Then, each point x ∈ R can be represented by the affine combination of points a and b as x = α(x)a + β (x)b, where

x−b α(x) = = a − b

x b a b

1 1 1 1

a a−x x , β (x) = a − b = a b

(7.4) 1 1 1 1

.

(7.5)

The convex combinations in formula (7.4) accentuating points x with α(x) ≥ 0 and β (x) ≥ 0 delineate the closed interval with endpoints a and b as the set  ∆ab = conv{a, b} = αa + β b : α, β ≥ 0, α + β = 1 . (7.6) Let a = (a1 , a2 ), b = (b1 , b2 ), and c = (c1 , c2 ) be the noncollinear points in the plane R2 . Then, each point x = (x1 , x2 ) ∈ R2 can be represented by the affine combination of points a, b, and c as x = α(x)a + β (x)b + γ(x)c, (7.7)

167

On Some Important Inequalities

where α (x) =

x1 b1 c1 a1 b1 c1

x2 b2 c2 a2 b2 c2

1 1 1 1 1 1

, β (x) =



a1 x1 c2 a1 b1 c1

a2 x2 c2 a2 b2 c2

1 1 1 1 1 1

, γ (x) =



a1 b1 x1 a1 b1 c1

a2 b2 x2 a2 b2 c2

1 1 1 1 1 1

.

(7.8)

The convex combinations in formula (7.7) emphasizing points x with α(x) ≥ 0, β (x) ≥ 0, and γ(x) ≥ 0 designate the triangle with vertices a, b, and c as the set  ∆abc = conv{a, b, c} = αa + β b + γc : α, β , γ ≥ 0, α + β + γ = 1 . (7.9) Let a0 = (a01 , . . . , a0n ), . . . , an = (an1 , . . . , ann ) be the points in the space Rn such that differences a1 − a0 , . . . , an − a0 are linearly independent. Then, each point x = (x1 , . . . , xn ) ∈ Rn can be represented by the affine combination of points ai as n

x = ∑ αi (x)ai ,

(7.10)

i=0

where a01 .. . αi (x) = x1 . .. an1

. . . a0n . .. . .. . . . xn . .. . .. . . . ann

1 . .. 1 .. . 1

a01 . . . · ai1 . .. an1

. . . a0n . .. . .. . . . ain . .. . .. . . . ann

1 .. . 1 . .. 1

−1 .

(7.11)

Since the differences a1 − a0 , . . . , an − a0 are linearly independent, the affine combination in formula (7.10) is unique for each x ∈ Rn . The affine hull of the set of simplex vertices covers the space Rn , i.e., Aa0 ...an = aff{a0 , . . . , an } = Rn . The convex combinations in formula (7.10) highlighting points x with all αi (x) ≥ 0 appoint the n-simplex with vertices ai as the set n o ∆a0 ...an = conv{a0 , . . . , an } = ∑ni=0 α i ai : α i ≥ 0, ∑ni=0 α i = 1 . (7.12) 7.1.3

COEFFICIENTS OF AFFINE AND CONVEX COMBINATIONS

Let a0 , . . . , an ∈ Rn be the points such that a1 − a0 , . . . , an − a0 are linearly independent. The existence and uniqueness of the representation of point x ∈ Rn by the affine combination of points a0 , . . . , an can be demonstrated in two simple steps. We first represent the difference x − a0 by the unique linear combination n

x − a0 = ∑ λi (ai − a0 ), i=1

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and then express the point x, which yields the affine combination   n n x = ∑ λi ai + 1 − ∑ λi a0 . i=1

i=1

We want to verify the coefficients in formula (7.11). According to the above consideration, we can suppose that  n  n n x = ∑ α j (x)a j = ∑ α j (x)a j1 , . . . , ∑ α j (x)a jn (7.13) j=0

with

j=0

j=0

∑nj=0 α j (x) = 1.

We still need the determinants a01 . . . a0n 1 a01 .. . . .. .. .. . . . . . Ai (x) = x1 . . . xn 1 , A = ai1 . . . . . ... ... .. .. an1 . . . ann 1 an1

. . . a0n . .. . .. . . . ain . .. . .. . . . ann

1 .. . 1 ... 1

.

(7.14)

By exposing the ith row of Ai (x) with coordinates of x in formula (7.13), and ∑nj=0 α j (x) instead of 1, we induce a01 ... a0n 1 .. . . . .. .. .. . n n n Ai (x) = ∑ α j (x)a j1 . . . ∑ α j (x)a jn ∑ α j (x) . j=0 j=0 j=0 .. . .. .. . . . . . an1 ... ann 1 After decomposition into n+1 summands by the ith row, it follows that a01 . . . a0n 1 a01 . . . a0n 1 .. . . .. . . .. .. .. .. . . . . . . n . . Ai (x) = ∑ α j (x) a j1 . . . a jn 1 = αi (x) ai1 . . . ain 1 = αi (x)A, . . . . j=0 . . ... ... . . ... ... .. . . an1 . . . ann 1 an1 . . . ann 1 and thus, we get the expression αi (x) = Ai (x)/A corresponding to formula (7.11). The coefficients αi (x) have a geometric meaning. To present it as simple as possible, we consider the simplex ∆a0 ...ai−1 xai+1 ...an as a fictive n-simplex. If x belongs to the facet ∆a0 ...ai−1 ai+1 ...an , we assume that ∆a0 ...ai−1 xai+1 ...an = ∆a0 ...ai−1 ai+1 ...an . In this case, the coefficient αi (x) and the n-volume of the facet are equal to zero. As for the coefficients’ geometric meaning, since Ai (x) = ±n!voln (∆a0 ...ai−1 xai+1 ...an ), A = ± n!voln (∆a0 ...an ),

(7.15)

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On Some Important Inequalities

it follows that αi (x) = ±

voln (∆a0 ...ai−1 xai+1 ...an ) . voln (∆a0 ...an )

(7.16)

The sign in formula (7.16) depends on the orientation in the space Rn , which refers to the ordered (n+1)-tuples (a0 , . . . , ai−1 , x, ai+1 , . . . , an ) and (a0 , . . . , an ). The plus sign is related to the same orientation, and the minus sign to the opposite. If we consider only the points x belonging to the n-simplex ∆a0 ...an , then the above ordered (n+1)-tuples have the same orientation and the coefficients αi (x) are nonnegative. In that case, the plus sign obviously stands in formula (7.16). The coefficients αi (x) in formula (7.11) indicate functions αi : Rn → R. After expanding the numerator determinant Ai (x) using the ith row containing coordinates of x, and rearranging, we reach the concise representation n

αi (x1 , . . . , xn ) = αi0 + ∑ αi j x j ,

(7.17)

j=1

where αi0 and αi j are the constants as the ratios of determinants without coordinates of x. Following the above formula, it is easy to show that functions αi are affine. Let us look at the representation of an affine function h : Rn → R. By combining the affinity of h with representations in formulas (7.10) and (7.17), we get n

h(x1 , . . . , xn ) = ∑ αi (x1 , . . . , xn )h(ai ) i=0 n 

 n = ∑ αi0 + ∑ αi j x j h(ai ) i=0

j=1

= κ0 + κ1 x1 + · · · + κn xn ,

(7.18)

where κ0 , . . . , κn are the constants. The above representation is usually called the standard form of h. It is not difficult to prove the uniqueness of this form. 7.1.4

SUPPORT AND SECANT HYPERPLANES

Let C ⊆ Rn be a convex set with the nonempty interior, let c be an interior point of C, and let f : C → R be a convex function. Then, the function f admits an affine function h1 : Rn → R, which meets the equality h1 (c) = f (c), and for all x ∈ C satisfies the inequality h1 (x) ≤ f (x). (7.19) In the space Rn+1 , we can imagine that the graph of h1 supports the graph of f at the point F = (c, f (c)). Accordingly, h1 is called the support hyperplane of f at c. The support hyperplane of f at c is not necessarily unique. Formula (7.19) can be called the support hyperplane inequality. The existence of support hyperplanes arises from the separating and supporting hyperplane theorems.

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Let ∆a0 ...an ⊂ Rn be an n-simplex, and let f : ∆a0 ...an → R be a convex function. Then, the function f goes with the affine function h2 : Rn → R determined by the equalities f (a0 ) = h2 (a0 ) , . . . , f (an ) = h2 (an ). It follows that the function h2 for all x ∈ ∆a0 ...an satisfies the inequality: f (x) ≤ h2 (x).

(7.20)

We can visualize that the graph of h2 is a roof of the graph of f fixed at the points F0 = (a0 , f (a0 )) , . . . , Fn = (an , f (an )). Suitably, h2 is called the secant hyperplane of f at vertices, and formula (7.20) can be called the secant hyperplane inequality. If we include c as an interior point of ∆a0 ...an and h1 as a support hyperplane of f at c, then for all x ∈ ∆a0 ...an , we have the support-secant hyperplane inequality: h1 (x) ≤ f (x) ≤ h2 (x).

(7.21)

The equation of the secant hyperplane can be easily determined. By applying the affinity of h2 to the affine combination in formula (7.10), and using the coincidences with f , we get the secant hyperplane equation: n

h2 (x) = ∑ αi (x) f (ai ).

(7.22)

i=0

If n = 1 (n = 2), we use terms “support” and “secant lines” (planes).

7.2

THE JENSEN INEQUALITY

The Jensen inequality is the most important. With more or less effort, almost all important and influential inequalities can be derived from Jensen’s. 7.2.1

DISCRETE AND INTEGRAL FORMS OF THE JENSEN INEQUALITY

Let X be a real vector space, and let ∑ni=1 λi xi be a linear combination of points xi ∈ X such that λ = ∑ni=1 λi = 6 0. The center of the given combination is defined as the point x ∈ X satisfying the discrete equation n

∑ λi (xi − x) = 0.

(7.23)

i=1

It follows that

n

λi xi . i=1 λ

x=∑

(7.24)

So the center x is the affine combination of points xi and coefficients κi = λi /λ . Accordingly, if the given combination is affine, then the combination center coincides with the combination itself, respectively, x = ∑ni=1 λi xi . This is especially true for convex combinations. Discrete form of Jensen’s inequality includes a real vector space, convex combination with its center, and convex function. Regarding the strict form, we present the following theorem.

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On Some Important Inequalities

Theorem 7.1. Let X be a real vector space, let ∑ni=1 λi xi be a convex combination of points xi ∈ X, let C ⊆ X be a convex set containing {x1 , . . . , xn }, and let f : C → R be a convex function. Then, we have the discrete inequality  n  n f ∑ λi xi ≤ ∑ λi f (xi ). (7.25) i=1

i=1

Proof. If n = 1, the trivial inequality f (x1 ) ≤ f (x1 ) represents formula (7.25) regardless of the convexity of f . If n ≥ 2, the method of mathematical induction enables us to perform formula (7.25) as follows. The case n = 2 is regarded as the induction base, wherein the convexity of f provides formula (7.25). The case n > 2 is considered as the induction step, wherein λ1 < 1 is assumed without a reduction of generality. The induction base and premise are included through the next procedure. The right side of the representation n

n

λi xi i=2 1 − λ1

∑ λi xi = λ1 x1 + (1 − λ1 ) ∑

i=1

can be regarded as a binomial convex combination. The induction base can be applied to the right side, and so gain the inequality  n   n  λi f ∑ λi xi ≤ λ1 f (x1 ) + (1 − λ1 ) f ∑ xi . i=1 i=2 1 − λ1 The induction premise can be applied to the (n−1)-member convex combination under the function f , and so obtain the inequality  n  n λi λi f ∑ xi ≤ ∑ f (xi ). 1 − λ 1 − λ1 1 i=2 i=2 The coupling of the above two inequalities gives formula (7.25). Let X be a set with a nonnegative measure µ such that µ(X) > 0, and let g : X → R be a µ-integrable function. The µ-integral arithmetic mean of the function g is defined as the number g ∈ R, which satisfies the integral equation Z


g(x) − g dµ(x) = 0.

