Volume 59, pp. 1-8, 2023.

Distribution results for a special class of matrix sequences: joining approximation theory and asymptotic linear algebra

Alec Jacopo Almo Schiavoni-Piazza and Stefano Serra-Capizzano

Abstract

In a recent paper, Lubinsky proved eigenvalue distribution results for a class of Hankel matrix sequences arising in several applications, ranging from Padé approximation to orthogonal polynomials and complex analysis. The results appeared in Linear Algebra and its Applications, and indeed many of the statements, whose origin belongs to the field of approximation theory and complex analysis, contain deep results in (asymptotic) linear algebra. Here we make an analysis of a part of these findings by combining his derivation with previous results in asymptotic linear algebra, showing that the use of an already available machinery shortens considerably the considered part of the derivations. Remarks and few additional results are also provided, in the spirit of bridging (numerical and asymptotic) linear algebra results and those coming from analysis and pure approximation theory.

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Key words

Hankel and Toeplitz matrix, matrix sequence, eigenvalue and singular value distribution, eigenvalue and singular value clustering

AMS subject classifications

15B05, 15A18, 47B35

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