Convergence of cascade algorithms in Sobolev spaces associated with inhomogeneous refinement equations
type: Item
instance of: scholarly article
title: Convergence of cascade algorithms in Sobolev spaces associated with inhomogeneous refinement equations
zbMATH Open document ID: 0953.65103
author: Song Li
published in: Journal of Approximation Theory
publication date: 2000-06-05T00:00:00Z
review text: The paper is concerned with the solutions \(\varphi= (\varphi_1,\dots, \varphi_r)'\) of multivariate refinement equations \[ \varphi(x)= \sum_{\alpha\in\mathbb{Z}^s} a(\alpha)\varphi(2x- \alpha)+ g(x)\qquad (x\in\mathbb{R}^s), \] where \(g:= (g_1,\dots, g_r)'\) and \(a\) is a finitely supported refinement mask such that \(a(\alpha)\in \mathbb{C}^{r\times r}\). The author gives a characterization for the strong convergence of the cascade algorithm \[ \varphi^{(n)}(x)= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha)\varphi^{(n-1)}(2x- \alpha)+ g(x)\qquad (x\in \mathbb{R}^s) \] in the Sobolev space \(W^k_2(\mathbb{R}^s)\) \((k\in\mathbb{N})\) in terms of the refinement mask \(a\), the inhomogeneous term \(g\) and the initial function vector \(\varphi^{(0)}\).
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1455763
zbMATH Keywords: hansition operator
zbMATH Keywords: wavelets
zbMATH Keywords: multivariate refinement equations
zbMATH Keywords: convergence
zbMATH Keywords: cascade algorithm
reviewed by: Gabriele Steidl
MaRDI profile type: Publication
full work available at URL: jath.1999.3436
OpenAlex ID: W2006779068
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DOI: 10.1006/JATH.1999.3436
description: scientific article; zbMATH DE number 1455763