(7.26)

X

A simple calculation gives R

g=

X

g(x)dµ(x) . µ(X)

(7.27)

If we include an affine function h : R → R, then we have the integral equality R  R h(g(x))dµ(x) X g(x)dµ(x) h = X . (7.28) µ(X) µ(X)

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This is easy to prove if we use the standard representation h(x) = κ0 + κ1 x. The question is what changes in formula (7.28) if we replace the affine function h with a convex function f . This leads to Jensen’s inequality. Integral form of Jensen’s inequality includes a measurable set X, integrable function g : X → R with its integral arithmetic mean g, and convex function f such that the composition f (g) : X → R is integrable. The next appropriate lemma and theorem provide the strict form. Lemma 7.1. Let X be a set with a nonnegative measure µ such that µ(X) > 0, and let g : X → R be a µ-integrable function. Then, the µ-integral arithmetic mean of g is in the convex hull of the image of g. Proof. Let g be the µ-integral arithmetic mean of g, and let I ⊆ R be an interval containing the image of g. Relying on proof by contradiction, we will suppose that g does not belong to I. Then, either g(x) − g > 0 or g(x)R− g < 0 for all x ∈ X. R Consequently, it implies either X (g(x) − g)dµ(x) > 0 or X (g(x) − g)dµ(x) < 0. Thus, g does not represent the µ-integral arithmetic mean of g. The conclusion is that g belongs to each interval containing the image of g. This must be true even for the smallest one, the convex hull of the image of g. Theorem 7.2. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let g : X → R be a µ-integrable function, let I ⊆ R be an interval containing the image of g, and let f : I → R be a convex function such that f (g) is µ-integrable. Then, we have the integral inequality R  R f (g(x))dµ(x) X g(x)dµ(x) f ≤ X . (7.29) µ(X) µ(X) R

Proof. The mean g = X g(x)dµ(x)/µ(X) belongs to the interval I by Lemma 7.1. Thus, g is either an interior or boundary point of I. If g is an interior point of I, then f as a convex function admits a support line h at g. Thus, h(g) = f (g), and for all x ∈ X, we have the inequality h(g(x)) ≤ f (g(x)).

(7.30)

By using the coincidence f (g) = h(g), affinity equality in formula (7.28), and support line inequality in formula (7.30), we obtain the multiple relation R  R  X g(x)dµ(x) X g(x)dµ(x) f =h µ(X) µ(X) R

=

X

R

≤ covering the inequality in formula (7.29).

X

h(g(x))dµ(x) µ(X) f (g(x))dµ(x) , µ(X)

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On Some Important Inequalities

If g is a boundary point of I, then either g(x) − g ≥ 0 or g(x) − g ≤ 0 for all x ∈ X. Integral equality in formula (7.26) implies g(x) − g = 0, and so g(x) = g for µ-almost all x ∈ X. The trivial inequality f (g) ≤ f (g) represents formula (7.29). Versions of the discrete and integrals forms of the inequalities in formulas (7.25) and (7.29) were proven by Jensen, the discrete version in [6], and integral version in [7]. In the period 1905–1906, Jensen also defined a convex function. 7.2.2

GENERALIZATIONS OF THE JENSEN INEQUALITY

For the sake of formulae conciseness, we will sometimes omit the variable x in integral expressions. This is the case in Theorem 7.3. Let g1 , . . . , gn : X → R be µ-integrable functions. The µ-integral arithmetic mean of the mapping g = (g1 , . . . , gn ) : X → Rn can be defined as the point g = (g1 , . . . , gn ) ∈ Rn whose coordinates satisfy the integral equations Z Z

g1 (x) − g1 dµ(x) = 0 , . . . , gn (x) − gn dµ(x) = 0. (7.31) X

X

Then, it follows that R g=

X

g1 (x)dµ(x) ,..., µ(X)

R X

 gn (x)dµ(x) . µ(X)

(7.32)

If we include an affine function h : Rn → R, then we have the integral equality
R R  R X h g1 (x), . . . , gn (x) dµ(x) X g1 (x)dµ(x) X gn (x)dµ(x) h ,..., = (7.33) µ(X) µ(X) µ(X) due to the representation h(t1 , . . . ,tn ) = κ0 + κ1t1 + · · · + κntn . Lemma 7.2. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let g1 , . . . , gn : X → R be µ-integrable functions, and let g = (g1 , . . . , gn ) : X → Rn be the corresponding Rn -valued mapping. Then, the µ-integral arithmetic mean of the mapping g is in the Cartesian product of the convex hulls of the images of g1 , . . . , gn . Proof. Let I1 , . . . , In be the convex hulls of images of g1 , . . . , gn , respectively. Since each gi ∈ Ii by Lemma 7.1, it follows that g = (g1 , . . . , gn ) ∈ I1 × · · · × In . Integral forms of Jensen’s inequality for convex functions of several variables are certainly very important. We present the following expansion of Theorem 7.2. Theorem 7.3. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let g1 , . . . , gn : X → R be µ-integrable functions, let I1 , . . . , In ⊆ R be intervals containing the images of g1 , . . . , gn , respectively, and let f : I1 ×· · ·×In → R be a convex function such that f (g1 , . . . , gn ) is µ-integrable. Then, we have the integral inequality R R  R f (g1 , . . . , gn )dµ X g1 dµ X gn dµ f ,..., ≤ X . (7.34) µ(X) µ(X) µ(X)

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Proof. If n = 1, Theorem 7.3 is reduced to Theorem 7.2. Therefore, we will assume that n ≥ 2 and consider three positions of the mean g = (g1 , . . . , gn ) located in the cuboid I = I1 × · · · × In . The first position refers to the interior of I, and the other two positions refer to the boundary (relative interior and vertices) of I. If g is an interior point of I (i.e., if g1 , . . . , gn are the interior points of I1 , . . . , In , respectively), then f as a convex function admits a support hyperplane h at g. The support hyperplane inequality applied to the points g(x) = (g1 (x), . . . , gn (x)) holds for all x ∈ X in the form

h g1 (x), . . . , gn (x) ≤ f g1 (x), . . . , gn (x) . (7.35) By relying on the coincidence f (g) = h(g), affinity equality in formula (7.33), and support hyperplane inequality in formula (7.35), we generate the connections R R R  R  X g1 dµ X gn dµ X g1 dµ X gn dµ f ,..., =h ,..., µ(X) µ(X) µ(X) µ(X) R h(g , . . . , g )dµ n 1 = X µ(X) R f (g1 , . . . , gn )dµ ≤ X , (7.36) µ(X) including the inequality in formula (7.34). If g is a boundary point of I so that g1 , . . . , gk are the interior points of I1 , . . . , Ik , respectively, and that gk+1 , . . . , gn are the boundary points of Ik+1 , . . . , In , respectively, where 1 ≤ k ≤ n − 1, then g is an interior point of the k-cuboid J = I1 × · · · × Ik × {gk+1 } × · · · × {gn } contained in the affine plane A = Rk × {gk+1 } × · · · × {gn } of dimension k. Therefore, we will use the restriction fJ = f /J. The convex function fJ : J → R admits a support hyperplane h˜ J : A → R at g as an interior point of J. The support hyperplane inequality holds for all x ∈ X in the form

h˜ J g1 (x), . . . , gk (x), gk+1 , . . . , gn ≤ fJ g1 (x), . . . , gk (x), gk+1 , . . . , gn . Let h˜ : Rn → R be any affine extension of h˜ J . By utilizing the connections h˜ /J = h˜ J , h˜ J (g) = fJ (g), and fJ = f /J, we get h˜ (g) = h˜ J (g) = fJ (g) = f (g). The numbers gk+1 , . . . , gn are the boundary points of the intervals Ik+1 , . . . , In , respectively, which guarantees that µ-almost all x ∈ X satisfy the equalities gk+1 (x) = gk+1 , . . . , gn (x) = gn . All considered indicates that µ-almost all x ∈ X satisfy the multiple relation

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On Some Important Inequalities



h˜ g1 (x), . . . , gn (x) = h˜ g1 (x), . . . , gk (x), gk+1 , . . . , gn
= h˜ J g1 (x), . . . , gk (x), gk+1 , . . . , gn
≤ fJ g1 (x), . . . , gk (x), gk+1 , . . . , gn
= f g1 (x), . . . , gk (x), gk+1 , . . . , gn
= f g1 (x), . . . , gn (x) , and so the support hyperplane inequality

h˜ g1 (x), . . . , gn (x) ≤ f g1 (x), . . . , gn (x) . The coincidence h˜ (g) = f (g) and the above µ-almost everywhere inequality encourage us to put h˜ instead of h into formula (7.36), and thus reach formula (7.34). Similar arguments apply to any k-cuboid of I. If g is a vertex of I (i.e., if g1 , . . . , gn are the boundary points of I1 , . . . , In , respectively), then g1 (x) = g1 , . . . , gn (x) = gn for µ-almost all x ∈ X. The trivial inequality f (g1 , . . . , gn ) ≤ f (g1 , . . . , gn ) represents formula (7.34). Some extensions and generalizations of different variants of Jensen’s inequality can be found in papers [13,16,17].

7.3

THE HERMITE-HADAMARD INEQUALITY

The Hermite-Hadamard inequality is the double inequality with harmonic form consisting of the integral member between two discrete members. 7.3.1

THE CLASSIC FORM OF THE HERMITE-HADAMARD INEQUALITY

In this subsection, we use the Riemann integral and a bounded closed interval [a, b] in the line R with a < b. By using the expression for the integral arithmetic mean Rb

g=

a

g(x)dx b−a

of an integrable function g : [a, b] → R with g(x) = x, we get the barycenter Rb

c=

a+b a xdx = b−a 2

(7.37)

of the interval [a, b], also called the midpoint. The classic form of the Hermite-Hadamard inequality is as follows. Theorem 7.4. Let [a, b] be a bounded closed interval in the line R, and let f : [a, b] → R be a convex function. Then, we have the double inequality   Rb f (x)dx a+b f (a) + f (b) f ≤ a ≤ . (7.38) 2 b−a 2

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First Proof. By applying the convexity of f to the right side of the convex combinations equality
1
a+b 1 = (1−t)a + tb + (1−t)b + ta , 2 2 2 as well as to convex combinations (1−t)a + tb and (1−t)b + ta, we obtain the multiple relation  
1
a+b 1 f ≤ f (1−t)a + tb + f (1−t)b + ta 2 2 2
1
1 ≤ (1−t) f (a) + t f (b) + (1−t) f (b) + t f (a) 2 2 f (a) + f (b) = . 2 Now we single out the double inequality  
1
f (a) + f (b) a+b 1 f ≤ f (1−t)a + tb + f (1−t)b + ta ≤ , 2 2 2 2 and then integrating over the unit interval [0, 1], and using equalities Z 1 0

Z
f (1−t)a + tb dt =

1

Rb


f (1−t)b + ta dt =

0

a

f (x)dx , b−a

we achieve the inequality in formula (7.38). Second Proof. Let h1 be a support line of f at the midpoint c = (a + b)/2, and let h2 be the secant line of f . By integrating the support-secant line inequality h1 (x) ≤ f (x) ≤ h2 (x) over the interval [a, b], dividing with b − a, and applying the affinity of h1 and h2 through formula (7.28) with g(x) = x, we get R b  Rb R b  a xdx a f (x)dx a xdx h1 ≤ ≤ h2 . b−a b−a b−a By using the barycenter representation in formula (7.37), we obtain   Rb   a+b a+b a f (x)dx h1 ≤ ≤ h2 . 2 b−a 2 By utilizing relations       a+b a+b a+b h2 (a)+h2 (b) f (a)+ f (b) h1 =f and h2 = = , 2 2 2 2 2 we attain the inequality in formula (7.38). The inequality of the first (last) two members in formula (7.38) is usually called the left-hand (right-hand) side of the Hermite-Hadamard inequality. The double inequality in formula (7.38) was discovered by Hermite in 1883, see [4]. Ten years later, the left-hand side of this inequality was rediscovered by Hadamard, see [3]. At that time, the notion of a convex function was not yet introduced.

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On Some Important Inequalities

7.3.2 GENERALIZATIONS OF THE HERMITE-HADAMARD INEQUALITY In the first part of this subsection, we use versions of the Riemann integral like multiple integrals in higher dimensions. Let ∆abc a triangle in the plane R2 . By using the expression for the integral arithmetic mean RR  RR  ∆abc g1 (x, y)dxdy ∆ g2 (x, y)dxdy g= , abc ar(∆abc ) ar(∆abc ) of the mapping g = (g1 , g2 ) : ∆abc → R2 with g1 (x, y) = x and g2 (x, y) = y, we get the barycenter  RR

∆abc x dxdy

d=

ar(∆abc )

RR

,

∆abc ydxdy

ar(∆abc )

 =

a+b+c 3

(7.39)

of the triangle ∆abc . The next is the generalization of the Hermite-Hadamard inequality to triangles. Lemma 7.3. Let ∆abc be a triangle in the plane R2 , and let f : ∆abc → R be a convex function. Then, we have the double inequality  f

a+b+c 3

RR

 ≤

∆abc

f (x, y) dxdy

ar(∆abc )

≤

f (a) + f (b) + f (c) . 3

(7.40)

First Proof. We put into practice the unit triangle in the plane R2 as the set ∆2 = {(t, s) : 0 ≤ t ≤ 1, 0 ≤ s ≤ 1 − t}, and thus, we have the correlation ∆abc = {(1−t−s)a + tb + sc : (t, s) ∈ ∆2 }. Now, we can employ the bijection gabc : ∆2 → ∆abc determined by gabc (t, s) = (1−t−s)a + tb + sc. Then, the convexity of f applied to the convex combination on the right side produces the inequality f (gabc (t, s)) ≤ (1−t−s) f (a) + t f (b) + s f (c) = g f (a) f (b) f (c) (t, s). If we put the vertices coordinates a = (a1 , a2 ), b = (b1 , b2 ), and c = (c1 , c2 ), and if we take the substitution
(x, y) = ga1 b1 c1 (t, s) , ga2 b2 c2 (t, s) = gabc (t, s),

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then the corresponding Jacobian outputs as ∂ x/∂t ∂ x/∂ s J = ∂ y/∂t ∂ y/∂ s

a1 a2 1 = b1 b2 1 c1 c2 1

b1 −a1 c1 −a1 = b2 −a2 c2 −a2

= ±2ar(∆abc ) = ±

ar(∆abc ) ar(∆2 )

because ar(∆2 ) = 1/2. It follows that |J| = ar(∆abc )/ar(∆2 ), and we accomplish the integral relation RR ∆abc

f (x, y) dxdy

ar(∆abc )

RR

=

∆2

f (gabc (t, s))dt ds ar(∆2 )

.

