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)

Dyadic (Walsh) analysis emerged as a new research area in applied mathematics and engineering in early seventies within attempts to provide answers to demands from practice related to application of spectral analysis of different classes of signals, including audio, video, sonar, and radar signals. In the meantime, it evolved in a mature mathematical discipline with fundamental results and important features providing basis for various applications. The book will provide fundamentals of the area through reprinting carefully selected earlier publications followed by overview of recent results concerning particular subjects in the area written by experts, most of them being founders of the field, and some of their followers. In this way, this first volume of the two volume book offers a rather complete coverage of the development of dyadic Walsh analysis, and provides a deep insight into its mathematical foundations necessary for consideration of generalizations and applications that are the subject of the second volume. The presented theory is quite sufficient to be a basis for further research in the subject area as well as to be applied in solving certain new problems or improving existing solutions for tasks in the areas which motivated development of the dyadic analysis. 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