Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets
type: Item
instance of: scholarly article
title: Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets
zbMATH Open document ID: 1035.42030
DOI: 10.1016/S0024-3795(02)00347-6
author: Ole Christensen
author: Alexander Lindner
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: Given a Hilbert space \(H\), a sequence \(\{ e_j : j \in J\} \subset H\) is a frame if there exist constants \(A, B >0\) such that for each \(f\in H\): \[ A \| f \| ^2 \leq \sum _{j \in J} | \langle f, e_j \rangle | ^2\leq B \| f \| ^2. \] A family \(\{ \phi_j : j \in J\} \subset H\) is a Riesz frame if it is complete and if there are constants \(A, B>0\) such that for any subset \(I \subset J\), \(\{ \phi_i : i \in I\}\) is a frame for the closure of its span with frame bounds \(A\) and \(B\). \(\{ \phi_j : j \in J\}\) is a Riesz sequence if there exist constants \(A, B>0\) such that for all subsets \(I \subset J\) and all square summable sequences \(\{c_i:i \in I\}\), \[ A \sum_{i \in I} | c_i| ^2 \leq \left\| \sum_{i \in I} c_i \phi_i \right\| ^2 \leq B \sum_{i \in I} | c_i| ^2. \] The authors prove that every Riesz frame can be partitioned into a finite union of Riesz sequences, and that the number of these sequences can be controlled by the ratio of the frame bounds. Various characterizations of Riesz frames are provided and the characterizing terms include decompositions into Riesz sequences, linearly independent sets, and conditions on the angles between the frame elements. It is also shown that compactly supported wavelets can be decomposed into a finite union of linearly independent sets.
reviewed by: Wojciech Czaja
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1864738
zbMATH Keywords: frame
zbMATH Keywords: Riesz basis
zbMATH Keywords: Riesz frame
zbMATH Keywords: wavelets
zbMATH Keywords: wavelet frame
zbMATH Keywords: linear independence
zbMATH Keywords: decomposition
zbMATH Keywords: angle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1864738