Abstract
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.
Highlights
In this paper, we consider positive Borel measures μ, which are finite and compactly supported in the complex plane
L2 (μ) is the following: For a certain measure μ, are polynomials dense in the space L2 (μ)? In other words, denote by P2 (μ) the closure of the polynomials in the space L2 (μ); the question is under what conditions the equality L2 (μ) = P2 (μ) is true
Lebesgue measure on an arbitrary domain G and L2 ( G ) the associated Hilbert space, the classical results of approximation by polynomials can be seen in, e.g., the work of Gaier [1], who explored the question of which assumptions on G will be assumed to have polynomials density in L2 ( G )
Summary
Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones, Escuela.
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