On the duality principle by Casazza, Kutyniok, and Lammers
type: Item
instance of: scholarly article
title: On the duality principle by Casazza, Kutyniok, and Lammers
zbMATH Open document ID: 1222.42031
author: Ole Christensen
author: Hong Oh Kim
author: Rae Young Kim
published in: The Journal of Fourier Analysis and Applications
publication date: 2011-08-30T00:00:00Z
review text: In this interesting paper, the authors study duality relations in the frame theory. Let \(\{f_i\}_{i\in I}\) be a frame for a separable Hilbert space \(H\), where \(I\) denotes a countable index set. Then the Riesz-dual sequence \(\{\omega_j\}_{j\in I}\) of \(\{f_i\}_{i\in I}\) is defined by \[ \omega_j = \sum_{i \in I} \langle f_i, e_j\rangle h_i \qquad (j \in I), \] where \(\{e_j\}_{j\in I}\) and \(\{h_i\}_{i\in I}\) are orthonormal bases of \(H\). The notion of a Riesz-dual sequence was introduced by \textit{P. G. Casazza}, \textit{G. Kutyniok} and \textit{M. C. Lammers} [J. Fourier Anal. Appl. 10, No. 4, 383--408 (2004; Zbl 1058.42020)]. In the paper under review, the authors solve the following problem: Let \(\{f_i\}_{i\in I}\) be a frame for \(H\) and \(\{\omega_j\}_{j\in I}\) a Riesz sequence in \(H\). Under what conditions do there exist orthonormal bases \(\{e_j\}_{j\in I}\) and \(\{h_i\}_{i\in I}\) for \(H\) such that \(\{\omega_j\}_{j\in I}\) is a Riesz-dual sequence of \(\{f_i\}_{i\in I}\)? It is shown that the Riesz-dual sequences \(\{\omega_j\}_{j\in I}\) can be characterized in terms of frame properties of an associated sequence \(\{n_i\}_{i\in I}\). Several examples illustrate the results. Finally, the authors improve the Cauchy-Schwarz inequality: If \(\{f_i\}_{i\in I}\) is a Bessel sequence with bound \(B\), then \[ |\langle f_i,f_j \rangle |^2 \leq B\,(B - \|f_i\|^2 - \|f_j\|^2) + \|f_i\|^2\, \|f_j\|^2. \]
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 5944318
zbMATH Keywords: frame theory
zbMATH Keywords: duality principle
zbMATH Keywords: Riesz sequence
zbMATH Keywords: Bessel sequence
zbMATH Keywords: Riesz-dual sequence
zbMATH Keywords: orthonormal basis
zbMATH Keywords: Gabor system
zbMATH Keywords: Wexler-Raz theorem
MaRDI profile type: Publication
full work available at URL: s00041-010-9151-4
OpenAlex ID: W1991463564
cites work: Duality principles in frame theory
cites work: Frames and bases. An introductory course
cites work: Gabor time-frequency lattices and the Wexler-Raz identity
cites work: Foundations of time-frequency analysis
cites work: Frames, bases and group representations
cites work: Duality and biorthogonality for Weyl-Heisenberg frames
cites work: Weyl-Heisenberg frames and Riesz bases in \(L_2(\mathbb{R}^d)\)
DOI: 10.1007/S00041-010-9151-4
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description: scientific article; zbMATH DE number 5944318