Multivariate polynomial interpolation and sampling in Paley-Wiener spaces

TL;DR

This paper establishes a link between exponential Riesz bases and polynomial interpolants for multivariate bandlimited functions, providing new recovery formulas with convergence guarantees and practical computational methods.

Contribution

It introduces a novel equivalence between Riesz bases and polynomial interpolants in multivariate bandlimited spaces, along with explicit recovery formulas and convergence analysis.

Findings

Existence of polynomial interpolants enables simple recovery formulas.

Recovery formulas demonstrate $L_2$ and uniform convergence on $ eal^d$.

Concrete examples of Riesz bases and data nodes are provided.

Abstract

In this paper, an equivalence between existence of particular exponential Riesz bases for multivariate bandlimited functions and existence of certain polynomial interpolants for these bandlimited functions is given. For certain classes of unequally spaced data nodes and corresponding $\ell_2$ data, the existence of these polynomial interpolants allows for a simple recovery formula for multivariate bandlimited functions which demonstrates $L_2$ and uniform convergence on $\mathbb{R}^d$. A simpler computational version of this recovery formula is also given, at the cost of replacing $L_2$ and uniform convergence on $\mathbb{R}^d$ with $L_2$ and uniform convergence on increasingly large subsets of $\mathbb{R}^d$. As a special case, the polynomial interpolants of given $\ell_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinant Riesz bases and unequally spaced data nodes are also given.