Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 2 Extensions and Generalizations: A Monograph ... in Mathematics for Engineering and Science) 9462391629, 9789462391628

Table of contents :
Preface
Organization of the Book
Contents
List of Contributors
1 Early Work in Gibbs Differentiation in China
1.1 Research in Butzer-Gibbs Calculus in China- A glance of p-type calculus
1.2 The Generalized Logical Derivative and Its Applications - Ren Fuxian, Su Weiyi, Zheng Weixing
1.2.1 Introduction
1.2.2 The p-adic Logical Derivatives
1.2.3 Bernstein Type Inequality and Bernstein Type Theorem
References
1.3 The Generalized Walsh Transform and An Extremum Problem - Zheng Weixing
1.3.1 Introduction
1.3.2 Properties ofWN-type functions
1.3.3 The Descent Order ofWN-type Functions
1.3.4 p-adic Logic Derivatives
1.3.5 Logic derivative
1.3.6 Adjoint logic derivative
1.3.7 A Class of Extremum Problems
References
1.4 The Logical Derivatives and Integrals
1.4.1 Introduction
1.4.2 Notation and important results
1.4.3 p-adic derivatives and integrals of functions in LqG, 1 . q . 2
References
1.5 The Logical Derivatives and Integrals Second Part
2 Gibbs-Butzer Calculus and Pseudo-differential Operators on Local Fields
2.1 Introduction
2.2 Notations
2.3 G-B type derivative and G-B type integrals
2.4 Function Spaces on Local Fields
2.5 A Comparison of Newton calculus with G-B calculus
2.6 Applications of Gibbs-Butzer-type Calculus
2.6.1 Application to Approximation Theory on Local Fields
2.6.2 Application to determine function space for which a function belongs to it
2.6.3 Application to partial differential equations with fractal boundaries
2.6.4 Application to medical science
References
3 Early Work in Differentiation in Soviet Union - SSSR
3.1 Interest in Dyadic Differentiation in Soviet Union
4 Research in Dyadic Differentiation in Russia
References
5 Generalized Derivatives and Integrals on Vilenkin Groups
5.1 A. The case of compact groups
5.2 B. The case of locally compact groups
References
6 Calculus on Walsh and Vilenkin Groups *
6.1 Introduction
6.2 Results related to Vilenkin systems
6.3 Results related to representative product systems
6.4 Results related to Walsh-Kaczmarz systems
6.5 Results related to two-dimensional Walsh-Paley systems
References
7 Gibbs Derivatives on Groups
7.1 Introduction
7.2 Basic Concepts
7.3 Gibbs Derivatives on Finite Non-Abelian Groups
7.4 Summary and Recent Work
References
8 My Research in Gibbs Derivatives on Finite Groups
References
9 Efficient Computation of Gibbs Derivatives on Finite Abelian Groups
9.1 Introduction
9.2 Definitions of Gibbs Derivatives on Finite Abelian Groups
9.3 Generalizations
9.4 Fast Fourier Transform on Finite Abelian Groups
9.5 Gibbs Derivatives on Finite Abelian Groups
9.6 Computing the Gibbs Derivatives on Finite Abelian Groups
9.6.1 Convolution-like algorithm
9.6.2 Computing in terms of partial Gibbs derivatives
9.7 GPGPU Computing Platform
9.8 Mapping the Algorithms to GPGPU Architecture
9.9 Experimental Results
References
10 Gibbs Derivative and Walsh Harmonizable DSP
10.1 Introduction
10.2 Q-loop and dyadic time
10.3 Gibbs derivative and Walsh functions
10.3.1 Walsh functions
10.3.2 Gibbs derivative
10.4 Dyadic system and dyadic flow
10.5 Dyadic stationary process
10.5.1 DSP in strict sense
10.5.2 DSP in wide sense and its W-harmonizability
10.5.3 Walsh harmonizability of DSP
10.5.4 DSP and LDP
10.5.5 Walsh Series and Approximation of DSP
10.6 Gibbs Differentiation of DSP
10.6.1 Gibbs Differentiability Conditions
10.6.2 A Linear Gibbs Differential Equation
10.7 Extensions and Comments
10.7.1 Chrestenson functions
10.7.2 A p-adic SP and C-harmonaizability
10.7.3 W-Harmonizable and C-Harmonizable DSPs
10.7.4 Open Problems
References
11 My Involvement in Gibbs Derivatives andWalsh Harmonizable Processes
11.0.1 Introduction
11.0.2 Walsh Harmonizable DSP
11.0.3 DSP and LDP
11.0.4 Walsh Series and Approximation of DSP
11.0.5 Gibbs Differentiability of DSP
11.0.6 A Linear Gibbs Differential Equation
11.0.7 Extensions and Comments
References
12 Open Problems in Theory and Applications of Dyadic Derivatives
References
References
References
12.1 Hardy Spaces
12.2 The one-dimensional dyadic derivative
References
List of Publications by the Authors of the Book
Index
Biographies of Authors