Convergence of cascade algorithms and smoothness of refinable distributions
type: Item
instance of: scholarly article
title: Convergence of cascade algorithms and smoothness of refinable distributions
zbMATH Open document ID: 1051.42028
DOI: 10.1142/S0252959903000372
author: Qiyu Sun
published in: Chinese Annals of Mathematics. Series B
publication date: 2004-05-27T00:00:00Z
review text: Let \(F\) be an \(N\times 1\) vector of functions in a function space \(X\). In this paper, the author considers the cascade algorithm \(\{ T^n F\}_{n=1}^\infty\), where the cascade operator \(T\) on \(X^N\) is defined to be \[ TF:=\sum_{j\in \mathbb{Z}^d} c(j) F(2\cdot-j), \] and \(c(j), j\in \mathbb{Z}^d\) are \(N\times N\) matrices and \(c(j)=0\) for all but finitely many \(j\in \mathbb{Z}^d\). The \(\{c(j)\}_{j\in \mathbb{Z}^d}\) is called the mask. Let \(\mathcal{D}_r\) denote the space of all compactly supported \(C^\infty\) functions \(h\) on \(\mathbb{R}^d\) such that \(\hat h(\xi)=O(| \xi| ^r)\) as \(\xi\to 0\). By introducing an interesting notion of the dependent ideal \(i_r(F):=\{ (h*F(j))_{j\in\mathbb{Z}^d} : h\in \mathcal{D}_r\}\) of \(F\), the author is able to obtain in Theorem~2.1 the asymptotic behavior of the cascade sequence \(T^n F\) in a family of function spaces without the assumption of stability on the initial function vector \(F\). As a consequence, the author characterizes the existence of a refinable function vector \(\Psi\) (that is, \(T \Psi=\Psi\)) in \(L^p\) and establishes the convergence of the cascade algorithm in various function spaces. A criterion for a refinable function vector \(\Psi\in L^{p, \gamma}\) has been given in Theorem~3.1. The author also discusses (mainly for the scalar case \(N=1\)) in Section 4 how to choose an appropriate initial function vector in a cascade algorithm.
reviewed by: Bin Han
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2068437
zbMATH Keywords: cascade algorithm
zbMATH Keywords: refinable function vector
zbMATH Keywords: stable shifts
zbMATH Keywords: fractional Sobolev space
MaRDI profile type: Publication
cites work: Ten Lectures on Wavelets
cites work: Q3975831
cites work: Theory of function spaces II
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description: scientific article; zbMATH DE number 2068437