Necessary conditions for interpolation by multivariate polynomials

ANTEZANA, JORGE; JORGE MARZO; JOAQUIM ORTEGA CERDÀ

Computational Methods and Function Theory

SPRINGER

Año: 2021

1617-9447

Let Ω be a smooth, bounded, convex domain in R^n and let L_k be a finite subset of Ω. We find necessary geometric conditions for L_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 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.