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...

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Autores: Antezana, Jorge, Marzo Sánchez, Jordi, Ortega Cerdà, Joaquim
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2021
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat: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
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spelling Necessary Conditions for Interpolation by Multivariate PolynomialsAntezana, JorgeMarzo Sánchez, JordiOrtega Cerdà, JoaquimAnàlisi harmònicaTeoria de l'aproximacióHarmonic analysisApproximation theoryLet $\Omega$ 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 $k$.Springer Verlag2021202220212021info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersion19 p.application/pdfhttps://hdl.handle.net/2445/179888Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésVersió postprint del document publicat a: https://doi.org/10.1007/s40315-021-00410-8Computational Methods And Function Theory, 2021https://doi.org/10.1007/s40315-021-00410-8(c) Springer Verlag, 2021info:eu-repo/semantics/openAccessoai:recercat.cat:2445/1798882026-05-29T05:05:01Z
dc.title.none.fl_str_mv Necessary Conditions for Interpolation by Multivariate Polynomials
title Necessary Conditions for Interpolation by Multivariate Polynomials
spellingShingle Necessary Conditions for Interpolation by Multivariate Polynomials
Antezana, Jorge
Anàlisi harmònica
Teoria de l'aproximació
Harmonic analysis
Approximation theory
title_short Necessary Conditions for Interpolation by Multivariate Polynomials
title_full Necessary Conditions for Interpolation by Multivariate Polynomials
title_fullStr Necessary Conditions for Interpolation by Multivariate Polynomials
title_full_unstemmed Necessary Conditions for Interpolation by Multivariate Polynomials
title_sort Necessary Conditions for Interpolation by Multivariate Polynomials
dc.creator.none.fl_str_mv Antezana, Jorge
Marzo Sánchez, Jordi
Ortega Cerdà, Joaquim
author Antezana, Jorge
author_facet Antezana, Jorge
Marzo Sánchez, Jordi
Ortega Cerdà, Joaquim
author_role author
author2 Marzo Sánchez, Jordi
Ortega Cerdà, Joaquim
author2_role author
author
dc.subject.none.fl_str_mv Anàlisi harmònica
Teoria de l'aproximació
Harmonic analysis
Approximation theory
topic Anàlisi harmònica
Teoria de l'aproximació
Harmonic analysis
Approximation theory
description 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 $k$.
publishDate 2021
dc.date.none.fl_str_mv 2021
2021
2021
2022
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/179888
url https://hdl.handle.net/2445/179888
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.1007/s40315-021-00410-8
Computational Methods And Function Theory, 2021
https://doi.org/10.1007/s40315-021-00410-8
dc.rights.none.fl_str_mv (c) Springer Verlag, 2021
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Springer Verlag, 2021
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 19 p.
application/pdf
dc.publisher.none.fl_str_mv Springer Verlag
publisher.none.fl_str_mv Springer Verlag
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,812455

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