Total positivity, Gramian matrices, and Schur polynomials

Díaz, Pablo (Universidad de Zaragoza) ; Mainar, Esmeralda (Universidad de Zaragoza) ; Rubio, Beatriz (Universidad de Zaragoza)

Resumen: 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

Idioma: Inglés

DOI: 10.3934/math.2025110

Año: 2025

Publicado en: AIMS Mathematics 10, 2 (2025), 2375-2391
ISSN: 2473-6988

Financiación: info:eu-repo/grantAgreement/ES/DGA/E41-23R
Financiación: info:eu-repo/grantAgreement/ES/MCIU/PID2022-138569NB-I00
Financiación: info:eu-repo/grantAgreement/ES/MICINN/RED2022-134176-T

Tipo y forma: Article (Published version)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)