Refinement differential equations and wavelets
type: Item
instance of: scholarly article
title: Refinement differential equations and wavelets
zbMATH Open document ID: 0929.34052
author: Jianhong Shen
published in: Methods and Applications of Analysis
publication date: 2000-01-16T00:00:00Z
review text: The author studies the refinement differential equation (RDE) of type \((P,H)\), \[ P(D)\phi (x)=2[H(E)\phi ](2x), \] where \(P(\lambda)\) is a polynomial in \(\lambda\), \(H(z)\) is a Laurent polynomial in \(z\), both with real coefficients, \(D=d /dx\) and \(E\) denotes the translation operator \(Ef(x)=f(x+1)\). The interesting results are derived for RDEs of type \((P(\lambda),1)\). New functions \(\text{kam}(x)\) and \(\Phi_{\theta}(x)\) are introduced as the atomic solutions to RDEs of type \((P(\lambda),1)\) when \(P(\lambda)\) has no purely imaginary roots. When \(P(\lambda)\) contains at least one purely imaginary root, it is possible to establish a set of linearly independent, periodic and \(C^{\infty}\) solutions based on the well-adapted structure of Dirichlet series. The general regular RDEs of type \((P(\lambda),H(z))\) are discussed as well. The main result is that a scaling function designed by the \((1,H(z))\) refinement equation is smoothed by the \((P(\lambda),1)\) RDE to yield a \(C^{\infty}\) solution to the original \((P(\lambda),H(z))\) RDE. The author discusses some related topics, including the probabilistic method, the generalized subdivision process and the application in the wavelet theory.
reviewed by: Jan Čermák
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 42A61
Mathematics Subject Classification ID: 42A85
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 1292481
zbMATH Keywords: refinement differential equations
zbMATH Keywords: wavelets
zbMATH Keywords: probability method
zbMATH Keywords: subdivision scheme
MaRDI profile type: Publication
DOI: 10.4310/MAA.1998.V5.N3.A5
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description: scientific article; zbMATH DE number 1292481