Necessary Conditions for Interpolation by Multivariate Polynomials

Abstract

Let Ω be a smooth, bounded, convex domain in Rn and let Λk be a finite subset of Ω. We find necessary geometric conditions for Λk 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 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 are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k.