(7.41)

By applying the convexity of f to the right side of the convex combinations equality a+b+c 1 1 1 = gabc (t, s) + gbca (t, s) + gcab (t, s), 3 3 3 3 as well as to the convex combinations of gabc (t, s), gbca (t, s), and gcab (t, s), we get the multiple relation   1 1 1 a+b+c f ≤ f (gabc (t, s)) + f (gbca (t, s)) + f (gcab (t, s)) 3 3 3 3 1 1 1 ≤ g f (a) f (b) f (c) (t, s)+ g f (b) f (c) f (a) (t, s)+ g f (c) f (a) f (b) (t, s) 3 3 3 f (a) + f (b) + f (c) = , 3 from which we extricate the double inequality   a+b+c f (gabc (t, s))+ f (gbca (t, s))+ f (gcab (t, s)) f (a)+ f (b)+ f (c) f ≤ ≤ . 3 3 3 By integrating over the unit triangle ∆2 , using equalities ZZ

ZZ

f (gabc (t, s))dt ds = ∆2

ZZ

f (gbca (t, s))dt ds = ∆2

f (gcab (t, s))dt ds, ∆2

and dividing with ar(∆2 ), we obtain  RR  a+b+c f (a) + f (b) + f (c) ∆2 f (gabc (t, s)) dt ds f ≤ ≤ . 3 ar(∆2 ) 3 Respecting the integral relation in formula (7.41), we have formula (7.40). Second Proof. Let h1 be a support plane of f at the barycenter d = (a + b + c)/3, and let h2 be the secant plane of f . By integrating the support-secant plane inequality h1 (x, y) ≤ f (x, y) ≤ h2 (x, y)

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On Some Important Inequalities

over the triangle ∆abc , dividing with ar(∆abc ), and applying the affinity of h1 and h2 through formula (7.33) with g1 (x, y) = x and g2 (x, y) = y, we get RR RR  RR ∆abc x dxdy ∆abc ydxdy ∆abc h1 (x, y) dxdy h1 , = ar(∆abc ) ar(∆abc ) ar(∆abc ) RR ∆abc f (x, y)dxdy ≤ ar(∆abc ) RR ∆abc h2 (x, y)dxdy ≤ ar(∆abc ) RR RR  ∆abc x dxdy ∆abc y dxdy = h2 , . ar(∆abc ) ar(∆abc ) By taking the odd members, and referring to the barycenter representation in formula (7.39), we obtain   RR   a+b+c a+b+c ∆abc f (x, y)dxdy h1 ≤ ≤ h2 . 3 ar(∆abc ) 3 By utilizing coincidences       a+b+c a+b+c a+b+c f (a)+ f (b)+ f (c) h1 =f and h2 = , 3 3 3 3 we attain the inequality in formula (7.40). The barycenter of the n-simplex ∆a0 ...an in the space Rn is the point  R . . .R a=

∆a0 ...an x1 dx1 . . . dxn

voln (∆a0 ...an )

R

,...,

R

...

∆a0 ...an xn dx1 . . . dxn



voln (∆a0 ...an )

=

∑ni=0 ai . n+1

(7.42)

We can realize the Hermite-Hadamard inequality for simplices. Theorem 7.5. Let ∆a0 ...an be an n-simplex in the space Rn , and let f : ∆a0 ...an → R be a convex function. Then, we have the double inequality  n  R . . .R ∆a0 ...an f (x1 , . . . , xn ) dx1 . . . dxn ∑n f (ai ) ∑i=0 ai ≤ ≤ i=0 . (7.43) f n+1 voln (∆a0 ...an ) n+1 To establish the relevant inequalities for convex functions defined on simplices, we will use the simplex representations presented in Section 7.1. Lemma 7.4. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let g : X → R be a bounded µ-integrable function, let [a, b] be an interval containing the image of g, let αa + β b be the convex combination such that R X

g(x)dµ(x) = α a + β b, µ(X)

(7.44)

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and let f : [a, b] → R be a convex function. Then, we have the double inequality R X

f (αa + β b) ≤

f (g(x)) d µ(x) ≤ α f (a) + β f (b). µ(X)

(7.45)

Proof. As regards the equality in formula (7.44), we will determine the convex combination αa + β b, which represents the µ-integral arithmetic mean R

g=

X

g(x)dµ(x) . µ(X)

By applying formula (7.4) to the point g, and utilizing the affinity of functions α and β , we obtain the affine combination g = α(g)a + β (g)b R X

=

α(g(x))dµ(x) a+ µ(X)

R X

β (g(x))dµ(x) b µ(X)

= αa+β b with coefficients R X

α=

R

α(g(x))dµ(x) and β = µ(X)

X

β (g(x))dµ(x) . µ(X)

(7.46)

These coefficients are nonnegative because g(x) ∈ [a, b] for all x ∈ X, and so the affine combination g = α a + β b is convex. To prove the inequality in formula (7.45), we will consider two cases of the combination g = αa + β b, referring to the number of positive coefficients. So the cases specify the interior (a, b) and boundary {a, b}. If g = αa + β b with α, β > 0, then g ∈ (a, b). By including a support line h1 of f at g, and the secant line h2 of f at a and b, we get the multiple inequality R

f (αa + β b) = f (g) = h1 (g) = R

≤

X

X h1 (g(x))dµ(x)

µ(X)

f (g(x))dµ(x) ≤ µ(X)

R

X h2 (g(x))dµ(x)

µ(X )

= h2 (g)

= α h2 (a) + β h2 (b) = α f (a) + β f (b), which contains formula (7.45). The composition f (g) is µ-integrable because it is bounded on X, and continuous µ-almost everywhere on X. If g = a, then the equation α = 1 via formula (7.46) implies that g(x) = a for µ-almost all x ∈ X (the same arises from the equation β = 0). It turns out that R X

f (g(x))dµ(x) = µ(X)

R X

f (a)dµ(x) = f (a), µ(X)

and so the trivial double inequality f (a) ≤ f (a) ≤ f (a) represents formula (7.45). A similar reasoning applies for g = b.

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On Some Important Inequalities

Lemma 7.5. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let g1 , g2 : X → R be the bounded µ-integrable functions, let ∆abc be a triangle containing the image of the mapping g = (g1 , g2 ) : X → R2 , let αa + β b + γc be the convex combination such that R R  X g1 (x)dµ(x) X g2 (x)dµ(x) = α a + β b + γ c, (7.47) , µ(X) µ(X) and let f : ∆abc → R be a convex function. Then, we have the double inequality R

f (αa + β b + γc) ≤

X

f (g1 (x), g2 (x))dµ(x) ≤ α f (a) + β f (b) + γ f (c). µ(X)

(7.48)

Theorem 7.6. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let g1 , . . . , gn : X → R be the bounded µ-integrable functions, let ∆a0 ...an be an n-simplex containing the image of the mapping g = (g1 , . . . , gn ) : X → Rn , let ∑ni=0 α i ai be the convex combination such that R R  n X g1 (x)dµ (x) X gn (x)dµ(x) ,..., = ∑ α i ai , (7.49) µ(X) µ(X) i=0 and let f : ∆a0 ...an → R be a convex function. Then, we have the double inequality  n  R n f (g1 (x), . . . , gn (x))dµ(x) f ∑ α i ai ≤ X ≤ ∑ α i f (ai ). µ(X) i=0 i=0

(7.50)

Proof. The representation in formula (7.49) can be realized by applying formula (7.10) to the µ-integral arithmetic mean R R  X g1 (x)dµ(x) X gn (x)dµ(x) g= ,..., , µ(X) µ(X) and employing the affinity of functions αi . This yields nonnegative coefficients R

αi =

X

αi (g(x))dx . µ(X)

(7.51)

To prove the inequality in formula (7.50), we will consider three cases of the convex combination g = ∑ni=0 α i ai , referring to the number of positive coefficients. The first case connotes the interior of ∆a0 ...an , and the other two cases connote the boundary (relative interior and vertices) of ∆a0 ...an . If g = ∑ni=0 α i ai with α 0 , . . . , α n > 0, then g is an interior point of ∆a0 ...an , and f as a convex function admits a support hyperplane h1 at g. By including the secant hyperplane h2 at vertices, we have the coincidences h1 (g) = f (g), f (a0 ) = h2 (a0 ), . . . , f (an ) = h2 (an ),

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and for all x ∈ X the support-secant hyperplane inequality h1 (g(x)) ≤ f (g(x)) ≤ h2 (g(x)), which is applied to the points g(x) = (g1 (x), . . . , gn (x)). By using the above coincidences and inequality, we can obtain the multiple relation R  n  h1 (g(x))dµ(x) f ∑ α i ai = f (g) = h1 (g) = X µ (X ) i=0 R

≤

f (g(x))dµ(x) ≤ µ(X)

X n

=

R

X h2 (g(x))dµ(x)

µ(X )

= h2 (g)

(7.52)

n

∑ α i h2 (ai ) = ∑ α i f (ai ),

i=0

i=0

covering the inequality in formula (7.50). If g = ∑ki=0 α i ai with α 0 , . . . , α k > 0, where 1 ≤ k ≤ n − 1, then g is an interior point of the k-face S = ∆a0 ...ak contained in the affine plane A = aff(S) of dimension k, and we rely on the restriction fS = f /S. The convex function fS : S → R admits a support hyperplane h˜ S : A → R at g, which provides the equality h˜ S (g) = fS (g), and for all s = (s1 , . . . , sn ) ∈ S, the support hyperplane inequality h˜ S (s) ≤ fS (s). (7.53) Let h˜ 1 : Rn → R be any affine extension of h˜ S . Now, we have the coincidences h˜ 1 (g) = h˜ S (g) = fS (g) = f (g). The system of equations α k+1 = 0, . . . , α n = 0 via formula (7.51) implies that g(x) ∈ ∆a0 ...ak ak+2 ...an ∩ . . . ∩ ∆a0 ...an−1 = ∆a0 ...ak = S for µ-almost all x ∈ X. Therefore, the support hyperplane inequality in formula (7.53) applies to µ-almost all x ∈ X in the form h˜ S (g(x)) ≤ fS (g(x)). Now, we include the secant hyperplane h2 of f at vertices. Then, µ-almost all x ∈ X satisfy the multiple relation h˜ 1 (g(x)) = h˜ S (g(x)) ≤ fS (g(x)) = f (g(x)) ≤ h2 (g(x)), and consequently, the support-secant hyperplane inequality h˜ 1 (g(x)) ≤ f (g(x)) ≤ h2 (g(x)).

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On Some Important Inequalities

The coincidence h˜ 1 (g) = f (g) and the above µ-almost everywhere inequality enable us to put h˜ 1 instead of h1 into formula (7.52), and thus reach formula (7.50). The same applies to other k-faces. If g = a0 , then the equation α 0 = 1 via formula (7.51) implies that g(x) = a0 for µ-almost all x ∈ X. It further produces the trivial inequality f (a0 ) ≤ f (a0 ) ≤ f (a0 ), representing formula (7.50). The same applies to other vertices. The inequality in formula (7.50) is reduced to the Hermite-Hadamard inequality in formula (7.43) if we take X = ∆a0 ...an and g1 (x) = x1 , . . . , gn (x) = xn . In this reduction, we have the correlations α 0 = . . . = α n = 1/(n + 1), dµ(x) = d1 . . . dxn , and µ(X) = voln (∆a0 ...an ). Formula (7.12) can be considered as the simplex definition in any real vector space of sufficiently large dimension. We will include this feature in the statement of the discrete version of the Hermite-Hadamard inequality. Theorem 7.7. Let X be a real vector space of dimension of at least n, let ∆a0 ...an be an n-simplex in X, let ∑mj=1 λ j x j be a convex combination of points x j ∈ ∆a0 ...an , let ∑ni=0 α i ai be the convex combination of vertices ai such that m

n

∑ λ j x j = ∑ α i ai ,

j=1

(7.54)

i=0

and let f : ∆a0 ...an → R be a convex function. Then, we have the double discrete inequality  n  m n f ∑ α i ai ≤ ∑ λ j f (x j ) ≤ ∑ α i f (ai ). i=0

j=1

(7.55)

i=0

Proof. By using the simplex definition in formula (7.12), we can represent the points x j as the convex combinations n

x j = ∑ α i j ai .

