On multi-dimensional sampling and interpolation

TL;DR

This paper explores the conditions for sampling and interpolation of multivariate functions with convex spectra, revealing that critical values differ between sampling and interpolation as the dimension increases.

Contribution

It provides sharp sufficient conditions for multivariate sampling and interpolation, highlighting the dimension-dependent growth of interpolation critical values.

Findings

Critical values for sampling remain constant with dimension.

Interpolation critical values grow linearly with the number of variables.

Classical one-dimensional theorems do not extend directly to higher dimensions.

Abstract

The paper discusses sharp sufficient conditions for interpolation and sampling for functions of n variables with convex spectrum. When n=1, the classical theorems of Ingham and Beurling state that the critical values in the estimates from above (from below) for the distances between interpolation (sampling) nodes are the same. This is no longer true for n>1. While the critical value for sampling sets remains constant, the one for interpolation grows linearly with the dimension.