A note on the description of frames of general form
type: Item
instance of: scholarly article
title: A note on the description of frames of general form
zbMATH Open document ID: 1044.42026
DOI: 10.1023/A:1021402315905
author: T. Yu. Kulikova
author: B. S. Kashin
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: A frame for a separable Hilbert space \(H\) is a collection \(\{f_i: i \in N\} \subset H\) such that there exist constants \(0 < A \leq B < \infty\) so that for all \(g \in H\), \[ A\| g\| _H^2 \leq \sum_{i \in N}| \langle g, f_i \rangle| ^2 \leq B\| g\| _H^2. \] The constants \(A, B\) are called the frame bounds. When \(A=B\) we say that the frame is tight. A theorem due to Naimark states that if \(\{f_i: i \in N\}\) is a tight frame for a Hilbert space \(H\), then there exists a Hilbert space \(H'\), \(H\subset H'\), and an orthonormal basis \(\{e_i: i \in N\}\) for \(H'\) such that \(\{f_i: i \in N\}\) is an orthogonal projection of \(\{e_i: i \in N\}\) onto \(H\). The paper under review extends this result to hold for general frames (with \(A \leq B\)) and for Riesz bases, i.e., images of orthonormal bases under bounded and invertible transformations. For related results see, e.g., \textit{P.G. Casazza, D. Han} and \textit{D. R. Larson} [Contemp. Math. 247, 149--182 (1999; Zbl 0947.46010)].
reviewed by: Wojciech Czaja
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1928353
zbMATH Keywords: frame
zbMATH Keywords: tight frame
zbMATH Keywords: Riesz basis
zbMATH Keywords: Hilbert space
zbMATH Keywords: projection
MaRDI profile type: Publication
full work available at URL: a:1021402315905
OpenAlex ID: W54839451
description: scientific article; zbMATH DE number 1928353