(7.56)

i=0

Then, using the assumption in formula (7.54), we obtain  n   n m m n  m ∑ α i ai = ∑ λ j x j = ∑ λ j ∑ α i j ai = ∑ ∑ λ j α i j ai , i=0

j=1

j=1

i=0

i=0

j=1

and since the convex combination of simplex vertices is unique, it follows that m

∑ λ j α i j = α i.

j=1

By applying Jensen’s inequality to the convex combination ∑mj=1 λ j x j and subsequently through formula (7.56), and utilizing the above correlation coefficients, we produce the multiple relation

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 f

n



∑ α i ai



m



∑ λ jx j

≤





=f

i=0

n

m

=∑

i=0

j=1 m

m

j=1

j=1

i=0

n

f (ai ) =

∑ λ jαi j

n

∑ λ j f (x j ) ≤ ∑ λ j ∑ α i j f (ai ) ∑ α i f (ai ),

j=1

i=0

including the double inequality in formula (7.55). Another Proof of the Case X = Rn . We will adapt Theorem 7.6 to the given discrete case. Let x j = (x j1 , . . . , x jn ) be the coordinate representations for j = 1, . . . , m. Let X = {x1 , . . . , xm } be an m-member set with the probability measure µ given by µ({x j }) = λ j , and let gi be the functions determined by gi (x j ) = x ji . Then, we have R

X g1 (x)dµ(x)

g =

µ(X) 

=

R

,...,

X gn (x)dµ(x)

m

=

µ(X) m

 ∑ g1 (x j )µ({x j }), . . . , ∑ gn (x j )µ({x j })

j=1





j=1

m

m

∑ λ j x j1 , . . . , ∑ λ j x jn

j=1

m

 =

j=1

∑ λ jx j

j=1

and R

f (g) = =

f (g1 (x), . . . , gn (x))dµ(x) µ(X) m
∑ f g1 (x j ), . . . , gn (x j ) µ({x j }) X

j=1 m

=

∑ λ j f (x j1 , . . . , x jn ) =

j=1

m

∑ λ j f (x j ).

j=1

Thus, equality in formula (7.49) turns into equality in formula (7.54), and inequality in formula (7.50) turns into inequality in formula (7.55). The inequality in formula (7.55) discusses the nature of the behavior of convex functions. It shows that the convex function values taken in the form of convex combinations grow from the center, across the middle, to the vertices as ends. Simple proof and refinement of the Hermite-Hadamard inequality was demonstrated in [2]. The Hermite-Hadamard inequality for the simplex, its refinements, and generalizations were considered in papers [1,10,11,14,15].

7.4

¨ THE ROGERS-HOLDER INEQUALITY

¨ The Rogers-Holder inequality occupies an important place in mathematical analysis, especially in the study and development of L p spaces. Although it comes down to nonnegative real-valued functions, complex-valued functions are used due to their wider applications. This is usually done because of L p spaces.

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On Some Important Inequalities

¨ 7.4.1 INTEGRAL AND DISCRETE FORMS OF THE ROGERS-HOLDER INEQUALITY Let X be a set with a nonnegative measure µ such that µ(X) > 0. Let p > 0 be a real number, and let Lµp (X ) = L p (µ) be the space of all µ-measurable functions f : X → C such that f p is µ-integrable (Lebesgue integrable with respect to the measure µ). Each function f ∈ L p (µ) has the fictive p-norm Z 1/p k f kp = | f (x)| p dµ(x) . (7.57) X

If p ≥ 1, the above definition gives the (proper) p-norm, the inequality | f (x)| ≤ | f (x)| p holds for every x ∈ X, and therefore, Lµp (X ) ⊆ L1µ (X). The fundamental space L1µ (X) contains all complex functions on X that are µ-integrable. ¨ Initial forms of the Rogers-Holder inequality use a pair of positive real numbers p and q such that p + q = pq. It is equivalent to conditions 1/p + 1/q = 1 and p, q > 1. The numbers p and q are called the conjugate exponents. ¨ All that is needed is prepared for Rogers-Holder inequalities. The first one that ¨ follows is the integral form of the Rogers-Holder inequality. The integral inequality in formula (7.34) adjusted to concave functions will serve as a basis of the proof. Theorem 7.8. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let p, q > 1 be the real numbers such that 1/p + 1/q = 1, let f : X → C be a function in Lµp (X ), and let g : X → C be a function in Lqµ (X). Then, we have the integral inequality Z 1/p  Z 1/q Z p q | f (x)g(x)|dµ(x) ≤ | f (x)| dµ(x) |g(x)| dµ(x) . (7.58) X

X

X

Proof. Since p, q > 1, the functions f and g are in L1µ (X), and therefore, their product f g is also in L1µ (X). Let I1 ⊆ R be an interval containing the image of | f | p , let I2 ⊆ R be an interval containing the image of |g|q , and let φ : I1 × I2 → R be a concave function. The generalized integral inequality in formula (7.34) for integer n = 2 adjusted to functions g1 = | f | p , g2 = |g|q and f = −φ takes the form
R R R  p q p q X φ | f (x)| , |g(x)| dµ(x) X | f (x)| d µ(x) X |g(x)| dµ(x) ≤φ , . µ(X) µ(X) µ(X) If we implement the concave function φ : [0, ∞) × [0, ∞) → R defined by φ (x, y) = x1/ p y1/q , then we obtain R X

| f (x)g(x)|dµ(x) ≤ µ(X)

R

X

| f (x)| p dµ(x) µ(X)

1/p  R

X

|g(x)|q dµ(x) µ(X)

1/q

After multiplying by µ(X), we achieve the inequality in formula (7.58).

.

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The integral inequality in formula (7.58) can be presented as the norm inequality k f gk1 ≤ k f k p kgkq .

(7.59)

¨ The integral form of the Rogers-Holder inequality can be transformed into discrete form by taking advantage of the counting measure. Theorem 7.9. Let xi , yi ∈ C be the complex numbers for i = 1, . . . , n, and let p, q > 1 be the real numbers such that 1/p + 1/q = 1. Then, we have the discrete inequality  n 1/p  n 1/q n p q |x y | ≤ |x | |y | . (7.60) ∑ ii ∑ i ∑ i i=1

i=1

i=1

Proof. Let X = {x1 , . . . , xn } be an n-member set with the counting measure µ as µ({xi }) = 1, let f be the function as f (xi ) = xi , and let g be the function as g(xi ) = yi . Then, the transformation of the left side in formula (7.58) shows Z X

n

n

| f (x)g(x)|dµ(x) = ∑ | f (xi )g(xi )| µ({xi }) = ∑ |xi yi |, i=1

i=1

pointing the left side in formula (7.60). The same applies to the right side. By promoting n-tuples x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ), the product x · y = (x1 y1 , . . . , xn yn ), and norms  kx k p =

n

∑ |xi |

p

1/p

 , kykq =

i=1

n q

1/q

∑ |yi |

i=1

n

, kx · yk1 = ∑ |xi yi |, i=1

the discrete inequality in formula (7.60) can be written as the norm inequality kx · yk1 ≤ kxk p kykq .

(7.61)

In the books, formulas (7.58) and (7.60) were proved by relying on the discrete form of the Young inequality. It says that the inequality ab ≤

1 p 1 q a + b p q

(7.62)

holds for every pair of nonnegative real numbers a and b. ¨ Inequalities in formulas (7.58) and (7.60) are usually called the Holder inequalities. Respecting the chronological publication order of paper [19] and paper [5], we ¨ preferred the name Rogers-Holder . In the case p = q = 2, these inequalities are also called the Cauchy-Bunyakovsky-Schwarz inequalities. As regards the above special case, the inequality for sums was published by Cauchy (1821), the corresponding inequality for integrals was proved by Bunyakovsky in (1859), and the improved inequality for integrals was given by Schwarz (1888).

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¨ 7.4.2 GENERALIZATIONS OF THE ROGERS-HOLDER INEQUALITY ¨ The main tool in the generalization of Rogers-Holder inequalities are positive real numbers p1 , . . . , pm , p such that ∑mj=1 1/p j = 1/p. The consequent relation p p +···+ =1 p1 pm is effective, implying that p/p j < 1 and thus p < p j for j = 1, . . . , m. Note that numbers p j may be less than or equal to 1. ¨ The generalized integral form of the Rogers-Holder inequality is as follows. Corollary 7.1. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let p1 , . . . , pm , p > 0 be the real numbers such that ∑mj=1 1/p j = 1/p, and let f j : X → C p be the functions in Lµ j (X) for j = 1, . . . , m. Then, we have the integral inequality Z 1/p  Z 1/p1  Z 1/pm p p1 pm | f1 . . . fm | dµ ≤ | f1 | dµ ... | fm | dµ . (7.63) X

X

X

Proof. Since p < p1 , . . . , p < pm , it follows that | f1 . . . fm | p ≤ | f1 | p1 . . . | fm | pm , and so the product f1 . . . fm is in Lµp (X). Let I j ⊆ R be an interval containing the image of | f j | p j for j = 1, . . . , m, and let φ : I1 × · · · × Im → R be a concave function. The generalized integral inequality in formula (7.34) for integer n = m with functions g1 = | f1 | p1 , . . . , gm = | fm | pm and f = −φ comes out as
R R R  pm dµ p1 | f1 | p1 dµ | fm | pm dµ X φ | f1 | , . . . , | fm | ≤φ X ,..., X . µ(X) µ(X) µ(X) If we include the concave function φ : [0, ∞) × · · · × [0, ∞) → R determined by p/p1

φ (x1 , . . . , xm ) = x1

p/pm

. . . xm

,

then we realize R X

| f1 . . . fm | p dµ ≤ µ(X)

R

X

| f1 | p1 dµ µ(X)

p/p1

R ...

X

| fm | pm dµ µ(X)

p/pm .

The multiplication by µ(X) and raising to the power 1/p gives formula (7.63). The integral inequality in formula (7.63) can be expressed as the fictive norm inequality k f1 . . . fm k p ≤ k f1 k p1 . . . k fm k pm . (7.64) If m = 2 and p = 1, formula (7.63) is reduced to formula (7.58), and formula (7.64) is reduced to formula (7.59). ¨ The following is the generalized discrete form of the Rogers-Holder inequality.

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Corollary 7.2. Let xi j ∈ C be the complex numbers for i = 1, . . . , n and j = 1, . . . , m, and let p1 , . . . , pm , p > 0 be the real numbers such that ∑mj=1 1/p j = 1/p. Then, we have the discrete inequality 

n

∑ |xi1 . . . xim | p

1/p

 ≤

i=1

n

∑ |xi1 | p1

1/p1

i=1

 ...

n

∑ |xim | pm

1/pm .

(7.65)

i=1

Proof. Let X = {x1 , . . . , xn } be an n-member set with the counting measure µ given by µ({xi }) = 1, and let f j be the functions determined by f j (xi ) = xi j . Then, the transformation of pth power of the left side in formula (7.63) produces Z X

n

n

| f1 (x) . . . fm (x)| p dµ(x) = ∑ | f1 (xi ) . . . fm (xi )| p µ({xi }) = ∑ |xi1 . . . xim | p i=1

i=1

as pth power of the left side in formula (7.65). The similar applies to the right side. By exploiting n-tuples x1 = (x11 , . . . , xn1 ), . . . , xm = (x1m , . . . , xnm ), the product x1 · . . . · xm = (x11 . . . x1m , . . . , xn1 . . . xnm ), and using the corresponding fictive norms, the discrete inequality in formula (7.65) can be presented as the fictive norm inequality kx1 · . . . · xm k p ≤ kx1 k p1 . . . kxm k pm .

(7.66)

Formulas (7.63) and (7.65) can be proved by applying the method of mathematical induction to integer m, wherein the case m = 2 serves as the induction base. These formulae can also be obtained by relying on the generalized discrete form of the Young inequality. It can be demonstrated that the inequality a1 . . . am ≤

p p1 /p p pm /p a +...+ am p1 1 pm

(7.67)

holds for every m-tuple of nonnegative real numbers a1 , . . . , am . More details and better insight into Young’s original inequalities can be found in the proceedings [20].

7.5

THE MINKOWSKI INEQUALITY

The Minkowski inequality is the key inequality of functional analysis. It has the greatest influence in the normed vector spaces.

