Author:
Antezana Jorge,Marzo Jordi,Ortega-Cerdà Joaquim
Abstract
AbstractLet $$\Omega $$
Ω
be a smooth, bounded, convex domain in $${\mathbb {R}}^n$$
R
n
and let $$\Lambda _k$$
Λ
k
be a finite subset of $$\Omega $$
Ω
. We find necessary geometric conditions for $$\Lambda _k$$
Λ
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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Theory and Mathematics,Analysis
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