Framelets and wavelets. Algorithms, analysis, and applications
type: Item
instance of: scholarly article
title: Framelets and wavelets. Algorithms, analysis, and applications
zbMATH Open document ID: 1387.42001
DOI: 10.1007/978-3-319-68530-4
author: Peng Zhang
published in: Applied and Numerical Harmonic Analysis
publication date: 2017-10-26T00:00:00Z
review text: This monograph provides a comprehensive introduction to the mathematical theory of framelets and discrete framelet transforms. In the classical wavelet theory, a \textit{wavelet} is a function set \(\Psi \subset L_2(\mathbb R)\), where the related affine system \[ \{\psi_{j,k}(x) = 2^{j/2}\,\psi(2^j x - k) :\, j,k\in \mathbb Z\,, \,\psi \in \Psi\} \] is a basis of \(L_2(\mathbb R)\). Now two function sets \(\Phi\), \(\Psi \subset L_2(\mathbb R)\) form a \textit{framelet}, if for \(J\in \mathbb Z\) the corresponding affine system \[ \{\varphi_{J,k} :\,k\in \mathbb Z\,,\,\varphi\in \Phi\} \cap\{\psi_{j,k} :\, k\in \mathbb Z\,,j \geq J\,, \, \psi \in \Psi\} \] is a frame of \(L_2(\mathbb R)\), where a frame is an ``overcomplete basis''. Thus a wavelet is merely a framelet without redundancy. The author has made important contributions to framelet theory. His book systematically introduces the theory of framelets from the perspective of discrete framelet transforms such that multiresolution analysis plays only a marginal role in this theory. This monograph consists of seven chapters and an appendix. Chapter 1 describes the properties of discrete framelet/wavelet transforms and the corresponding implementation. Chapter 2 presents algorithms for constructing wavelet filter banks with/without symmetry. Chapter 3 develops general algorithms for constructing all possible dual framelet filter banks and tight framelet filter banks with/without symmetry such that the high-pass filters have short filter supports. In Chapter 4, the author investigates affine systems and dual framelets in the univariate case. In Chapter 5, he studies special refinable vector functions and their relations to affine systems derived from refinable functions. Multivariate scalar framelets/wavelets are studied in Chapter 7, where subdivision schemes and their applications to curve/surface generation are discussed. The Appendix presents some basic results on function spaces and Fourier analysis. The the…
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 6798658
zbMATH Keywords: monograph
zbMATH Keywords: framelets
zbMATH Keywords: wavelets
zbMATH Keywords: discrete framelet transforms
zbMATH Keywords: wavelet filter banks
zbMATH Keywords: framelet filter banks
zbMATH Keywords: dual framelets
zbMATH Keywords: refinable vector functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-68530-4
OpenAlex ID: W2781502722
description: scientific article; zbMATH DE number 6798658