Interpolation by multivariate polynomials in convex domains

TL;DR

This paper establishes geometric conditions for interpolation by multivariate polynomials in convex domains, linking these conditions to the equilibrium potential and analyzing the case of the unit ball.

Contribution

It provides asymptotic necessary geometric conditions for interpolation sets in multivariate polynomial spaces, connecting them to sampling set conditions and equilibrium potentials.

Findings

Density conditions match known sampling set conditions

No orthogonal reproducing kernel family exists in the unit ball for large k

Results are asymptotic in the polynomial degree k

Abstract

Let $\Omega$ be a convex open set in $\mathbb R^n$ and let $\Lambda_k$ be a finite subset of $\Omega$. We find necessary geometric conditions for $\Lambda_k$ to be interpolating for the space of multivariate polynomials of degree at most $k$. Our results are asymptotic in $k$. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy, and they are expressed in terms of the equilibrium potential of the convex set. Moreover, we prove that in the particular case of the unit ball, for $k$ large enough, there is no family of orthogonal reproducing kernels in the space of polynomials of degree at most $k$.