Compactly supported tight and sibling frames with maximum vanishing moments
type: Item
instance of: scholarly article
title: Compactly supported tight and sibling frames with maximum vanishing moments
zbMATH Open document ID: 1016.42023
DOI: 10.1016/S1063-5203(02)00510-9
author: Wenjie He
author: Charles K. Chui
author: Joachim Stöckler
published in: Applied and Computational Harmonic Analysis
publication date: 2003-04-02T00:00:00Z
review text: Two families \[ \Psi = \{ \psi_{i,j,k} = 2^{j/2} \psi_i (2^j\cdot - k): 1 \leq i \leq N, j,k \in {\mathbf Z} \} \] and \[ \tilde \Psi = \{ \tilde\psi_{i,j,k} = 2^{j/2} \tilde\psi_i (2^j\cdot - k): 1 \leq i \leq N, j,k \in {\mathbf Z} \} \] are called \textit{sibling frames} if the frame generators \(\psi_i, \tilde\psi_i, i=1, \ldots, N\), are generated by the same refinable function \(\phi\), if they are Bessel families, and if the duality relation \(\langle f,g \rangle = \sum_{i=1}^N \sum_{j,k \in {\mathbb{Z}}} \langle f, \psi_{i,j,k} \rangle \langle \tilde\psi_{i,j,k}, g \rangle\) holds for all \(f,g \in L^2({\mathbb{R}})\). One of the main results of the paper under review is that there exist two compactly supported sibling frames with the maximal number of vanishing moments, which can be chosen to be symmetric or antisymmetric. The proof of this result is constructive. The authors also provide the characterization of sibling frames in terms of the \textit{vanishing moment recovery function}, which is then used in the construction of a tight frame for \(L^2({\mathbf R})\) with two compactly supported generators that both have the maximal number of vanishing moments. This tight frame does not have symmetry or antisymmetry. The results obtained in this paper are related to the work of \textit{I. Daubechies, B. Han, A. Ron} and \textit{Z. Shen} [``Framelets: MRA-based constructions of wavelet frames'', Appl. Comput. Harmon. Anal. 14, No.~1, 1-46 (2003; Zbl 1035.42031)].
reviewed by: Wojciech Czaja
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 1891402
zbMATH Keywords: sibling frame
zbMATH Keywords: tight frame
zbMATH Keywords: unitary extension
zbMATH Keywords: vanishing moment recovery
zbMATH Keywords: inter-orthogonality
zbMATH Keywords: matrix factorization
describes a project that uses: DT-CWT
MaRDI profile type: Publication
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full work available at URL: s1063-5203(02)00510-9
OpenAlex ID: W1993077418
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description: scientific article; zbMATH DE number 1891402