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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| Language: | English |
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AIMS Press
2025-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2025110 |
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| author | Pablo Díaz Esmeralda Mainar Beatriz Rubio |
| author_facet | Pablo Díaz Esmeralda Mainar Beatriz Rubio |
| author_sort | Pablo Díaz |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-bb80f693443e4d808f941dcb3666d768 |
| institution | DOAJ |
| issn | 2473-6988 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-bb80f693443e4d808f941dcb3666d7682025-08-20T03:17:09ZengAIMS PressAIMS Mathematics2473-69882025-02-011022375239110.3934/math.2025110Total positivity, Gramian matrices, and Schur polynomialsPablo Díaz0Esmeralda Mainar1Beatriz Rubio2Departamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, EspañaDepartamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, EspañaDepartamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, EspañaThis 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.https://www.aimspress.com/article/doi/10.3934/math.2025110gramian matricesbidiagonal decompositionsschur functionsselberg-like integralstotally positive matrices |
| spellingShingle | Pablo Díaz Esmeralda Mainar Beatriz Rubio Total positivity, Gramian matrices, and Schur polynomials AIMS Mathematics gramian matrices bidiagonal decompositions schur functions selberg-like integrals totally positive matrices |
| title | Total positivity, Gramian matrices, and Schur polynomials |
| title_full | Total positivity, Gramian matrices, and Schur polynomials |
| title_fullStr | Total positivity, Gramian matrices, and Schur polynomials |
| title_full_unstemmed | Total positivity, Gramian matrices, and Schur polynomials |
| title_short | Total positivity, Gramian matrices, and Schur polynomials |
| title_sort | total positivity gramian matrices and schur polynomials |
| topic | gramian matrices bidiagonal decompositions schur functions selberg-like integrals totally positive matrices |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025110 |
| work_keys_str_mv | AT pablodiaz totalpositivitygramianmatricesandschurpolynomials AT esmeraldamainar totalpositivitygramianmatricesandschurpolynomials AT beatrizrubio totalpositivitygramianmatricesandschurpolynomials |