Total positivity, Gramian matrices, and Schur polynomials

This paper investigated the total positivity of Gramian matrices associated with general bases of functions. It demonstrated that the total positivity of collocation matrices for totally positive bases extends to their Gramian matrices. Additionally, a bidiagonal decomposition of these Gramian matri...

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Bibliographic Details
Published inAIMS mathematics Vol. 10; no. 2; pp. 2375 - 2391
Main Authors Díaz, Pablo, Mainar, Esmeralda, Rubio, Beatriz
Format Journal Article
LanguageEnglish
Published AIMS Press 01.02.2025
Subjects
bidiagonal decompositions
gramian matrices
schur functions
selberg-like integrals
totally positive matrices
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Summary:This paper investigated the total positivity of Gramian matrices associated with general bases of functions. It demonstrated that the total positivity of collocation matrices for totally positive bases extends to their Gramian matrices. Additionally, a bidiagonal decomposition of these Gramian matrices, derived from integrals of symmetric functions, was presented. This decomposition enables the design of algorithms with high relative accuracy for solving linear algebra problems involving totally positive Gramian matrices. For polynomial bases, compact and explicit formulas for the bidiagonal decomposition were provided, involving integrals of Schur polynomials. These integrals, known as Selberg-like integrals, arise naturally in various contexts within Physics and Mathematics.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2025110