Necessary Conditions for Interpolation by Multivariate Polynomials
Let $\Omega$ be a smooth, bounded, convex domain 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 den...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/179888 |
| Acceso en línea: | https://hdl.handle.net/2445/179888 |
| Access Level: | acceso abierto |
| Palabra clave: | Anàlisi harmònica Teoria de l'aproximació Harmonic analysis Approximation theory |
| Sumario: | Let $\Omega$ be a smooth, bounded, convex domain 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 are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most $k$. |
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