Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1 Foundations (Atlantis Studies in Mathematics for Engineering and Science, 12) [1st ed. 2015] 9462391599, 9789462391598
Dyadic (Walsh) analysis emerged as a new research area in applied mathematics and engineering in early seventies within
205 19 8MB
English Pages 480 [474] Year 2016
Table of contents :
Preface
Organization of the Book
Reprinted Papers
List of Reprinted Papers
List of Translated Papers
Contents
List of Contributors
1 Early History of Walsh Analysis
1.1 History of the American School of Walsh Analysis
2 My involvement in Walsh and Dyadic Analysis
References
3 The Origins of the Dyadic Derivative due to James Edmund Gibbs
3.1 Definition of the Gibbs Derivative
3.2 Further Development
References
4 Early Contributions from the Aachen School to Dyadic Walsh Analysis with Applications to Dyadic PDEs and Approximation Theory
4.1 Butzer-Wagner Dyadic Derivative
4.2 Further Developments
4.3 A Dyadic PDE, the Wave Equation
4.4 Applications to Approximation by Walsh Polynomials and Walsh-Fourier Series
4.5 Later Contributions from Aachen
4.6 Decimal and Dyadic Systems
4.7 Further Suggestions for Studying Our Monograph
Acknowledgment
References
5 Dyadic Derivative, Summation, Approximation *
5.1 Introduction
5.2 Dyadic Derivative
5.2.1 Dyadic derivative of functions of one variable
5.2.2 Dyadic derivative in the multivariable case
5.2.3 Dyadic derivative on the real line
5.2.4 Dyadic derivative on groups
5.3 Summation, Strong summation
5.3.1 (C,1) summability of WalshFourier series
5.3.2 Strong summability
5.3.3 Summability of multivariable WalshFourier series
5.3.4 Dyadic Ces.aro and Copson operators
5.3.5 Multipliers
References
6 How I Started My Research in Walsh and Dyadic Analysis
References
7 My Involvement with the Dyadic Derivative
References
8 Hardy Spaces in the Theory of Dyadic Derivative
8.1 Introduction
8.2 One-dimensional Hardy Spaces
8.3 The One-dimensional Dyadic Derivative
8.4 d-dimensional Hardy Spaces
8.4.1 The Hardy Spaces H p [0,1)d
8.4.2 The Hardy Spaces Hp[0,1)d
8.5 More-dimensional dyadic derivative
8.5.1 Restricted dyadic derivative
8.5.2 Unrestricted dyadic derivative
References
9 Term by Term Dyadic Differentiation of Walsh Series
9.1 Introduction
9.2 The original problem
9.3 Gap series
9.4 Rapidly converging series
9.5 Term-by-term strong dyadic differentiation
References
10 Why I got Interested in Dyadic Differentiation
11 Dyadic Derivative and Walsh-Fourier Transform
11.1 Introduction
11.2 Notations and Definitions
11.3 Lemmas
11.4 Dyadic Differentiation and Integration of Walsh-Fourier Transform
References
12 How I started my research in Walsh and dyadic analysis
References
Index
Preface
Organization of the Book
Reprinted Papers
List of Reprinted Papers
List of Translated Papers
Contents
List of Contributors
1 Early History of Walsh Analysis
1.1 History of the American School of Walsh Analysis
2 My involvement in Walsh and Dyadic Analysis
References
3 The Origins of the Dyadic Derivative due to James Edmund Gibbs
3.1 Definition of the Gibbs Derivative
3.2 Further Development
References
4 Early Contributions from the Aachen School to Dyadic Walsh Analysis with Applications to Dyadic PDEs and Approximation Theory
4.1 Butzer-Wagner Dyadic Derivative
4.2 Further Developments
4.3 A Dyadic PDE, the Wave Equation
4.4 Applications to Approximation by Walsh Polynomials and Walsh-Fourier Series
4.5 Later Contributions from Aachen
4.6 Decimal and Dyadic Systems
4.7 Further Suggestions for Studying Our Monograph
Acknowledgment
References
5 Dyadic Derivative, Summation, Approximation *
5.1 Introduction
5.2 Dyadic Derivative
5.2.1 Dyadic derivative of functions of one variable
5.2.2 Dyadic derivative in the multivariable case
5.2.3 Dyadic derivative on the real line
5.2.4 Dyadic derivative on groups
5.3 Summation, Strong summation
5.3.1 (C,1) summability of WalshFourier series
5.3.2 Strong summability
5.3.3 Summability of multivariable WalshFourier series
5.3.4 Dyadic Ces.aro and Copson operators
5.3.5 Multipliers
References
6 How I Started My Research in Walsh and Dyadic Analysis
References
7 My Involvement with the Dyadic Derivative
References
8 Hardy Spaces in the Theory of Dyadic Derivative
8.1 Introduction
8.2 One-dimensional Hardy Spaces
8.3 The One-dimensional Dyadic Derivative
8.4 d-dimensional Hardy Spaces
8.4.1 The Hardy Spaces H p [0,1)d
8.4.2 The Hardy Spaces Hp[0,1)d
8.5 More-dimensional dyadic derivative
8.5.1 Restricted dyadic derivative
8.5.2 Unrestricted dyadic derivative
References
9 Term by Term Dyadic Differentiation of Walsh Series
9.1 Introduction
9.2 The original problem
9.3 Gap series
9.4 Rapidly converging series
9.5 Term-by-term strong dyadic differentiation
References
10 Why I got Interested in Dyadic Differentiation
11 Dyadic Derivative and Walsh-Fourier Transform
11.1 Introduction
11.2 Notations and Definitions
11.3 Lemmas
11.4 Dyadic Differentiation and Integration of Walsh-Fourier Transform
References
12 How I started my research in Walsh and dyadic analysis
References
Index
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