Extensions of Co-Compact Gabor Frames on Locally Compact Abelian Groups and Applications

Abstract

This paper addresses the extensions of co-compact Gabor Bessel sequences to tight frames and dual pairs on locally compact abelian (LCA) groups, and its applications to the $L^2({\mathbb{R}}^d)$-setting. Firstly, we present a method to construct co-compact Gabor frames on LCA groups under a mild condition. This condition is optimal in the sense that it reduces to the usual density condition for lattice-based Gabor frames in $L^2({\mathbb{R}}^d)$. Secondly, we obtain an extension theorem of a co-compact Gabor Bessel sequence (a pair of co-compact Gabor Bessel sequences) to a tight co-compact Gabor frame (a pair of dual co-compact Gabor frames). Finally, as an application, we derive a strategy to obtain co-compact Gabor frames for $L^2({\mathbb{R}}^d)$, and establish an extension theorem of dual co-compact Gabor frames for $L^2({\mathbb{R}}^d)$ with $C_c^{\infty }({\mathbb{R}}^d)$-window functions. An example is also provided. It demonstrates that, for general co-compact (i.e., at least one of time and frequency translations is not a lattice) Gabor frames, the product of the sizes of time and frequency translations can take an arbitrary positive number.