Multivariate refinement equations and convergence of cascade algorithms in \(L_p(0<p<1)\) spaces
type: Item
instance of: scholarly article
title: Multivariate refinement equations and convergence of cascade algorithms in \(L_p(0<p<1)\) spaces
zbMATH Open document ID: 1027.42034
author: Song Li
publication date: 2003-09-25T00:00:00Z
review text: We consider the solutions of refinement equations written in the form \[ \varphi(x)= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha) \varphi(Mx- \alpha)+ g(x),\qquad x\in \mathbb{R}^s, \] where the vector of functions \(\varphi=(\varphi_1,\dots, \varphi_r)^T\) is unknown, \(g\) is a given vector of compactly supported functions on \(\mathbb{R}^s\), \(a\) is a finitely supported sequence of \(r\times r\) matrices called the refinement mask, and \(M\) is an \(s\times s\) dilation matrix with \(m= |\text{det }M|\). Inhomogeneous refinement equations appear in the constructions of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask \(a\), \(g\), and dilation \(M\) generates a sequence \(\varphi_n\), \(n= 1,2,\dots\), by the iterative process \[ \varphi_n(x)= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha) \varphi_{n-1}(Mx- \alpha)+ g(x),\qquad x\in\mathbb{R}^s, \] from a starting vector of the function \(\varphi_0\). We characterize the \(L_p\)-convergence \((0<p<1)\) of the cascade algorithm in terms of the \(p\)-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 1921181
zbMATH Keywords: multivariate refinement equations
zbMATH Keywords: multiwavelets
zbMATH Keywords: wavelets
zbMATH Keywords: cascade algorithm
zbMATH Keywords: \(L_p\)-convergence
zbMATH Keywords: refinement mask
zbMATH Keywords: homogeneous refinement equation
MaRDI profile type: Publication
full work available at URL: s10114-002-0216-4
OpenAlex ID: W1977040520
cites work: Q3143213
cites work: Inhomogeneous refinement equations
cites work: Distributional Solutions of Nonhomogeneous Discrete and Continuous Refinement Equations
cites work: Stationary subdivision
cites work: Analysis of uniform binary subdivision schemes for curve design
cites work: Subdivision schemes in \(L_ p\) spaces
cites work: Multivariate Refinement Equations and Convergence of Subdivision Schemes
cites work: Vector subdivision schemes and multiple wavelets
cites work: Convergence of multidimensional cascade algorithm
cites work: Refinable Function Vectors
cites work: Convergence of cascade algorithms associated with nonhomogeneous refinement equations
cites work: Convergence of cascade algorithms in Sobolev spaces associated with inhomogeneous refinement equations
cites work: Compression of Wavelet Decompositions
cites work: Q4273944
cites work: Stability of the shifts of a finite number of functions
cites work: Q3272731
cites work: Norms concerning subdivision sequences and their applications in wavelets
cites work: The \(p\)-norm joint spectral radius for even integers
cites work: Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces
DOI: 10.1007/S10114-002-0216-4
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1921181