Regularity of refinable function vectors
type: Item
instance of: scholarly article
title: Regularity of refinable function vectors
zbMATH Open document ID: 0914.42025
DOI: 10.1007/BF02649113
author: Albert Cohen
author: Ingrid Daubechies
author: Gerlind Plonka
published in: The Journal of Fourier Analysis and Applications
publication date: 1999-06-02T00:00:00Z
full work available at URL: https://eudml.org/doc/59510
review text: Compactly supported solutions of refinement equations \(\phi(x)=\sum_{n=0}^N P_n\phi(2x-n)\) are considered for vector valued functions \(\phi\) and constant square matrices \(P_n\). As in the scalar case, the smoothness of \(\phi\) and the decay of the Fourier transform \(\widehat{\phi}\) are characterized by properties of the refinement mask \(P(u)={1\over 2}\sum_n P_ne^{-inu}\). In the scalar case, the approximation order can only be \(m\) if \(P(u)\) factors as \([(1+e^{-iu})/2]^mP^{(m)}(u)\) where \(P^{(m)}\) is \(2\pi\)-periodic, and \(P^{(m)}(0)=1\). The factorization property needed in the vector case is however much more involved: a factorization of the form \[ P(u)=2^{-m}C_0(2u)\cdots C_{m-1}(2u) P^{(m)}(u) C_{m-1}(u)^{-1}\cdots C_0(u)^{-1} \] is needed. From the relation \(\widehat{\phi}(u)=P(u/2)\widehat{\phi}(u/2)\), one gets the well-known infinite product representation of \(\widehat{\phi}\). The convergence of the infinite product in the matrix case is again much more intricate than in the scalar case and needs more conditions on \(P(u)\), for example \(P(0)\) should be diagonalizable with spectral radius at most 1. Finally, some conditions are needed to make \(\widehat{\phi}(u)\) decay for \(| u| \to\infty\). These decay properties allow to prove uniqueness of the solution of the refinement equations in a large function class and also the convergence of the cascade and subdivision algorithm can be derived. Several examples illustrate these results. The scaling functions of \textit{G. C. Donovan, J. S. Geronimo, D. P. Hardin} and \textit{P. R. Massopust} [SIAM J. Math. Anal. 27, No. 4, 1158-1192 (1996; Zbl 0873.42021)] are a special case.
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 1038865
zbMATH Keywords: refinable function vectors
zbMATH Keywords: refinable mask
zbMATH Keywords: Fourier transform
zbMATH Keywords: spectral radius
zbMATH Keywords: wavelets
zbMATH Keywords: refinement equations
zbMATH Keywords: scaling functions
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1038865