Vector-valued nonstationary Gabor frames
type: Item
instance of: scholarly article
title: Vector-valued nonstationary Gabor frames
zbMATH Open document ID: 1405.42057
author: Linlin Song
author: Qiaofang Lian
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-10-18T00:00:00Z
review text: Gabor systems and, in particular, Gabor frames (cf. [\textit{O. Christensen}, An introduction to frames and Riesz bases. Boston, MA: Birkhäuser (2003; Zbl 1017.42022)] or [\textit{K. Gröchenig}, Foundations of time-frequency analysis. Boston, MA: Birkhäuser (2001; Zbl 0966.42020)] for definitions) provide a fixed time-frequency resolution and sampling strategy. For that reason various adapted and adaptive representations have been introduced in processing signals with variable time-frequency characteristics, cf. [\textit{P. Balazs} et al., ``Adapted and adaptive linear time-frequency representations: a synthesis point of view'', IEEE Signal Process. Mag. 30, 20--31 (2013)] from the list of references. For example, nonstationary Gabor (NSG) systems generalize Gabor systems by allowing for different window functions and irregular sampling points. If an NSG system constitutes a frame, it is called an NSG frame. Different NSG frames are considered in [\textit{P. Balazs} et al., J. Comput. Appl. Math. 236, No. 6, 1481--1496 (2011; Zbl 1236.94026)], [\textit{M. Dörfler} and \textit{E. Matusiak}, Int. J. Wavelets Multiresolut. Inf. Process. 12, No. 3, Article ID 1450032, 18 p. (2014; Zbl 1345.42030)], [\textit{M. Dörfler} and \textit{E. Matusiak}, Adv. Comput. Math. 41, No. 2, 293--316 (2015; Zbl 1345.42031)] and used in corresponding adaptive transforms. In this paper the notion of NSG system for \(L^2 (\mathbb{R})\) is extended to the space of vector-valued signals \(L^2 (\mathbb{R}, \mathbb{C}^L)\) whose elements can be understood as vectors \( \mathbf{f} = (f_1, \dots, f_L)\) with \(f_l \in L^2 (\mathbb{R})\). The common concepts from the theory of Gabor systems are introduced in such vector-valued context. The authors proceed with considering the case when all components of window functions belong to the Wiener space (cf. e.g. [\textit{K. Gröchenig}, Foundations of time-frequency analysis. Boston, MA: Birkhäuser (2001; Zbl 0966.42020)]). The Walnut's representation of…
reviewed by: Nenad Teofanov
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 6957995
zbMATH Keywords: adaptive representation
zbMATH Keywords: Gabor frames
zbMATH Keywords: vector-valued space
zbMATH Keywords: nonstationary Gabor frames
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.09.019
OpenAlex ID: W2891545608
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DOI: 10.1016/J.JMAA.2018.09.019
Wikidata QID: Q129253661
description: scientific article; zbMATH DE number 6957995