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On Some Important Inequalities

7.5.1 INTEGRAL AND DISCRETE FORMS OF THE MINKOWSKI INEQUALITY Minkowski inequalities demonstrate that p-norms satisfy the triangle inequality. These inequalities are fundamental in the field of functional analysis. Let x, y ∈ C be a pair of complex numbers. The prominent triangle inequality for absolute value as |x + y| ≤ |x| + |y| (7.68) can be obtained by applying Jensen’s inequality in formula (7.25) to the convex combination x/2 + y/2, and convex function ϕ : C → R given by ϕ(z) = 2|z|. The following is the integral form of the Minkowski inequality. The integral inequality in formula (7.34) will be the mainstay of the proof. Theorem 7.10. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let p ≥ 1 be a real number, and let f , g : X → C be the functions in Lµp (X). Then, we have the integral inequality Z 1/p Z 1/p Z 1/p | f (x) + g(x)| p dµ(x) ≤ | f (x)| p dµ(x) + |g(x)| p dµ(x) . X

X

X

(7.69)
p Proof. A slightly modified formula (7.69) with the integrand | f (x)| + |g(x)| on the left side will be discussed first. The sum | f | + |g| is in Lµp (X) because
p
| f (x)| + |g(x)| ≤ 2 p−1 | f (x)| p + |g(x)| p for every x ∈ X. The above inequality can be obtained by applying Jensen’s inequality in formula (7.25) to the convex combination | f (x)|/2+|g(x)|/2, and convex function ϕ : [0, ∞) → R given by ϕ(t) = 2 pt p . Let I1 ⊆ R be an interval containing the image of | f | p , let I2 ⊆ R be an interval containing the image of |g| p , and let φ : I1 × I2 → R be a concave function. The generalized integral inequality in formula (7.34) for integer n = 2 with functions g1 = | f | p , g2 = |g| p , and f = −φ arises in the form
R R R  p p p p X φ | f (x)| , |g(x)| dµ(x) X | f (x)| dµ(x) X |g(x)| dµ(x) ≤φ , . µ(X) µ(X) µ(X) If we utilize the concave function φ : [0, ∞) × [0, ∞) → R defined by
p φ (x, y) = x1/p + y1/p , then we gain R X


p R 1/ p R 1/p  p p p | f (x)| + |g(x)| dµ(x) X | f (x)| dµ (x) X |g(x)| dµ(x) ≤ + . µ(X) µ(X) µ(X)

After multiplying by µ(X), and raising to the power 1/p, we have Z 1/p Z 1/p Z 1/p
p p p | f (x)| + |g(x)| dµ (x) ≤ | f (x)| dµ(x) + |g(x)| dµ(x) . X

X

X

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This inequality can be extended to the
left by using the integrand | f (x) + g(x)| p p because | f (x) + g(x)| p ≤ | f (x)| + |g(x)| for every x ∈ X. The collection Lµp (X) with p ≥ 1 becomes the normed vector space if the integral inequality in formula (7.69) is expressed as the norm inequality k f + gk p ≤ k f k p + kgk p .

(7.70)

The discrete form of the Minkowski inequality is as follows. Theorem 7.11. Let xi , yi ∈ C be the complex numbers for i = 1, . . . , n, and let p ≥ 1 be a real number. Then, we have the discrete inequality !1/p !1/p   n

∑ |xi + yi | p

1/p

n

≤

i=1

∑ |xi | p

n

+

i=1

∑ |yi | p

.

(7.71)

i=1

Proof. The integral inequality in formula (7.69) is transformed into discrete inequality in formula (7.71) by exploring the counting measure as in Theorem 7.9. By using n-tuples x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ), and required p-norms, the discrete inequality in formula (7.71) can be presented as the norm inequality kx + yk p ≤ kxk p + kyk p .

(7.72)

In the books, formulas (7.69) and (7.71) were proved by relying on the triangle in¨ equality and corresponding Rogers-Holder inequalities. 7.5.2

GENERALIZATIONS OF THE MINKOWSKI INEQUALITY

To generalize the integral form of the Minkowski inequality, we need a real number p ≥ 1, and complex functions in the space Lµp (X). It is as follows. Corollary 7.3. Let X be a set with a nonnegative measure µ such that µ(X) > 0, let p ≥ 1 be a real number, and let f1 , . . . , fm : X → C be the functions in Lµp (X). Then, we have the integral inequality Z 1/p  Z 1/p Z 1/p p p p | f1 + · · · + fm | dµ ≤ | f1 | dµ +···+ | fm | dµ . (7.73) X

X

X


p Proof. The modified formula (7.73) having the integrand | f1 | + · · · + | fm | on the p left side will be considered first. The sum | f1 | + · · · + | fm | is in Lµ (X) because
p
| f1 (x)| + · · · + | fm (x)| ≤ m p−1 | f1 (x)| p + · · · + | fm (x)| p for every x ∈ X. The above inequality can be obtained by applying the discrete form of Jensen’s inequality to the convex combination | f1 (x)|/m + · · · + | fm (x)|/m, and convex function ϕ : [0, ∞) → R given by ϕ(t) = m pt p .

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On Some Important Inequalities

Let I j ⊆ R be an interval containing the image of | f j | p for j = 1, . . . , m, and let φ : I1 × · · · × Im → R be a concave function. The generalized integral inequality in formula (7.34) for integer n = m with functions g1 = | f1 | p , . . . , gm = | fm | p , and f = −φ appears as
R R R  p p | f1 | p dµ | fm | p dµ X φ | f 1 | , . . . , | f m | dµ ≤φ X ,..., X . µ(X) µ(X) µ(X) If we employ the concave function φ : [0, ∞) × · · · × [0, ∞) → R defined by 1/p 1/p
p φ (x1 , . . . , xm ) = x1 + · · · + xm , then we obtain R X


p R p  R p p p p | f1 | + · · · + | fm | dµ X | f 1 | dµ X | f m | dµ ≤ +···+ . µ(X) µ(X) µ(X)

After multiplying by µ(X), and raising to the power 1/p, we attain Z 1/p  Z 1/ p Z 1/p
p ≤ | f1 | p dµ +···+ | fm | p dµ . | f1 | + · · · + | fm | dµ X

X

X

p This inequality can be extended to the left by using the
p integrand | f1 + · · · + fm | p because | f1 (x) + · · · + fm (x)| ≤ | f1 (x)| + · · · + | fm (x)| for every x ∈ X.

The integral inequality in formula (7.73) can be represented as the norm inequality k f1 + · · · + fm k p ≤ k f1 k p + · · · + k fm k p .

(7.74)

The following is the generalized discrete form of the Minkowski inequality. Corollary 7.4. Let xi j ∈ C be the complex numbers for i = 1, . . . , n and j = 1, . . . , m, and let p ≥ 1 be a real number. Then, we have the discrete inequality  n 1/p  n 1/p  n 1/p p p p |x + · · · + x | ≤ |x | + · · · + |x | . (7.75) im ∑ i1 ∑ i1 ∑ im i=1

i=1

i=1

Proof. The integral inequality in formula (7.73) is transformed into discrete inequality in formula (7.75) by exploring the counting measure as in Corollary 7.2. By using n-tuples x j = (x1 j , . . . , xn j ) for j = 1, . . . , m, and required p-norms, the discrete inequality in formula (7.75) can be presented as the norm inequality kx1 + · · · + xm k p ≤ kx1 k p + · · · + kxm k p .

(7.76)

Formulas (7.74) and (7.76) can be easily achieved by applying the method of mathematical induction to integer m, wherein the case m = 2 is as the induction base. In doing so, formulas (7.73) and (7.76) would also be verified as equivalents of (7.74) and (7.76), respectively. Original details concerning Minkowski’s inequalities can be seen in his book [9].

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BIBLIOGRAPHY [1] M. Bessenyei, The Hermite-Hadamard inequality on simplices, Amer. Math. Monthly 115 (2008), 339–345. [2] A. Farissi, Simple proof and refinement of Hermite-Hadamard inequality, J. Math. Inequal. 4 (2010), 365–369. ´ ´ es ´ des fonctions entieres ` et en particulier d’une fonc[3] J. Hadamard, Etude sur les propriet ´ par Riemann, J. Math. Pures Appl., 58 (1893), 171–215. tion consideree ´ ´ [4] Ch. Hermite, Sur deux limites d’une integrale definie, Mathesis, 3 (1883), 82. ¨ [5] O. Holder , Uber einen Mittelwertsatz, Nachr. Ges. Wiss. Goettingen (1889), 38–47. [6] J. L. W. V. Jensen, Om konvekse Funktioner og Uligheder mellem Middelværdier, Nyt Tidsskr. Math. B 16 (1905), 49–68. ´ alites ´ entre les valeurs moyennes, [7] J. L. W. V. Jensen, Sur les fonctions convexes et les ineg Acta Math. 30 (1906), 175–193. [8] E. J. McShane, Jensen’s inequality, Bulletin of the American Mathematical Society, 43 (1937), 521–527. ¨ [9] H. Minkowski, Theorie der Konvexen Korper , Insbesondere Begrundung ihres ¨ Oberflac ¨ henbegriffs, Gesammelte Abhandlungen II, Leipzig, (1911). [10] F.-C. Mitroi and C. I. Spiridon, Refinements of Hermite-Hadamard inequality on simplices, Math. Rep. (Bucur.)15 (2013), arXiv:1105.5043. [11] E. Neuman, Inequalities involving multivariate convex functions II, Proc. Amer. Math. Soc. 109 (1990), 965–974. [12] C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Canadian Mathematical Society, Springer, New York, (2006). ´ Extension of Jensen’s inequality to affine combinations, J. Inequal. Appl. [13] Z. Pavic, 2014 (2014), Article 298. ´ Geometric and analytic connections of the Jensen and Hermite-Hadamard in[14] Z. Pavic, equality, Math. Sci. Appl. E-Notes 4 (2016), 69–76. ´ Improvements of the Hermite-Hadamard inequality for the simplex, J. Inequal. [15] Z. Pavic, Appl. 2017 (2017), Article 3. ´ J. Pecari ˇ c´ and I. Peric, ´ Integral, discrete and functional variants of Jensen’s [16] Z. Pavic, inequality, J. Math. Inequal. 5 (2011), 253–264. ´ The Jensen and Hermite-Hadamard inequality on the triangle, J. Math. Inequal. [17] Z. Pavic, 11 (2017), 1099–1112. [18] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York and London, (1973). [19] L. J. Rogers, An extension of a certain theorem in inequalities, Messenger of Math. 17 (1888), 145–150. [20] W. H. Young, On classes of summable functions and their Fourier series, Proceedings of the Royal Society A 87 (1912), 225–229.

of Young’s 8 Refinements Integral Inequality via Fundamental Inequalities and Mean Value Theorems for Derivatives1 Feng Qi Henan Polytechnic University Inner Mongolia University for Nationalities Tianjin Polytechnic University

Wen-Hui Li Zhengzhou University of Science and Technology

Guo-Sheng Wu Sichuan Technology and Business University

Bai-Ni Guo Henan Polytechnic University

CONTENTS 8.1 Young’s Integral Inequality and Several Refinements................................... 194 8.1.1 Young’s Integral Inequality............................................................... 195 8.1.2 Refinements of Young’s Integral Inequality via Lagrange’s Mean Value Theorem .................................................................................. 197 8.1.3 Refinements of Young’s Integral Inequality via ˇ ˇ v’s Integral Inequalities................ 201 Hermite-Hadamard’s and Ceby se 8.1.4 Refinements of Young’s Integral Inequality via Jensen’s Discrete and Integral Inequalities.................................................................... 202 ¨ 8.1.5 Refinements of Young’s Integral Inequality via Holder’ s Integral Inequality .......................................................................................... 203

1 Dedicated

to people facing and fighting COVID-19.

