Multivariate Dynamical Sampling in $l^2(\mathbb{Z}^2)$ and Shift-Invariant Spaces Associated with Linear Canonical Transform

TL;DR

This paper studies multivariate dynamical sampling in two-dimensional spaces linked with the linear canonical transform, establishing conditions for stable recovery of signals from dynamical measurements.

Contribution

It provides necessary and sufficient conditions for stable signal recovery in 2D dynamical sampling related to the linear canonical transform, a novel extension in this area.

Findings

Derived conditions for stable recovery in $l^2( Z^2)$ and shift-invariant spaces.

Connected dynamical sampling with the two-dimensional linear canonical transform.

Presented an example illustrating the main theoretical results.

Abstract

In this paper, we investigate the multivariate dynamical sampling problem in $l^2(\mathbb{Z}^2)$ associated with the two-dimensional discrete time non-separable linear canonical transform (2D-DT-NS-LCT) and shift-invariant spaces associated with the two-dimensional non-separable linear canonical transform (2D-NS-LCT), respectively. Specifically, we derive a sufficient and necessary condition under which a sequence in $l^2(\mathbb{Z}^2)$ (or a function in a shift-invariant space) can be stably recovered from its dynamical sampling measurements associated with the 2D-DT-NS-LCT (or the 2D-NS-LCT). We also present a simple example to elucidate our main results.