Sampling and Recovery of Multidimensional Bandlimited Functions via Frames

TL;DR

This paper develops multidimensional oversampling formulas for bandlimited functions using exponential frames, analyzes their stability, and generalizes classical theorems to higher dimensions.

Contribution

It introduces a general multidimensional oversampling formula, explores stability under perturbations, and extends Kadec's theorem to higher dimensions.

Findings

Derived a nonuniform oversampling formula for multidimensional bandlimited functions.

Proved stability of the sampling formula under data perturbations.

Extended Kadec's 1/4 theorem to higher dimensions.

Abstract

In this paper, we investigate frames for $L_2[-\pi,\pi]^d$ consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally managable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's $1/4$ Theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for $L_2[-\pi,\pi]$, and a well known theorem of Levinson is recovered as a corollary.