Vector-valued nonstationary Gabor frames
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 the mixed frame operator is derived. As a consequence, a sufficient condition for a vector-valued NSG system to be a frame for \(L^2 (\mathbb{R}, \mathbb{C}^L)\) is obtained. Furthermore, the existence of painless vector-valued NSG frames and of vector-valued NSG frames with fast decaying window functions is shown.
scientific article; zbMATH DE number 6957995
scientific article