Fritz Gesztesy, Lance L. Littlejohn, Mateusz Piorkowski, Jonathan Stanfill
{"title":"The Jacobi Operator on $$(-1,1)$$ and Its Various m-Functions","authors":"Fritz Gesztesy, Lance L. Littlejohn, Mateusz Piorkowski, Jonathan Stanfill","doi":"10.1007/s11785-024-01576-4","DOIUrl":null,"url":null,"abstract":"<p>We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression </p><span>$$\\begin{aligned} \\tau _{\\alpha ,\\beta } =&- (1-x)^{-\\alpha } (1+x)^{-\\beta }(d/dx) \\big ((1-x)^{\\alpha +1}(1+x)^{\\beta +1}\\big ) (d/dx), \\\\&\\alpha , \\beta \\in {\\mathbb {R}}, \\, x \\in (-1,1), \\end{aligned}$$</span><p>in <span>\\(L^2\\big ((-1,1); (1-x)^{\\alpha } (1+x)^{\\beta } dx\\big )\\)</span>, <span>\\(\\alpha , \\beta \\in {\\mathbb {R}}\\)</span>. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general <span>\\(\\eta \\)</span>-periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced <i>m</i>-functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.</p>","PeriodicalId":50654,"journal":{"name":"Complex Analysis and Operator Theory","volume":"34 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complex Analysis and Operator Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11785-024-01576-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression
in \(L^2\big ((-1,1); (1-x)^{\alpha } (1+x)^{\beta } dx\big )\), \(\alpha , \beta \in {\mathbb {R}}\). In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general \(\eta \)-periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced m-functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.
期刊介绍:
Complex Analysis and Operator Theory (CAOT) is devoted to the publication of current research developments in the closely related fields of complex analysis and operator theory as well as in applications to system theory, harmonic analysis, probability, statistics, learning theory, mathematical physics and other related fields. Articles using the theory of reproducing kernel spaces are in particular welcomed.