Construction of wavelets through Walsh functions
type: Item
instance of: scholarly article
title: Construction of wavelets through Walsh functions
zbMATH Open document ID: 1418.42002
DOI: 10.1007/978-981-13-6370-2
author: A. H. Siddiqi
author: Yuri A. Farkov
author: P. Manchanda
published in: Industrial and Applied Mathematics
publication date: 2019-02-15T00:00:00Z
review text: This book focuses on the fertile interplay between Walsh functions and wavelet analysis. It contains nine chapters and an appendix. Chapter 1 discusses the basic results of Walsh analysis and wavelets on the real line. Chapter 2 deals with the growth and decay of Walsh-Fourier coefficients for classes of functions, convergence and summability of Walsh-Fourier series and the rate of approximation for different classes of functions and different means of Walsh-Fourier series, and applications to signal and image processing. Chapter 3 presents the Haar system and its generalization the Haar-Fourier series, the relation between the Haar and Walsh functions, nonuniform Haar wavelets, generalized Haar wavelets and frames, as well as applications of Haar wavelets to the solution of initial value problems, boundary value problems and integral equations. Chapter 4 deals with the construction of dyadic wavelets and frames through Walsh functions. The first section covers basic results needed in subsequent chapters, the second section explains orthogonal wavelets and multiresolution analysis on the half line, the third section covers orthogonal wavelets with compact support on the half line, the fourth section presents estimates for the smoothness of the scaling functions, and the fifth and last section presents approximation properties of dyadic wavelets. Chapter 5 is devoted to orthogonal and periodic wavelets on Vilenkin groups. The first section deals with multiresolution analysis on Vilenkin groups, section 2 introduces compactly supported orthogonal $p$-wavelets, section 3 presents periodic wavelets on Vilenkin groups, and section 4 introduces periodic wavelets related to the Vilenkin-Christenson transform; finally, section 5 discusses applications to the coding of fractal functions. Chapter 6 studies Haar-Vilenkin wavelets and their properties. Its five sections are titled as follows: Introduction, Haar-Vilenkin wavelets, Approximation by Haar-Vilenkin wavelets, Converg…
reviewed by: Richard A. Zalik
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 7024146
zbMATH Keywords: Walsh functions
zbMATH Keywords: Walsh-Fourier series
zbMATH Keywords: rate of approximation
zbMATH Keywords: Haar-Fourier series
zbMATH Keywords: nonuniform Haar wavelets
zbMATH Keywords: generalized Haar wavelets and frames
zbMATH Keywords: dyadic wavelets and frames
zbMATH Keywords: orthogonal and periodic wavelets on Vilenkin groups
zbMATH Keywords: Vilenkin-Christenson transform
zbMATH Keywords: approximation by Haar-Vilenkin wavelets
zbMATH Keywords: nonuniform multiresolution analysis
zbMATH Keywords: vector-valued nonuniform wavelet packets and bases
zbMATH Keywords: orthogonal wavelets and multiresolution analysis on the half line
zbMATH Keywords: orthogonal vector-valued wavelets on the half line
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-981-13-6370-2
OpenAlex ID: W2943556297
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description: scientific article; zbMATH DE number 7024146