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Construction of Wavelets Through Walsh Functions

Authors: Prof. Yu. A. Farkov, Prof. Pammy Manchanda, Prof. Abul Hasan Siddiqi

Publisher: Springer Singapore

Book Series : Industrial and Applied Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book focuses on the fusion of wavelets and Walsh analysis, which involves non-trigonometric function series (or Walsh–Fourier series). The primary objective of the book is to systematically present the basic properties of non-trigonometric orthonormal systems such as the Haar system, Haar–Vilenkin system, Walsh system, wavelet system and frame system, as well as updated results on the book’s main theme.
Based on lectures that the authors presented at several international conferences, the notions and concepts introduced in this interdisciplinary book can be applied to any situation where wavelets and their variants are used. Most of the applications of wavelet analysis and Walsh analysis can be tried for newly constructed wavelets. Given its breadth of coverage, the book offers a valuable resource for theoreticians and those applying mathematics in diverse areas. It is especially intended for graduate students of mathematics and engineering and researchers interested in applied analysis.

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Table of Contents

Frontmatter

Chapter 1. Introduction to Walsh Analysis and Wavelets

Abstract

The trigonometric Fourier series has played a very significant role in solving problems of science and technology. The concept of non-trigonometric Fourier series such as Haar–Fourier series and Walsh–Fourier series were introduced by Haar [1] and Walsh [2], respectively; Kaczmarz, Steinhaus, and Paley studied some aspects of Walsh system between 1929 and 1931.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 2. Walsh–Fourier Series

Abstract

In this chapter, we study the growth and decay of Walsh–Fourier coefficients for classes of functions, conditions under which Walsh–Fourier series converges pointwise and converges absolutely, summability of Walsh–Fourier series and degree (order of approximation) for different classes of functions and different means of Walsh–Fourier series.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 3. Haar–Fourier Analysis

Abstract

It is well known that the history of Walsh series began with Haar’s(Hungarian Mathematician Alfred Haar) dissertation of 1909 Zur Theories Orthogonal Function system in which Haar system was introduced. Supervisor of Haar, David Hilbert at Göttingen university asked him to find an orthonormal system on the interval whose Fourier series of continuous functions converges uniformly. He constructed the system which is under discussion in this chapter, now known as Haar system provided answer to the problem posed by Hilbert.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 4. Construction of Dyadic Wavelets and Frames Through Walsh Functions

Abstract

Walsh system of functions \(\{w_{l}: l \in \mathbb {Z}_{+}\}\) on half line \(\mathbb {R}_{+}=[0,\infty )\) is determined by the equations.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 5. Orthogonal and Periodic Wavelets on Vilenkin Groups

Abstract

As noted in Chap. 1, the Walsh function can be identified with characters of the Cantor dyadic group. This fact was first recognized by Gelfand in the 1940s, who offered to Vilenkin study series with respect to characters of a large class of abelian groups which includes the Cantor group as special case see Vilenkin [1], Fine [2], Agaev, Vilenkin, Dzhafarli, Rubinshtein [3]. For wavelets on Vilenkin groups most of the results relate to the locally compact group \(G_{p}\), which is defined by a fixed integer \(p\ge 2.\) The group \(G_{p}\) has a standard interpretation on \(\mathbb {R_{+}}\). Since the case \(p=2\) corresponds to the Cantor group \(\mathscr {C}\), all the results on wavelets on \(\mathbb {R_{+}}\) presented in Chap. 4 can be rewritten for wavelets on \(\mathscr {C}\). In this section, necessary and sufficient conditions are given for refinable functions to generate an MRA in the space \(L^{2}(G_{p})\). The partition of unity property, the linear independence, the stability, and the orthogonality of “integer shifts” of refinable functions in \(L^{2}(G_{p})\) are also considered.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 6. Haar–Vilenkin Wavelet

Abstract

It can be extended to \(\mathbb {R}\) by the periodicity of period 1. Each Haar function is continuous from the right and the Haar system H is orthonormal on [0, 1).

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 7. Construction Biorthogonal Wavelets and Frames

Abstract

In this chapter, basic properties of biorthogonal wavelets on positive real lines and Vilenkin groups, frames on Cantor group, Parseval frames on Vilenkin group, and application of biorthogonal dyadic wavelets to image processing are presented and these results are discussed in more detail Farkov (Facta Univers (Nis) ser. Elec Eng 21: 309–325, 2008), Farkov, Maksimov, and Stroganov (Int. J. Wavelets Multiresolution Inf Process 9: 485–499, 2011), Farkov (J Math Sci 187: 22–34, 2012).

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 8. Wavelets Associated with Nonuniform Multiresolution Analysis on Positive Half Line

Abstract

Gabardo and Nashed considered a generalization of the notion of multiresolution analysis, which is called nonuniform multiresolution analysis (NUMRA) and is based on the theory of spectral pairs. In this set up, the associated subspace \(V_{0}\) of \(L^{2}(\mathbb {R})\) has, as an orthonormal basis, a collection of translates of the scaling function \(\phi \) of the form \(\{\phi (x-\lambda )\}_{\lambda \in \varLambda }\) where \(\varLambda = \{0,r/N\}+ 2\mathbb {Z}\), \(N\ge 1\) is an integer and r is an odd integer with \(1\le r\le 2N-1\) such that r and N are relatively prime and \(\mathbb {Z}\) is the set of all integers. The main results of Gabardo and Nashed deal with necessary and sufficient condition for the existence of associated wavelets and extension of Cohen’s theorem.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Chapter 9. Orthogonal Vector-Valued Wavelets on

Abstract

We have considered the notion of vector-valued multiresolution analysis (VMRA) on positive half line \(\mathbb {R}_+\) and studied associated vector-valued wavelets and wavelet packets. Xia and Suter in 1996 generalized the concept of multiresolution analysis (MRA) on \(\mathbb {R}\) to vector-valued multiresolution analysis (VMRA) on \(\mathbb {R}\) and studied associated vector-valued wavelets. Farkov (Farkov in Orthogonal p-wavelets on \(R^{+}\). St. Petersburg University Press, Saint Petersburg, p. 426, 2005) introduced MRA on \(\mathbb {R}_+\). In this chapter, we have introduced vector-valued multiresolution analysis (VMRA) on \(\mathbb {R}_+\), where the associated subspace \(V_0\) of \(L^{2}(\mathbb {R}_+,\mathbb {C}^N)\) has an orthonormal basis, a family of translates of a vector-valued function \(\varPhi \), i.e, \(\{\varPhi (x\ominus l)\}_{l\in \mathbb {Z}_+}\), where \(\mathbb {Z}_+\) is the set of nonnegative integers. The necessary and sufficient condition for the existence of associated vector-valued wavelets has been obtained and the construction of vector-valued multiresolution analysis (VMRA) on \(\mathbb {R}_{+}\) has been presented.

Yu. A. Farkov, Pammy Manchanda, Abul Hasan Siddiqi

Backmatter

Title: Construction of Wavelets Through Walsh Functions
Authors: Prof. Yu. A. Farkov
Prof. Pammy Manchanda
Prof. Abul Hasan Siddiqi
Copyright Year: 2019
Publisher: Springer Singapore
Electronic ISBN: 978-981-13-6370-2
Print ISBN: 978-981-13-6369-6
DOI: https://doi.org/10.1007/978-981-13-6370-2

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