New stable biorthogonal spline-wavelets on the interval
type: Item
instance of: scholarly article
title: New stable biorthogonal spline-wavelets on the interval
zbMATH Open document ID: 1190.42021
author: Miriam Primbs
published in: Results in Mathematics
publication date: 2010-05-03T00:00:00Z
review text: In this interesting paper, the author constructs new stable biorthogonal spline-wavelet bases on \([0,\,1]\). This construction is based on the biorthogonal multiresolution analysis (MRA) on \(\mathbb R\) presented by \textit{A. Cohen}, \textit{I. Daubechies}, and \textit{J.-C. Feauveau} [Commun. Pure Appl. Math. 45, No. 5, 485--560 (1992; Zbl 0776.42020)]. For a first construction of biorthogonal spline-wavelets on \([0,\,1]\) [see \textit{W. Dahmen}, \textit{A. Kunoth}, and \textit{K. Urban}, Appl. Comput. Harm. Anal. 6, No. 2, 132--196 (1999; Zbl 0922.42021)]. The aim of this new approach to biorthogonal spline-wavelets on \([0,\,1]\) are better Riesz bounds and better condition numbers of the corresponding wavelet transform matrices. For the construction of the primal MRA, the author chooses the Schoenberg spline bases on \([0,\,1]\) concerning equidistant, dyadic partitions of \([0,\,1]\). The full degree of polynomial reproduction is preserved on the primal side. The construction of boundary scaling functions on the dual side guarantees polynomial reproduction on the dual side too. The MRA spaces satisfy certain Jackson and Bernstein inequalities such that the associated wavelets are \(L^2([0,\,1])\)-stable with nice Riesz bounds. The biorthogonal wavelets are computed by the method of stable completion. The corresponding decomposition and reconstruction transforms are sparse matrices with low condition numbers. The author derives also biorthogonal spline-wavelet bases on \([0,\,1]\) with complementary or homogeneous boundary conditions.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 5703374
zbMATH Keywords: wavelets
zbMATH Keywords: spline-wavelets
zbMATH Keywords: biorthogonal spline-wavelets
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: Riesz bounds
zbMATH Keywords: condition numbers
zbMATH Keywords: polynomial reproduction
zbMATH Keywords: method of stable completion
zbMATH Keywords: boundary scaling function
zbMATH Keywords: Jackson inequality
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: Riesz stability
zbMATH Keywords: wavelet algorithms
MaRDI profile type: Publication
full work available at URL: s00025-009-0008-6
OpenAlex ID: W2059967877
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DOI: 10.1007/S00025-009-0008-6
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description: scientific article; zbMATH DE number 5703374