Wavelets through a looking glass. The world of the spectrum
type: Item
instance of: scholarly article
title: Wavelets through a looking glass. The world of the spectrum
zbMATH Open document ID: 1012.42023
DOI: 10.1007/978-0-8176-8144-9
author: Ola Bratteli
author: Palle Jorgensen
published in: Applied and Numerical Harmonic Analysis
publication date: 2002-10-20T00:00:00Z
review text: This is a quite unusual wavelet book, with a fresh view on the subject. While the vast majority of wavelet books concentrate on multiresolution analysis and its applications, this book treats the topic from an operator theoretic point of view, with focus on techniques having a geometric or spectral theoretic flavor. In fact, while parts of the book could be used in a wavelet course, other parts would be suitable in a course directed towards operator theory. Each chapter (and some of the sections) starts with an informal tutorial, which explains the main ideas in less technical terms. Furthermore, Chapter 1 contains a section with terminology, which explains the main concepts and the meaning of the key words in, e.g., engineering, physics, and mathematics. Both features certainly helps the reader. Each chapter concludes with exercises of varying difficulty. Chapter 1 contains introductory material, like Fourier series, multiresolution analysis, and the cascade algorithm. The transfer operator, one of the central topics in the book, is introduced: given a trigonometric polynomial \(m_0\) and a positive integer \(N\), it is defined on certain functions \(f\) on the torus \( T\) by \[ R_{m_0}f(z)= \frac 1{N} \sum_{w\in { T}, w^N=z}|m_0(w)|^2f(w). \] A large part of the book concerns this operator, e.g., analysis of properties of \(R_{m_0}\) on certain function spaces based on assumptions on \(m_0\). Chapter 2 deals with homotopy theory and convergence properties of the cascade algorithm, which turns out to be related with the spectrum of a certain transfer operator. The chapter concludes with some open problems concerning vector-valued versions of the scaling identity. Chapters 3 and 4 deal with a more general definition of transfer operators than stated above (Ruelle operators) and discuss their spectral properties, e.g., properties of the restriction of the transfer operator to \(C({ T})\). Chapter 6 demonstrates how biorthogonal wavelets might be turned into orthogon…
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 94A12
zbMATH DE Number: 1817634
zbMATH Keywords: wavelets
zbMATH Keywords: cascade approximation
zbMATH Keywords: cycle
zbMATH Keywords: homotopy
zbMATH Keywords: transfer operator
zbMATH Keywords: Ruelle operator
zbMATH Keywords: \(C^*\)-algebra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-8144-9
OpenAlex ID: W175341177
description: scientific article; zbMATH DE number 1817634