193

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8.1.6 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Lagrange’s Type Remainder ............................... 205 8.1.7 Refinements of Young’s Integral Inequality via Taylor’s Mean ¨ Value Theorem of Cauchy’s Type Remainder and Holder’ s Integral Inequality............................................................................. 207 8.1.8 Refinements of Young’s Integral Inequality via Taylor’s Mean ˇ ˇ v’s Value Theorem of Cauchy’s Type Remainder and Ceby se Integral Inequality............................................................................. 209 8.1.9 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Cauchy’s Type Remainder and Jensen’s Inequalities........................................................................................ 210 8.1.10 Refinements of Young’s Integral Inequality via Taylor’s Mean Value Theorem of Cauchy’s Type Remainder and Integral Inequalities of Hermite-Hadamard Type for the Product of Two Convex Functions.............................................................................. 211 8.1.11 Three Examples Showing Refinements of Young’s Integral Inequality .......................................................................................... 213 8.1.11.1 First Example ..................................................................... 213 8.1.11.2 Second Example ................................................................. 214 8.1.11.3 Third Example.................................................................... 214 ´ 8.2 New Refinements of Young’s Integral Inequality via Polya’ s Type Integral Inequalities ....................................................................................... 214 8.2.1 Refinements of Young’s Integral Inequality in Terms of Bounds of the First Derivative ....................................................................... 214 8.2.2 Refinements of Young’s Integral Inequality in Terms of Bounds of the Second Derivative ................................................................... 216 8.2.3 Refinements of Young’s Integral Inequality in Terms of Bounds of Higher-Order Derivatives ............................................................. 217 8.2.4 Refinements of Young’s Integral Inequality in Terms of L p -Norms .......................................................................................... 218 8.2.5 Three Examples for New Refinements of Young’s Integral Inequalities....................................................................................... 221 8.2.5.1 First Example ..................................................................... 221 8.2.5.2 Second Example ................................................................. 222 8.2.5.3 Third Example.................................................................... 223 8.3 More Remarks ............................................................................................... 223 Acknowledgments.................................................................................................. 224 Bibliography .......................................................................................................... 224

8.1

YOUNG’S INTEGRAL INEQUALITY AND SEVERAL REFINEMENTS

In the first part of this paper, we mainly review several refinements of Young’s integral inequality via several mean value theorems, such as Lagrange’s and Taylor’s mean value theorems of Lagrange’s and Cauchy’s type remainders, and

195

Refinements of Young Integral Inequality

ˇ ˇ v’s integral inequality, Hermitevia several fundamental inequalities, such as Ceby se ¨ Hadamard’s type integral inequalities, Holder’ s integral inequality, and Jensen’s discrete and integral inequalities, in terms of higher-order derivatives and their norms, and simply survey several applications of several refinements of Young’s integral inequality. 8.1.1

YOUNG’S INTEGRAL INEQUALITY

One of the fundamental and general inequalities in mathematics is Young’s integral inequality below. Theorem 8.1 ([55]). Let h(x) be a real-valued, continuous, and strictly increasing function on [0, c] with c > 0. If h(0) = 0, a ∈ [0, c], and b ∈ [0, h(c)], then Z a

Z b

h(x) d x + 0

h−1 (x) d x ≥ ab,

(8.1)

0

where h−1 denotes the inverse function of h. The equality in (8.1) is valid if and only if b = h(a). Proof. This proof is adapted from the proof of [25, Section 2.7, Theorem 1]. Set Z a f (a) = ab − h(x) d x

(8.2)

0

and consider b > 0 as a parameter. Since one obtains   > 0, f 0 (a) = 0,   < 0,

f 0 (a) = b − h(a) and h is strictly increasing, 0 < a < h−1 (b); a = h−1 (b); a > h−1 (b).

This means that f (a) has a maximum of f at a = h−1 (b). Therefore, it follows that
f (a) ≤ max{ f (x)} = f h−1 (b) . (8.3) Integrating by parts gives Z
f h−1 (b) = bh−1 (b) −

h−1 (b)

Z h−1 (b)

h(x) d x =

0

xh0 (x) d x.

0

Substituting y = h(x) into the above integral yields
Z b −1 f h−1 (b) = h (y) d y.

(8.4)

0

Putting (8.2) and (8.4) into (8.3) results in (8.1). The proof of Theorem 8.1 is complete.

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Remark 8.1. The geometric interpretation of Young’s integral inequality (8.1) can be demonstrated by Figures 8.1 and 8.2. In Figure 8.1, we have Z a

A +C =

Z b

h(x) d x,

A + B = ab,

B=

0

h−1 (x) d x,

0

Z a

A + B +C =

Z b

h(x) d x + 0

h−1 (x) d x ≥ ab = A + B.

0

Therefore, the inequality (8.1) means that the area Z a

C=

h−1 (b)

  h(x) d x − b a − h−1 (b) ≥ 0.

In Figure 8.2, we have Z a

A=

Z b

h(x) d x,

A + B = ab,

B +C =

0

h−1 (x) d x,

0

Z a

A + B +C =

Z b

h(x) d x + 0

h−1 (x) d x ≥ ab = A + B.

0

Figure 8.1 Geometric interpretation of the inequality (8.1).

Figure 8.2 Geometric interpretation of the inequality (8.1).

(8.5)

197

Refinements of Young Integral Inequality

Therefore, the inequality (8.1) means that the area   Z C = b h−1 (b) − a −

h−1 (b)

h(x) d x ≥ 0.

(8.6)

a

Remark 8.2. We notice that two expressions (8.5) and (8.6) are of the same form Z a

C=

h−1 (b)

h(x) d x − ab + bh−1 (b) ≥ 0,

(8.7)

no matter which of a and h−1 (b) is smaller or bigger. Remark 8.3. When p > 1, taking h(x) = x p−1 in (8.1) derives 1 p 1 q a + b ≥ ab p q for a, b ≥ 0 and p, q > 1 satisfying 1p + 1q = 1. Further replacing a p and bq by x and y, respectively, leads to x y x1/p y1/q ≤ + (8.8) p q for x, y ≥ 0 and a, q > 1 satisfying 1p + q1 = 1. Perhaps this is why the weighted arithmetic-geometric inequality (8.8) is also called Young’s inequality in [13,14,21] and closely related references therein. Remark 8.4. The inequality n

cos(kθ ) > −1, k k=1

∑

n ≥ 2,

θ ∈ [0, π]

is also called Young’s inequality in [2,3] and closely related references therein. Remark 8.5. In [25, Secton 2.7] and [26, Chapter XIV], a plenty of refinements, extensions, generalizations, and applications of Young’s integral inequality (8.1) were collected, reviewed, and surveyed. For some new and recent development on this topic after the year 1990, please refer to the papers [4,43,47,51,56] and closely related references therein. 8.1.2

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA LAGRANGE’S MEAN VALUE THEOREM

In 2008, Hoorfar and Qi refined Young’s integral inequality (8.1) via Lagrange’s mean value theorem for derivatives. Theorem 8.2 ([18, Theorem 1]). Let h(x) be a differentiable and strictly increasing function on [0, c] for c > 0, and let h−1 be the inverse function of h. If h(0) = 0, a ∈ [0, c], b ∈ [0, h(c)], and h0 (x) is strictly monotonic on [0, c], then Z b 2 Z a 2 m M a − h−1 (b) ≤ h(x) d x + h−1 (x) d x − ab ≤ a − h−1 (b) , 2 2 0 0

(8.9)

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where 
m = min h0 (a), h0 h−1 (b) and 
M = max h0 (a), h0 h−1 (b) . The equalities in (8.9) are valid if and only if b = h(a). Proof. This is a modification of the proof of [18, Theorem 1] in [18, Section 2]. Changing the variable of integration by x = h(y) and integrating by parts yield Z a

Z b

h(x) d x + 0

h−1 (x) d x =

0

Z h−1 (b)

Z a

h(x) d x + 0

Z a

=

yh0 (y) d y

0

h(x) d x + bh−1 (b) −

Z h−1 (b)

0

= bh−1 (b) + Z a

= ab +

h(x) d x 0

Z a

h−1 (b)

h−1 (b)

h(x) d x

[h(x) − b] d x.

(8.10)

From the last line in (8.10), we can see that if h−1 (b) = a, then those equalities in (8.9) hold. If h−1 (b)
< a, since h(x) is strictly increasing, then h(x) − b > 0 for x ∈ −1 h (b), a . By Lagrange’s mean value theorem for derivatives, we can see that there exists ξ = ξ (x), satisfying h−1 (b) < ξ < x ≤ a, such that   0 < h(x) − b = x − h−1 (b) h0 (ξ ). By virtue of monotonicity of h0 (x) on [0, c], we reveal that 

0 < m = min h0 (a), h0 h−1 (b) < h0 (ξ ) < max h0 (a), h0 h−1 (b) = M. Consequently, we have     0 < m x − h−1 (b) < h(x) − b < M x − h−1 (b) . As a result, we have Z a

m

h−1 (b)

Z   −1 x − h (b) d x
a, we can derive inequalities in (8.11) by a similar argument as above. Substituting the double inequality (8.11) into the equality (8.10) leads to the double inequality (8.9). The proof of Theorem 8.2 is complete.

Refinements of Young Integral Inequality

199

Remark 8.6. The geometric interpretation of the double inequality (8.9) is that the areas C in Figures 8.3–8.6 satisfy 2 2 M m (8.12) a − h−1 (b) . a − h−1 (b) ≤ C ≤ 2 2 When h0 (x) is strictly increasing, the double inequality (8.12) can be equivalently written as Ra
  h0 h−1 (b)  h0 (a)  h−1 (b) h(x) d x −1 a − h (b) ≤ ≤ a − h−1 (b) 1 − 2 a − h (b) 2 and


R h−1 (b)   h(x) d x h0 h−1 (b)  −1 h0 (a)  −1 a h (b) − a ≤ ≤ h (b) − a − 1 2 h (b) − a 2 corresponding to Figures 8.3 and 8.4, respectively. Remark 8.7. If Q is a convex function on J, then   Z µ τ +µ 1 Q(τ) + Q(µ) Q ≤ Q(x) d x ≤ ; 2 µ −τ τ 2

Figure 8.3 Geometric interpretation of the double inequality (8.9).

Figure 8.4 Geometric interpretation of the double inequality (8.9).

(8.13)

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if Q is a concave function on J, then the double inequality (8.13) is reversed, where J ⊆ R is a nonempty interval and τ, µ ∈ J with τ < µ. The double inequality (8.13) is called Hermite-Hadamard’s integral inequality for convex functions [7,38,45]. When a > h−1 (b), as shown in Figures 8.3 and 8.5, and h0 (x) is strictly increasing, i.e., the function h(x) is convex, as shown in Figures 8.3 and 8.4, applying the double inequality (8.13) yields   Ra a + h−1 (b) h(a) + b h−1 (b) h(x) d x h ≤ ≤ . −1 2 a − h (b) 2 Substituting this into the third line in (8.10) gives Z a

Z b

h(x) d x + 0

h−1 (x) d x ≤ ab +

0

 h(a) − b  a − h−1 (b) 2

and Z a 0

   a + h−1 (b)  h(x) d x + h (x) d x ≥ bh (b) + h a − h−1 (b) 2 0       a + h−1 (b) = ab + h − b a − h−1 (b) . 2 Z b

−1

−1

Equivalently speaking, it follows that the area C satisfies        a + h−1 (b) h(a) − b  h − b a − h−1 (b) ≤ C ≤ a − h−1 (b) . 2 2 Similarly, we can discuss other cases, corresponding to Figures 8.5 and 8.6, that the derivative h0 (x) is strictly decreasing.

Figure 8.5 Geometric interpretation of the double inequality (8.9).

Refinements of Young Integral Inequality

201

Figure 8.6 Geometric interpretation of the double inequality (8.9).

Remark 8.8. Mercer has applied and employed the double inequality (8.9) in the paper [23] and in the Undergraduate Texts in Mathematics [24]. 8.1.3 REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA ˇ ˇ HERMITE-HADAMARD’S AND CEBY SEV’S INTEGRAL INEQUALITIES ˇ c´ and Pecari ˇ c´ refined Young’s integral In 2009 and 2010, among other things, Jakseti inequality (8.1) and Hoorfar–Qi’s double inequality (8.9) in [19,20]. Theorem 8.3 ([19, Theorem 2.1] and [20, Theorem 2.3]). Let h(x) be a differentiable and strictly increasing function on [0, c] for c > 0, h(0) = 0, a ∈ [0, c], b ∈ [0, h(c)], and h−1 be the inverse function of h. Denote   α = min a, h−1 (b) and β = max a, h−1 (b) . (8.14) 1. If h0 (x) is increasing on [α, β ] and b < h(a), or if h0 (x) is decreasing on [α, β ] and b > h(a), then     Za Z b   a + h−1 (b) a − h−1 (b) h −b ≤ h(x) d x + h−1 (x) d x − ab 2 0 0  1 ≤ a − h−1 (b) [h(a) − b]. 2 (8.15) 2. If h0 (x) is increasing on [α, β ] and b > h(a), or if h0 (x) is decreasing on [α, β ] and b < h(a), then the inequality (8.15) is reversed. 3. The equality in (8.15) is valid if and only if h(x) = λ x for λ > 0 or b = h(a). Proof. This is the outline of proofs of [19, Theorem 2.1] and [20, Theorem 2.3].

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From the third line in (8.10), it follows that Z a

Z b

h(x) d x + 0

  Z h−1 (x) d x − ab = b h−1 (b) − a +

a

h−1 (b)

0

h(x) d x.

(8.16)

Considering monotonicity of h0 (x) and applying the double inequality (8.13) to the integrand in the last term of (8.16), we can derive the double inequality (8.15). The last term in (8.10) can be rewritten as Z a h−1 (b)

[h(x) − b] d x =

Z a



h−1 (b)

Z a

=


 h(x) − h h−1 (b) d x

Z x

h−1 (b) h−1 (b)

h0 (u) d u d x =

Z a h−1 (b)

(a − u)h0 (u) d u. (8.17)

Let f , g : [µ, ν] → R be the integrable functions satisfying that they are both increasing or both decreasing. Then Z ν

Z ν

f (x) d x µ

g(x) d x ≤ (ν − µ)

Z ν

µ

f (x)g(x) d x.

