This book discusses, develops and applies the theory of Vilenkin-Fourier series connected to modern harmonic analysis.
The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, code theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin-Fourier series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group.
The first part of the book can be used as an introduction to the subject, and the following chapters summarize the most recent research in this fascinating area and can be read independently. Each chapter concludes with historical remarks and open questions. The book will appeal to researchers working in Fourier and more broad harmonic analysis and will inspire them for their own and their students' research. Moreover, researchers in applied fields will appreciate it as a sourcebook far beyond the traditional mathematical domains.
Author(s): Lars-Erik Persson, George Tephnadze, Ferenc Weisz
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 632
City: Cham
Preface
How to Read the Book?
Acknowledgements
Contents
1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces
1.1 Introduction
1.2 Vilenkin Groups and Functions
1.3 The Representation of the Vilenkin Groups on the Interval [0,1)
1.4 Convex Functions and Classical Inequalities
1.5 Lebesgue Spaces
1.6 Dirichlet Kernels
1.7 Lebesgue Constants
1.8 Vilenkin-Fourier Coefficients
1.9 Partial Sums
1.10 Final Comments and Open Questions
2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-Fourier Series
2.1 Introduction
2.2 Conditional Expectation Operators
2.3 Martingales and Maximal Functions
2.4 Calderon-Zygmund Decomposition
2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series
2.6 Almost Everywhere Divergence of Vilenkin-Fourier Series
2.7 Final Comments and Open Questions
3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces
3.1 Introduction
3.2 Vilenkin-Fejér Kernels
3.3 Approximation of Vilenkin-Fejér Means
3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means
3.5 Approximate Identity
3.6 Final Comments and Open Questions
4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces
4.1 Introduction
4.2 Well-Known and New Examples of Nörlund and TMeans
4.3 Regularity of Nörlund and T Means
4.4 Kernels of Nörlund Means
4.5 Kernels of T Means
4.6 Norm Convergence of Nörlund and T Means in Lebesgue Spaces
4.7 Almost Everywhere Convergence of Nörlund and T Means
4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points
4.9 Riesz and Nörlund Logarithmic Kernels and Means
4.10 Final Comments and Open Questions
5 Theory of Martingale Hardy Spaces
5.1 Introduction
5.2 Martingale Hardy Spaces and Modulus of Continuity
5.3 Atomic Decomposition of the Martingale Hardy Spaces Hp
5.4 Interpolation Between Hardy Spaces Hp
5.5 Bounded Operators on Hp Spaces
5.6 Examples of p-Atoms and Hp Martingales
5.7 Final Comments and Open Questions
6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces
6.1 Introduction
6.2 Estimations of Vilenkin-Fourier Coefficients in Hp Spaces
6.3 Hardy and Paley Type Inequalities in Hp Spaces
6.4 Maximal Operators of Partial Sums on Hp Spaces
6.5 Convergence of Partial Sums in Hp Spaces
6.6 Convergence of Subsequences of Partial Sums in Hp Spaces
6.7 Strong Convergence of Partial Sums in Hp Spaces
6.8 Final Comments and Open Questions
7 Vilenkin-Fejér Means in Martingale Hardy Spaces
7.1 Introduction
7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces
7.3 Convergence of Vilenkin-Fejér Means in Hp Spaces
7.4 Convergence of Subsequences of Vilenkin-Fejér Means in Hp Spaces
7.5 Strong Convergence of Vilenkin-Fejér Means in Hp Spaces
7.6 Final Comments and Open Questions
8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces
8.1 Introduction
8.2 Maximal Operators of Nörlund Means on Hp Spaces
8.3 Maximal Operators of T Means on Hp Spaces
8.4 Strong Convergence of Nörlund Means in Hp Spaces
8.5 Strong Convergence of T Means in Hp Spaces
8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces
8.7 Strong Convergence of Riesz and Nörlund Logarithmic Means in Hp Spaces
8.8 Final Comments and Open Questions
9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces
9.1 Introduction
9.2 Variable Lebesgue Spaces
9.3 Doob's Inequality in Variable Lebesgue Spaces
9.4 The Maximal Operator Us
9.5 The Maximal Operator Vα,s
9.6 Variable Martingale Hardy Spaces
9.7 Atomic Decomposition of Variable Hardy Spaces
9.8 Martingale Inequalities in Variable Spaces
9.9 Partial Sums of Vilenkin-Fourier Series in Variable Lebesgue Spaces
9.10 The Maximal Fejér Operator on Hp(·)
9.11 Final Comments and Open Questions
10 Appendix: Dyadic Group and Walsh and Kaczmarz Systems
10.1 Introduction
10.2 Walsh Group and Walsh and Kaczmarz Systems
10.3 Estimates of the Walsh-Fejér Kernels
10.4 Walsh-Fejér Means in Hp
10.5 Modulus of Continuity in Hp and Walsh-Fejér Means
10.6 Riesz and Nörlund Logarithmic Means in Hp
10.7 Maximal Operators of Kaczmarz-Fejér Means on Hp
10.8 Modulus of Continuity in Hp and Kaczmarz-Fejér Means
10.9 Final Comments and Open Questions
References
Notations
Index
