Pairs of dual periodic frames
type: Item
instance of: scholarly article
title: Pairs of dual periodic frames
zbMATH Open document ID: 1266.42073
author: Ole Christensen
author: Say Song Goh
published in: Applied and Computational Harmonic Analysis
publication date: 2012-09-06T00:00:00Z
reviewed by: Jakob Lemvig
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 6078405
zbMATH Keywords: periodic frames
zbMATH Keywords: Gabor frames
zbMATH Keywords: wavelet frames
zbMATH Keywords: dual pairs of frames
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: generalized shift-invariant systems
MaRDI profile type: Publication
full work available at URL: j.acha.2011.12.003
OpenAlex ID: W2144041641
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DOI: 10.1016/J.ACHA.2011.12.003
review text: The authors consider constructions of dual pairs of frames in the setting of the Hilbert space of periodic functions \(L^2(0,2\pi)\). The frames constructed are generalized shift-invariant systems, whose generators are given explicitly as trigonometric polynomials which provides an efficient computation procedure of the coefficients in the series expansions.NEWLINENEWLINEThe main result (Theorem 3.1) is the following general setup for constructions of dual pairs of frames for \(L^2(0,2\pi)\): Let \(\{\psi_k\}_{k \in I}\) be a collection of trigonometric polynomials with real-valued Fourier coefficients, and consider any sequence \(\{L_k\}_{k\in I}\) of positive integers. Assume that \(\{T_{2\pi/L_k}^\ell \psi_k\}_{k \in I,\ell=1,\dotsc,L_k}\) forms a Bessel sequence. Under a partition of unity condition \(\sum_k \sqrt{L_k}\hat{\psi_k}(n)=1\), \(n \in \mathbb{Z}\), and two assumptions on the support sets \(S_k:=\operatorname{supp}\,\hat{\psi_k} \subset \mathbb{Z}\) (bounding the number of overlapping \(S_k\)'s and bounding a growth rate of certain unions of the \(S_k\)'s), the systems \(\{T_{2\pi/L_k}^\ell \psi_k\}_{k \in I,\ell=1,\dotsc,L_k}\) and \(\{T_{2\pi/L_k}^\ell \widetilde{\psi}_k\}_{k \in I,\ell=1,\dotsc,L_k}\) form a pair of dual frames, whenever \(\{T_{2\pi/L_k}^\ell \widetilde{\psi}_k\}_{k \in I,\ell=1,\dotsc,L_k}\) is a Bessel sequence. Here, \(\widetilde{\psi}_k\) is defined as a finite linear combination of the \(\psi_k\); in particular, \(\widetilde{\psi}_k\) is also a trigonometric polynomial for each \(k \in I\). The result is similar in spirit of those in [\textit{O. Christensen}, Appl. Comput. Harmon. Anal. 20, No. 3, 403--410 (2006; Zbl 1106.42030)], where dual pairs of Gabor frames are constructed for \(L^2(\mathbb{R})\). Note, however, that the technical Bessel condition on the dual frame in the above general setup is not necessary in the setting of, e.g., Gabor and wavelet systems (cf. Corollary 3.1 and 3.2).NEWLINENEWLINEFinally, the construc…
description: scientific article; zbMATH DE number 6078405