A characterization of polynomial density on curves via matrix algebra

Escribano Iglesias, M. del Carmen and Gonzalo Palomar, Raquel and Torrano Gimenez, Emilio (2019). A characterization of polynomial density on curves via matrix algebra. "Mathematics", v. 7 (n. 12); pp. 1-12. ISSN 2227-7390. https://doi.org/10.3390/math7121231.

Description

Title: A characterization of polynomial density on curves via matrix algebra
Author/s:
  • Escribano Iglesias, M. del Carmen
  • Gonzalo Palomar, Raquel
  • Torrano Gimenez, Emilio
Item Type: Article
Título de Revista/Publicación: Mathematics
Date: 2019
ISSN: 2227-7390
Volume: 7
Subjects:
Freetext Keywords: Hermitian moment problem, Orthogonal polynomials, Smallest eigenvalue, Measures, Polynomial density
Faculty: E.T.S. de Ingenieros Informáticos (UPM)
Department: Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L2(m), with m a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure m. To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, g(M) and l(M), associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index g and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index g that will allow us to give an alternative proof of Thomson's theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

Funding Projects

TypeCodeAcronymLeaderTitle
Government of SpainMTM2016-80582-RUnspecifiedUniversidad Politécnica de MadridTopología digital y matrices infinitas: aplicaciones al procesamiento de imágenes digitales

More information

Item ID: 67075
DC Identifier: http://oa.upm.es/67075/
OAI Identifier: oai:oa.upm.es:67075
DOI: 10.3390/math7121231
Official URL: https://www.mdpi.com/2227-7390/7/12/1231
Deposited by: Memoria Investigacion
Deposited on: 17 May 2021 07:43
Last Modified: 17 May 2021 07:43
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