(8.18)

µ

If one of the function f or g is nonincreasing and the other nondecreasing, then the ˇ ˇ v’s integral inequality in (8.18) is reversed. The inequality (8.18) is called Ceby se inequality in the literature [26, Chapter IX] and [35,39]. Applying (8.18) to the last term in (8.17) leads to the right-hand side of the inequality (8.15). The proof of Theorem 8.3 is complete. Remark 8.9. The double inequality (8.15) can be geometrically interpreted as        a + h−1 (b) 1 −1 a − h (b) h − b ≤ C ≤ a − h−1 (b) [h(a) − b], 2 2 where C denotes the area shown in Figures 8.1–8.6. 8.1.4

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA JENSEN’S DISCRETE AND INTEGRAL INEQUALITIES

In [20, Theorem 2.6], Jensen’s discrete and integral inequalities were employed to establish the following inequalities which refine Young’s integral inequality (8.1) and Hoorfar–Qi’s double inequality (8.9). Theorem 8.4 ([20, Theorem 2.6]). Let h(x) be a differentiable and strictly increasing function on [0, c] for c > 0, and let h−1 be the inverse function of h. If h(0) = 0, a ∈ [0, c], b ∈ [0, h(c)], and h0 (x) is convex on [α, β ], then  2   Za Z b a − h−1 (b) 0 a + 2h−1 (b) h ≤ h(x) d x + h−1 (x) d x − ab 2 3 0 0  2  
a − h−1 (b) h0 (a) ≤ + h0 h−1 (b) . (8.19) 3 2 If h0 (x) is concave, then the double inequality (8.19) is reversed.

Refinements of Young Integral Inequality

203

Proof. This is the outline of the proof of [20, Theorem 2.6]. Changing the variable of the last term in (8.17) results in Z a h−1 (b)

(a − u)h0 (u) d u =

Z 1

2
a − h−1 (b) (1 − x)h0 xa + (1 − x)h−1 (b) d x. (8.20)

0

If f is a convex function on an interval I ⊆ R and if n ≥ 2 and xk ∈ I for 1 ≤ k ≤ n, then ! n n 1 1 f pk xk ≤ n pk f (xk ), (8.21) ∑ ∑ n ∑k=1 pk k=1 ∑k=1 pk k=1 where pk > 0 for 1 ≤ k ≤ n. If f is concave, the inequality (8.21) is reversed. The inequality (8.21) is called Jensen’s discrete inequality for convex functions in the literature [25, Section 1.4] and [26, Chapter I]. Applying (8.21) to the third factor in the integrand of the right-hand side in (8.20) arrives at the right inequality in (8.19). Let φ be a convex function on [µ, ν], f ∈ L1 (µ, ν), and σ be a nonnegative measure. Then, ! Rν Rν µ f (x) d σ µ φ ( f (x)) d σ Rν Rν φ ≤ . (8.22) µ dσ µ dσ If φ is a concave function, then the inequality (8.22) is reversed. The inequality (8.22) is called Jensen’s integral inequality for convex functions in the literature [26, p. 10, (7.15)]. Applying (8.22) yields ! Ra Z a [1 − h−1 (b)]2 0 h−1 (b) (a − x)x d x 0 (a − x)h (x) d x ≥ h Ra 2 h−1 (b) h−1 (b) (a − x) d x   − 1 2 [1 − h (b)] 0 a + 2h−1 (b) = h . 2 3 The proof of Theorem 8.4 is complete. Remark 8.10. The double inequality (8.19) can be geometrically interpreted as  2   2   
a − h−1 (b) 0 a + 2h−1 (b) a − h−1 (b) h0 (a) 0 −1 h ≤C ≤ + h h (b) , 2 3 3 2 where C denotes the area shown in Figures 8.1–8.6. 8.1.5

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA ¨ LDER’S INTEGRAL INEQUALITY HO

¨ In [20, Theorem 2.1], Holder’ s integral inequality was utilized to present the following inequalities, which refine Young’s integral inequality (8.1) and Hoorfar–Qi’s double inequality (8.9), for the normed spaces.

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Theorem 8.5 ([20, Theorem 2.1]). Let h(x) be a differentiable and strictly increasing function on [0, c] for c > 0, and let h−1 be the inverse function of h. If h(0) = 0, a ∈ [0, c], b ∈ [0, h(c)], and h0 (x) is almost everywhere continuous with respect to Lebesgue measure on [α, β ], then the double inequality Cu kh0 kv ≤

Z a

Z b

h(x) d x + 0

0

h−1 (x) d x − ab ≤ Cp kh0 kq

(8.23)

is valid for all u, v and p, q satisfying 1. 2.

1 1 u + v = 1 for u, v ∈ (−∞, 0) ∪ (0, 1), or (u, v) = (1, −∞), or (u, v) = (−∞, 1); 1 1 p + q = 1 for 1 < p, q < ∞, or (p, q) = (+∞, 1), or (p, q) = (1, +∞);

where

" #  a−h−1 (b) r+1 1/r    6 0, ±∞; , r=  r+1 Cr =  a − h−1 (b) ,  r = +∞;    0, r = −∞

and kh0 kr =

 1/r Rβ 0   r dt  [h (t)] ,  α

r 6= 0, ±∞;

 sup{h0 (t),t ∈ [α, β ]}, r = +∞;    inf{h0 (t),t ∈ [α, β ]}, r = −∞.

Proof. This is the outline of the proof of [20, Theorem 2.1]. Let 1p + 1q = 1 with p > 0 and p 6= 1, let f and g be the real functions on [µ, ν], and let | f | p and |g|q be integrable on [µ, ν]. 1.

If p > 1, then Z ν

| f (x)g(x)| d x ≤

Z

µ

ν

p

| f (x)| d x

1/p Z

µ

ν

q

|g(x)| d x

1/q .

(8.24)

µ

The equality in (8.24) holds if and only if A| f (x)| p = B|g(x)|q almost everywhere for two constants A and B. 2. If 0 < p < 1, then the inequality (8.24) is reversed. ¨ The inequality (8.24) is called Holder’ s integral inequality in the lierature [26, Chapter V] and [44,48,49]. From (8.17), it follows that, 1.

by a property of definite integrals, we have Z a h−1 (b)

(a − u)h0 (u) d u =

Z α

|a − u|h0 (u) d u

β

Z ≤ h−1 (b) − a

α

β

h0 (u) d u = C∞ kh0 k1 ;

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Refinements of Young Integral Inequality

2. by a property of definite integrals, we have Z a h−1 (b)

3.

(a − u)h0 (u) d u =

Z α

|a − u|h0 (u) d u ≤ C1 kh0 k∞ ;

β

¨ by Holder’ s integral inequality (8.24), we have Z a

0

h−1 (b)

(a − u)h (u) d u =

Z α

|a − u|h0 (u) d u

β

≤

Z

α

β

1/q Z α 1/p |a − u|q d u [h0 (u)] p d u = Cq kh0 k p . β

The rest proofs are straightforward. The proofs of the double inequality (8.23) and Theorem 8.5 are thus complete. Remark 8.11. The double inequality (8.23) can be geometrically interpreted as Cu kh0 kv ≤ C ≤ C p kh0 kq , where C denotes the area shown in Figures 8.1–8.6. 8.1.6

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA TAYLOR’S MEAN VALUE THEOREM OF LAGRANGE’S TYPE REMAINDER

In [50, Theorem 3.1], making use of Taylor’s mean value theorem of Lagrange’s type remainder, Wang, Guo, and Qi refined the above inequalities of Young’s type via higher-order derivatives. Theorem 8.6 ([50, Theorem 3.1]). Let h(0) = 0 and h(x) be strictly increasing on [0, c] for c > 0, let h(n) (x) for n ≥ 0 be continuous on [0, c], let h(n+1) (x) be finite and strictly monotonic on (0, c), and let h−1 be the inverse function of h. For a ∈ [0, c] and b ∈ [0, h(c)], 1. if b < h(a), then n

(k)

∑h ≤

k=1 Z a

Z b

(8.25) h−1 (x) d x − ab    n+2
a − h−1 (b) k+1 a − h−1 (b) −1 h (b) + Mn (a, b) , (k + 1)! (n + 2)!

h(x) d x + 0 n

≤

   n+2
a − h−1 (b) k+1 a − h−1 (b) h (b) + mn (a, b) (k + 1)! (n + 2)! −1

∑ h(k) k=1

0

where 
mn (a, b) = min h(n+1) h−1 (b) , h(n+1) (a) and 
Mn (a, b) = max h(n+1) h−1 (b) , h(n+1) (a) ;

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2. if b > h(a), then a. when n = 2` for ` ≥ 0, the double inequality (8.25) is valid; b. when n = 2` + 1 for ` ≥ 0, we have    n+2 n
a − h−1 (b) k+1 a − h−1 (b) (k) −1 − Mn (a, b) ∑ h h (b) (k + 1)! (n + 2)! k=1 ≤

Z a

Z b

0 n

≤

h−1 (x) d x − ab    n+2
a − h−1 (b) k+1 a − h−1 (b) −1 h (b) − mn (a, b) ; (k + 1)! (n + 2)! (8.26)

h(x) d x +

∑ h(k) k=1

0

3. if, and only if, b = h(a), those equalities in (8.25) and (8.26) hold. Proof. This is the outline of the proof of [50, Theorem 3.1]. Let f (x) be a function having finite nth derivative f (n) (x) everywhere in an open interval (µ, ν) and assume that f (n−1) (x) is continuous on the closed interval [µ, ν]. Then, for a fixed point x0 ∈ [µ, ν] and every x ∈ [µ, ν] with x 6= x0 , there exists a point x1 interior to the interval joining x and x0 such that n−1

f (x) = f (x0 ) + ∑ k=1

f (k) (x0 ) f (n) (x1 ) (x − x0 )k + (x − x0 )n . k! n!

(8.27)

Formula (8.27) is called Taylor’s mean value theorem of Lagrange’s type remainder in the literature [6, p. 113, Theorem 5.19]. Applying (8.27) in the last term of (8.10) reveals Z a Z a 
 [h(x) − b] d x = h(x) − h h−1 (b) d x −1 −1 h (b) h (b)
n h(k) h−1 (b) Z a  k =∑ x − h−1 (b) d x k! h−1 (b) k=1 Z a  n+1 1 + h(n+1) (ξ ) x − h−1 (b) dx (n + 1)! h−1 (b)   n
a − h−1 (b) k+1 (k) −1 = ∑ h h (b) (k + 1)! k=1  n+1 Z a x − h−1 (b) (n+1) + h (ξ ) d x, (n + 1)! h−1 (b) where ξ is a point interior to the interval joining x and h−1 (b). The rest proofs are straightforward discussions on various cases of the factor h(n+1) (ξ ). The proof of Theorem 8.6 is complete. Remark 8.12. The double inequalities (8.25) and (8.26) can be geometrically interpreted as

Refinements of Young Integral Inequality n

(k)

∑h k=1

207

   n+2
a − h−1 (b) k+1 a − h−1 (b) + mn (a, b) ≤C h (b) (k + 1)! (n + 2)!    n+2 n
a − h−1 (b) k+1 a − h−1 (b) (k) −1 ≤ ∑ h h (b) + Mn (a, b) (k + 1)! (n + 2)! k=1 −1

and n

∑h k=1

(k)

   n+2
a − h−1 (b) k+1 a − h−1 (b) h (b) − Mn (a, b) ≤C (k + 1)! (n + 2)!    n+2 n
a − h−1 (b) k+1 a − h−1 (b) (k) −1 ≤ ∑ h h (b) − mn (a, b) , (k + 1)! (n + 2)! k=1 −1

where C denotes the area shown in Figures 8.1–8.6. 8.1.7

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA TAYLOR’S MEAN VALUE THEOREM OF CAUCHY’S TYPE REMAINDER AND ¨ LDER’S INTEGRAL INEQUALITY HO

In [50, Theorem 3.2], employing Taylor’s mean value theorem of Cauchy’s type ¨ remainder and Holder’ s integral inequality, Wang, Guo, and Qi refined the above inequalities of Young’s type via norms of higher-order derivatives. Theorem 8.7 ([50, Theorem 3.2]). Let n ≥ 0 and h(x) ∈ Cn+1 [0, c] such that h(0) = 0, h(n+1) (x) ≥ 0 on [α, β ], and h(x) is strictly increasing on [0, c] for c > 0, let h−1 be the inverse function of h, and let a ∈ [0, c] and b ∈ [0, h(c)]. Then 1. when b > h(a) and n = 2` for ` ≥ 0 or when b < h(a), we have Z a Z b

Cu,n

h(n+1) ≤ h(x) d x + h−1 (x) d x − ab v (n + 1)! 0 0   n
a − h−1 (b) k+1 − ∑ h(k) h−1 (b) (k + 1)! k=1

C p,n

h(n+1) ; ≤ q (n + 1)!

