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Open AccessFeature PaperArticle

A Characterization of Polynomial Density on Curves via Matrix Algebra

by Carmen Escribano 1,2,*,†, Raquel Gonzalo 1,† and Emilio Torrano 1,†
1
Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones, Escuela Técnica Superior de Ingenieros Informáticos, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain
2
Center for Computational Simulation, Universidad Politécnica de Madrid, 28660 Madrid, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2019, 7(12), 1231; https://doi.org/10.3390/math7121231
Received: 18 October 2019 / Revised: 29 November 2019 / Accepted: 2 December 2019 / Published: 12 December 2019
In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( 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 γ 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 γ 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. View Full-Text
Keywords: Hermitian moment problem; orthogonal polynomials; smallest eigenvalue; measures; polynomial density Hermitian moment problem; orthogonal polynomials; smallest eigenvalue; measures; polynomial density
MDPI and ACS Style

Escribano, C.; Gonzalo, R.; Torrano, E. A Characterization of Polynomial Density on Curves via Matrix Algebra. Mathematics 2019, 7, 1231.

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