{"title":"Darboux transformations and the algebra D(W)","authors":"Ignacio Bono Parisi, Ines Pacharoni","doi":"10.1016/j.laa.2025.01.002","DOIUrl":null,"url":null,"abstract":"<div><div>The problem of finding weight matrices <span><math><mi>W</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> of size <span><math><mi>N</mi><mo>×</mo><mi>N</mi></math></span> such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner Problem, and it is closely related to Darboux transformations of some differential operators.</div><div>This paper aims to study Darboux transformations between weight matrices and to establish a direct connection with the structure of the algebra <span><math><mi>D</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span> of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to <em>W</em> as eigenfunctions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 203-232"},"PeriodicalIF":1.0000,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525000023","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
The problem of finding weight matrices of size such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner Problem, and it is closely related to Darboux transformations of some differential operators.
This paper aims to study Darboux transformations between weight matrices and to establish a direct connection with the structure of the algebra of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to W as eigenfunctions.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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