Smooth multiwavelet duals of Alpert bases by moment-interpolating refinement
type: Item
instance of: scholarly article
title: Smooth multiwavelet duals of Alpert bases by moment-interpolating refinement
zbMATH Open document ID: 0967.65120
author: R. Smith
published in: Applied and Computational Harmonic Analysis
publication date: 2001-08-21T00:00:00Z
review text: The authors construct smooth multiwavelet bases of \(L^{2}[0,1]\) and \(L^{2}(R) \) which are duals of a generalized class of Alpert multiwavelets [cf. \textit{B. K. Alpert}, SIAM J. Math. Anal. 24, No. 1, 246-262 (1993; Zbl 0764.42017); \textit{B. Alpert, G. Beylkin, R. Coifman} and \textit{V. Rokhlin}, SIAM J. Sci. Comput. 14, No. 1, 159-184 (1993; Zbl 0771.65088)]. One begins by observing that the Alpert bases can be thought of as the first few Legendre polynomials cut off at the endpoints of \([0,1]\). By shifting to other unit intervals one then gets a basis for \(L^{2}(R)\) that gives rise to higher order multiwavelets such that inner products of \(f\) with these wavelets represent local moments of \(f\). The moment interpolation problem is that of starting with a sequence of moments at given scales, and determining local polynomials having a number of wavelet coefficients equal to the corresponding moments. The degree of the polynomial and the number of moments/wavelet coefficients that must coincide are chosen so that the solution is uniquely determined. The moment imputation problem is that of computing the moments/wavelet coefficients of the local polynomials at the next scale and interpolating these moments to define new finer scale polynomials. The process implicitly defines an operator, independent of scale, that maps moments from one scale to the next. This moment interpolation scheme is said to be \(C^{r}\) if the interated refinement applied to a bounded moment sequence at the starting scale produces a \(C^{r}\) limit function. Here the authors show that when the wavelet coefficients are with respect to the Alpert wavelets, this refinement process produces continuous limit functions. More precise regularity estimates are addressed in a related paper authored by \textit{T. P.-Y. Yu} [Parametric families of Hermite subdivision schemes in dimension 1, Available at \url{http://www.rpi.edu/~yut/Papers/hermite1.pdf} (1999)]. In a different direction, Hermi…
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65D05
zbMATH DE Number: 1512546
zbMATH Keywords: wavelets
zbMATH Keywords: multiwavelets
zbMATH Keywords: refinement
zbMATH Keywords: interpolation
zbMATH Keywords: moment interpolation problem
zbMATH Keywords: moment imputation problem
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: polynomial imputation
zbMATH Keywords: algorithm
reviewed by: Joseph Lakey
describes a project that uses: WaveLab
MaRDI profile type: Publication
OpenAlex ID: W1995861918
cites work: Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations
cites work: A Class of Bases in $L^2$ for the Sparse Representation of Integral Operators
cites work: Q4849988
cites work: Biorthogonal bases of compactly supported wavelets
cites work: Q3960030
cites work: Two-Scale Difference Equations. I. Existence and Global Regularity of Solutions
cites work: Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals
cites work: A practical guide to splines
cites work: Symmetric iterative interpolation processes
cites work: Q4279332
cites work: Interpolation through an iterative scheme
cites work: Q4026009
cites work: Analysis of uniform binary subdivision schemes for curve design
cites work: Analysis of Hermite-interpolatory subdivision schemes
cites work: Q3793199
cites work: Q4000004
cites work: A family of Hermite interpolants by bisection algorithms
cites work: Uniform refinement of curves
cites work: Simple Regularity Criteria for Subdivision Schemes
cites work: Embedded image coding using zerotrees of wavelet coefficients
DOI: 10.1006/ACHA.2000.0315
description: scientific article; zbMATH DE number 1512546