Nonuniform Sampling and Recovery of Bandlimited Functions in Higher Dimensions

TL;DR

This paper establishes conditions for sampling and reconstructing multivariate bandlimited functions at nonuniform points in higher dimensions, providing recovery guarantees and approximation rates.

Contribution

It introduces new sufficient conditions for sampling multivariate bandlimited functions at scattered points, extending classical results to higher dimensions and nonuniform sampling schemes.

Findings

Recovery results with explicit approximation orders

Conditions ensuring Riesz basis formation for exponential functions

Extension of sampling theory to higher-dimensional nonuniform settings

Abstract

We provide sufficient conditions on a family of functions $(\phi_\alpha)_{\alpha\in A}:\mathbb{R}^d\to\mathbb{R}$ for sampling of multivariate bandlimited functions at certain nonuniform sequences of points in $\mathbb{R}^d$. We consider interpolation of functions whose Fourier transform is supported in some small ball in $\mathbb{R}^d$ at scattered points $(x_j)_{j\in\mathbb{N}}$ such that the complex exponentials $\left(e^{-i\langle x_j,\cdot\rangle}\right)_{j\in\mathbb{N}}$ form a Riesz basis for the $L_2$ space of a convex body containing the ball. Recovery results as well as corresponding approximation orders in terms of the parameter $\alpha$ are obtained.