2. when b > h(a) and n = 2` + 1 for ` ≥ 0, we have −

Z a Z b

C p,n

h(n+1) ≤ h(x) d x + h−1 (x) d x − ab q (n + 1)! 0 0   n
a − h−1 (b) k+1 (k) −1 − ∑ h h (b) (k + 1)! k=1

(n+1) Cu,n

; ≤− h v (n + 1)!

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where α, β are defined as in (8.14), " r(n+1)+1 #1/r  −1 (b)  a−h   , r 6= 0, ±∞;  r(n+1)+1 Cr,n =  a − h−1 (b) n+1 ,  r = +∞;    0, r = −∞,  1/r R β  (n+1) r    (t) dt , r 6= 0, ±∞;  α h

(n+1)

h

=  (n+1) r  sup h (t),t ∈ [α, β ] , r = +∞;    (n+1)  inf h (t),t ∈ [α, β ] , r = −∞, and u, v, p, q satisfy 1. u < 1 and u 6= 0 with 1u + 1v = 1, or (u, v) = (−∞, 1), or (u, v) = (1, −∞); 2. 1 < p, q < ∞ with 1p + q1 = 1, or (p, q) = (+∞, 1), or (p, q) = (1, +∞). Proof. This is the outline of the proof of [50, Theorem 3.2]. If f (x) ∈ Cn+1 [µ, ν] and x0 ∈ [µ, ν], then n

f (x) =

∑ k=0

f (k) (x0 ) 1 (x − x0 )k + k! n!

Z x

(x − t )n f (n+1) (t) dt.

(8.28)

x0

Formula (8.28) is called Taylor’s mean value theorem of Cauchy’s type remainder in the literature [5, p. 279, Theorem 7.6] and [27, p. 6, 1.4.37]. Applying formula (8.28) to the integrand in the last term of (8.10) yields Z a h−1 (b)

[h(x) − b] d x =

Z a h−1 (b) n

=

 (k)

∑h k=1


 h(x) − h h−1 (b) d x  
a − h−1 (b) k+1 h (b) (k + 1)! −1

Z a

1 h−1 (b) n!

+

Z x h−1 (b)

(x − t )n h(n+1) (t) dt d x

 
a − h−1 (b) k+1 (k) −1 h h (b) ∑ (k + 1)! k=1 n

=

Z a

1 −1 n! h (b)

+ n

=

∑h k=1

+

(k)

Z a

(x − t)n h(n+1) (t) d x dt

t

 
a − h−1 (b) k+1 h (b) (k + 1)! −1

1 (n + 1)!

Z a h−1 (b)

(a − t )n+1 h(n+1) (t) dt

209

Refinements of Young Integral Inequality n

=

(k)

∑h k=1

( +

 
a − h−1 (b) k+1 h (b) (k + 1)! −1

Rβ 1 (n+1)! α (−1)n R β (n+1)! α

|a − t |n+1 h(n+1) (t) dt, b < h(a); |a − t |n+1 h(n+1) (t) dt, b > h(a).

¨ Discussing and making use of Holder’ s integral inequality (8.24) as in the proof of Theorem 8.5, we can complete the proof of Theorem 8.7. Remark 8.13. Two double inequalities in Theorem 8.7 can be geometrically interpreted as  k+1 −1 n


Cu,n

h(n+1) ≤ C − ∑ h(k) h−1 (b) a − h (b) v (n + 1)! (k + 1)! k=1

C p,n

h(n+1) ≤ q (n + 1)! and −

 k+1 −1 n


C p,n

h(n+1) ≤ C − ∑ h(k) h−1 (b) a − h (b) q (n + 1)! (k + 1)! k=1

Cu,n

h(n+1) , ≤− v (n + 1)!

where C denotes the area shown in Figures 8.1–8.6. 8.1.8

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA TAYLOR’S MEAN VALUE THEOREM OF CAUCHY’S TYPE REMAINDER AND ˇ ˇ CEBY SEV’S INTEGRAL INEQUALITY

Theorem 8.8 ([50, Theorem 3.3]). Let n ≥ 0 and h(x) ∈ Cn+1 [0, c] such that h(0) = 0 and h(x) is strictly increasing on [0, c] for c > 0, let h−1 be the inverse function of h, let a ∈ [0, c] and b ∈ [0, h(c)], and let ` ≥ 0 be an integer. Then 1. when a. either h(a) > b and h(n+1) (x) is increasing on [α, β ]; b. or h(a) < b, h(n+1) (x) is increasing on [α, β ], and n = 2` + 1; c. or h(a) < b, h(n+1) (x) is decreasing on [α, β ], and n = 2`; the inequality Z a

Z b

h(x) d x + 0

0

−1

n

h (x) d x − ab − ∑ h k=1

(k)

 
a − h−1 (b) k+1 h (b) (k + 1)! −1

 n+1  (n)
 a − h−1 (b) ≤ h (a) − h(n) h−1 (b) (n + 2)! is valid;

(8.29)

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Topics in Contemporary Mathematical Analysis and Applications

2. when a. either h(a) > b and h(n+1) (x) is decreasing on [α, β ]; b. or h(a) < b, h(n+1) (x) is increasing on [α, β ], and n = 2`; c. or h(a) < b, h(n+1) (x) is decreasing on [α, β ], and n = 2` + 1; the inequality (8.29) is reversed; where α, β are defined as in (8.14). Proof. This is the outline of the proof of [50, Theorem 3.3]. This follows from applying formula (8.28) as in the proof of Theorem 8.7 and ˇ ˇ v’s integral inequality (8.18) to the integral applying Ceby se Z a h−1 (b)

(a − t )n+1 h(n+1) (t) dt

(8.30)

in the proof of Theorem 8.7. The proof of Theorem 8.8 is complete. Remark 8.14. The inequality (8.29) can be geometrically interpreted as n

(k)

C− ∑ h k=1

 
a − h−1 (b) k+1 h (b) (k + 1)! −1

 n+1  (n)
 a − h−1 (b) ≤ h (a) − h(n) h−1 (b) , (n + 2)! where C denotes the area shown in Figures 8.1–8.6. 8.1.9

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA TAYLOR’S MEAN VALUE THEOREM OF CAUCHY’S TYPE REMAINDER AND JENSEN’S INEQUALITIES

Theorem 8.9 ([50, Theorem 3.4]). Let h(x) ∈ Cn+1 [0, c] such that h(0) = 0 and h(x) is strictly increasing on [0, c] for c > 0, let h−1 be the inverse function of h, and let a ∈ [0, c] and b ∈ [0, h(c)]. If h(n+1) (x) is convex on [α, β ], where α, β are defined as in (8.14), then 1. when h(a) > b or when h(a) < b and n = 2`, we have  n+2   −1 a − h−1 (b) (n+1) a + (n + 2)h (b) h n+2 n+3  
a − h−1 (b) k+1 ≤ h(x) d x + h (x) d x − ab − ∑ h h (b) (k + 1)! 0 0 k=1
(n+1) (n+1) −1  n+2 h (a) + (n + 2)h h (b) ≤ a − h−1 (b) ; (8.31) (n + 3)! Z a

Z b

−1

n

(k)

−1

Refinements of Young Integral Inequality

211

2. when h(a) < b and n = 2` + 1, the double inequality (8.31) is reversed, where ` ≥ 0 is an integer. If h(n+1) (x) is concave on [α, β ], all the above inequalities are reversed for all corresponding cases. Proof. This is the outline of the proof of [50, Theorem 3.4]. Considering the integral (8.30) and substituting integral variables give Z a h−1 (b) Z 1

 n+2 (a − t )n+1 h(n+1) (t) dt = a − h−1 (b)


(1 − s)n+1 h(n+1) sa + (1 − s)h−1 (b) d s.

×

0


Applying Jensen’s inequalities (8.21) and (8.22) to h(n+1) sa + (1 − s)h−1 (b) in the above equation yield the double inequality (8.31) and its reversed version. The proof of Theorem 8.9 is complete. Remark 8.15. The double inequality (8.31) can be geometrically interpreted as  n+2   −1 a − h−1 (b) (n+1) a + (n + 2)h (b) h n+2 n+3   n
a − h−1 (b) k+1 (k) −1 ≤ C − ∑ h h (b) (k + 1)! k=1
 n+2 h(n+1) (a) + (n + 2)h(n+1) h−1 (b) −1 ≤ a − h (b) , (n + 3)! where C denotes the area shown in Figures 8.1–8.6. 8.1.10

REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY VIA TAYLOR’S MEAN VALUE THEOREM OF CAUCHY’S TYPE REMAINDER AND INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR THE PRODUCT OF TWO CONVEX FUNCTIONS

Theorem 8.10 ([50, Theorem 3.5]). Let n ≥ 0 and h(x) ∈ Cn+1 [0, c] such that h(0) = 0 and h(x) is strictly increasing on [0, c] for c > 0, let h−1 be the inverse function of h, let a ∈ [0, c] and b ∈ [0, h(c)], and let h(n+1) (x) be nonnegative and convex on [α, β ], where α, β are defined as in (8.14). If h(a) > b, then
  n+2    a − h−1 (b) 2h(n+1) (a) + h(n+1) h−1 (b) 1 (n+1) a + h−1 (b) h − (n + 1)! 2n 2 6   Z a Z b n
a − h−1 (b) k+1 −1 (k) −1 ≤ h(x) d x + h (x) d x − ab − ∑ h h (b) (k + 1)! 0 0 k=1  
n+2 (n+1) a − h−1 (b) h (a) + 2h(n+1) h−1 (b) ≤ . (8.32) (n + 1)! 6

212

Topics in Contemporary Mathematical Analysis and Applications

If h(a) < b and n = 2` for ` ≥ 0, then  −1 n+2 
   h (b) − a 2h(n+1) (a) + h(n+1) h−1 (b) 1 (n+1) a + h−1 (b) h − (n + 1)! 2n 2 6   Z a Z b n
a − h−1 (b) k+1 −1 (k) −1 ≤ h(x) d x + h (x) d x − ab − ∑ h h (b) (k + 1)! 0 0 k=1  −1 n+2 (n+1)
h (b) − a h (a) + 2h(n+1) h−1 (b) ≤ . (8.33) (n + 1)! 6 If a < h−1 (b) and n = 2` + 1 for ` ≥ 0, the double inequality (8.33) is reversed. Proof. This is the outline of the proof of [50, Theorem 3.5]. Let f (x) and g(x) be nonnegative and convex functions on [µ, ν]. Then,  2f

   µ +ν µ +ν 1 1 g − M(µ, ν) − N(µ, ν) 2 2 6 3 Z ν 1 1 1 ≤ f (x)g(x) d x ≤ M(µ, ν) + N(µ, ν), (8.34) ν −µ µ 3 6

where M(µ, ν) = f (µ)g(µ) + f (ν)g(ν) and N(µ, ν) = f (µ)g(ν) + f (ν)g(µ). The double inequality (8.34) can be found in [28,52–54] and closely related references therein. Applying (8.34) in the integral (8.30) arrives at the double inequalities in (8.32) and (8.33). The proof of Theorem 8.10 is complete. Remark 8.16. The double inequalities (8.32) and (8.33) can be geometrically interpreted as  n+2 
   a − h−1 (b) 2h(n+1) (a) + h(n+1) h−1 (b) 1 (n+1) a + h−1 (b) h − (n + 1)! 2n 2 6  k+1  n+2 (n+1)
− 1 − 1 n
a − h (b) a − h (b) h (a) + 2h(n+1) h−1 (b) ≤ C − ∑ h(k) h−1 (b) ≤ (k + 1)! (n + 1)! 6 k=1

and  −1 n+2 
   h (b) − a 2h(n+1) (a) + h(n+1) h−1 (b) 1 (n+1) a + h−1 (b) h − (n + 1)! 2n 2 6  k+1  −1 n+2 (n+1)
− 1 n
a − h (b) h (b) − a h (a) + 2h(n+1) h−1 (b) (k) −1 ≤ C − ∑ h h (b) ≤ , (k + 1)! (n + 1)! 6 k=1

where C denotes the area shown in Figures 8.1–8.6.

213

Refinements of Young Integral Inequality

8.1.11 THREE EXAMPLES SHOWING REFINEMENTS OF YOUNG’S INTEGRAL INEQUALITY 8.1.11.1

First Example

In [18, Section 3], the double inequality (8.9) was applied to obtain the estimate √ √ 4 4 125 4
2 9.000042866 . . . = 3−2 5 27 Z 3p Z 3p 4 4 4 4 < x +1 dx+ x −1 dx−9 0

1

√ 27 4
2 < √ 3−2 5 4 2 823 = 9.000042871 . . .

whose gap between the upper and lower bounds is 0.000000005 . . . and which refines a known result Z 3p Z 3p 4 4 4 4 9< x +1 dx+ x − 1 d x < 9.0001 0

1

In [20, Example 2.5] and [20, Remark 2.7], it was obtained that 9.000042866