Matrix-Valued Gabor Frames over LCA Groups for Operators

TL;DR

This paper introduces matrix-valued Gabor frames over LCA groups controlled by operators, providing conditions for their existence, properties, and stability, extending frame theory to operator-controlled matrix-valued systems.

Contribution

It develops the theory of matrix-valued Gabor frames over LCA groups with operator control, including existence conditions, characterizations, and stability analysis.

Findings

Existence of $ ext{lambda}$-tight $( heta, heta^*)$-Gabor frames for hyponormal operators.

Characterization of matrix-valued $( heta, heta^*)$-Gabor frames.

Stability of these frames under small perturbations.

Abstract

G\v avruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely $K$-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator $K$. For a locally compact abelian group G and a positive integer $n$, we study frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space $L^2(G, \mathbb{C}^{n\times n})$ , where a bounded linear operator $\Theta$ on $L^2(G, \mathbb{C}^{n\times n})$ controls not only lower but also the upper frame condition. We term such frames matrix-valued $(\Theta, \Theta^*)$-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued $(\Theta, \Theta^*)$- Gabor frames in terms of hyponormal operators. It is shown that if $\Theta$ is adjointable hyponormal operator, then $L^2(G, \mathbb{C}^{n\times n})$ admits a $\lambda$-tight $(\Theta, \Theta^*)$-Gabor frame for every positive real number $\lambda$. A characterization of matrix-valued $(\Theta, \Theta^*)$-Gabor frames is given. Finally, we show that matrix-valued $(\Theta, \Theta^*)$-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.