Extensions of Bessel sequences to dual pairs of frames
type: Item
instance of: scholarly article
title: Extensions of Bessel sequences to dual pairs of frames
zbMATH Open document ID: 1323.42027
author: Ole Christensen
author: Hong Oh Kim
author: Rae Young Kim
published in: Applied and Computational Harmonic Analysis
publication date: 2013-03-05T00:00:00Z
review text: The authors study extensions of Bessel sequences in separable Hilbert spaces. They prove that given two Bessel sequences \(\{f_i\}_{i\in I}\) and \(\{g_i\}_{i\in I}\) in the Hilbert space \(\mathcal{H}\), there exist Bessel sequences \(\{p_j\}_{j\in J}\) and \(\{q_j\}_{j\in J}\) such that the systems \(\{h_t\}_{t\in T}:=\{f_i\}_{i\in I}\cup\{p_j\}_{j\in J}\) and \(\{k_t\}_{t\in T}:=\{g_i\}_{i\in I}\cup\{q_j\}_{j\in J}\) form a pair of dual frames for \(\mathcal{H}\), meaning that \(\{h_t\}_{t\in T}\) and \(\{k_t\}_{t\in T}\) are frames and satisfy the reconstruction formula \[ f=\sum_{t\in T}\langle f,h_t \rangle k_t,\qquad\forall\,\,\,f\in\mathcal{H}. \] The result is more flexible than the one proven by \textit{P. G. Casazza} and \textit{N. Leonhard} [Contemp. Math. 451, 11--31 (2008; Zbl 1210.42047)] and \textit{D. Li} and \textit{W. Sun} [Acta Math. Sin., Engl. Ser. 25, No. 2, 287--292 (2009; Zbl 1177.41030)], which says that any Bessel sequence can be extended to a tight frame. Additionally, in contrast to the extension to a tight frame, the extension to a pair of dual frames does not involve the square root of any operator and therefore it might be more efficient from the calculational point of view. In fact, this is illustrated by the authors with concrete examples. When the initial pair of Bessel sequences has a particular structure it might be convenient to have the extensions of them to be of the same shape. The authors prove that this is possible when having Gabor sequences. More precisely, they show that two Gabor Bessel sequences in \(L^2(\mathbb{R})\) can be extended to a pair of dual frames by adding two Gabor systems. Furthermore, when each of the original Gabor sequences is generated by compactly supported functions, the generators of the Gabor systems one needs to add to have a pair of frame duals can be chosen to be compactly supported as well. Under certain conditions, a result in this spirit is obtained for wavelet systems.
reviewed by: Victoria Paternostro
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 6141631
zbMATH Keywords: Bessel sequences
zbMATH Keywords: dual frame pairs
zbMATH Keywords: Gabor frames
zbMATH Keywords: wavelet frames
MaRDI profile type: Publication
full work available at URL: j.acha.2012.04.003
OpenAlex ID: W2121688150
DOI: 10.1016/J.ACHA.2012.04.003
description: scientific article; zbMATH DE number 6141631