Discrete frames of non-uniform shifts in frequency domain

TL;DR

This paper investigates conditions under which collections of matrix-valued functions, generated by non-uniform shifts, form stable frames in Hilbert spaces, including criteria and perturbation results.

Contribution

It provides necessary and sufficient conditions for matrix-valued non-uniform discrete frames and extends frame theory to non-uniform shift scenarios.

Findings

Characterization of frame conditions via Fourier transform of window functions

Necessary and sufficient conditions for matrix-valued discrete Bessel sequences

Perturbation results ensuring stability of non-uniform frames

Abstract

Frames in separable Hilbert spaces gives stable analysis and reconstruction of each vector in the underlying space. In this paper, we study frame conditions for a collection of matrix-valued functions obtained by non-uniform shifts. We give necessary and sufficient conditions for the existence of matrix-valued discrete Bessel sequence over non-uniform displacement parameters in terms of the Fourier transform of window functions. We also present perturbation results for matrix-valued non-uniform discrete frames.