arXiv 2101.080642021-01-20math.CA

Interpolation by multivariate polynomials in convex domains

Jorge Antezana, Jordi Marzo, Joaquim Ortega-Cerdà

备注：: 17 pages
发表信息：: Comput. Methods Funct. Theory 21 (2021), no. 4, 831-849
DOI：: 10.1007/s40315-021-00410-8

英文摘要：

Let $Ω$ be a convex open set in $\mathbb R^n$ and let $Λ_k$ be a finite subset of $Ω$. We find necessary geometric conditions for $Λ_k$ 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 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 $k$.

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