Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics
We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^{\alpha}dx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],$$ where $\alpha>-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Our main objective is to calculat...
| Main Authors: | , , , |
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| 格式: | info:eu-repo/semantics/article |
| 语言: | English |
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2024
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| 在线阅读: | http://hdl.handle.net/10835/15245 |
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| author | Littlejohn, Lance L. Mañas Mañas, Juan Francisco Moreno Balcázar, Juan José Wellman, Richard |
| author_facet | Littlejohn, Lance L. Mañas Mañas, Juan Francisco Moreno Balcázar, Juan José Wellman, Richard |
| author_sort | Littlejohn, Lance L. |
| collection | DSpace |
| description | We consider the following discrete Sobolev inner product involving the Gegenbauer weight
$$(f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^{\alpha}dx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],$$
where $\alpha>-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Our main objective is to calculate the exact value
$$r_0 = \lim_{n\rightarrow \infty}\frac{\log \left(\max_{x\in [-1,1]} |\widetilde{Q}_n^{(\alpha,M,j)}(x)|\right)}{\log \widetilde{\lambda}_n}, \quad \alpha\ge -1/2,$$
where $\{\widetilde{Q}_n^{(\alpha,M,j)}\}_{n\geq0}$ is the sequence of orthonormal polynomials with respect to this Sobolev inner product. These polynomials are eigenfunctions of a differential operator and the obtaining of the asymptotic behavior of the corresponding eigenvalues, $\widetilde{\lambda}_n$ , is the principal key to get the result. This value $r_0$ is related to the convergence of a series in a left--definite space. In addition, to complete the asymptotic study of this family of nonstandard polynomials we give the Mehler--Heine formulae for the corresponding orthogonal polynomials. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-15245 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2024 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-152452024-01-18T08:16:37Z Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics Littlejohn, Lance L. Mañas Mañas, Juan Francisco Moreno Balcázar, Juan José Wellman, Richard Sobolev orthogonality differential operators asymptotics We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^{\alpha}dx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],$$ where $\alpha>-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Our main objective is to calculate the exact value $$r_0 = \lim_{n\rightarrow \infty}\frac{\log \left(\max_{x\in [-1,1]} |\widetilde{Q}_n^{(\alpha,M,j)}(x)|\right)}{\log \widetilde{\lambda}_n}, \quad \alpha\ge -1/2,$$ where $\{\widetilde{Q}_n^{(\alpha,M,j)}\}_{n\geq0}$ is the sequence of orthonormal polynomials with respect to this Sobolev inner product. These polynomials are eigenfunctions of a differential operator and the obtaining of the asymptotic behavior of the corresponding eigenvalues, $\widetilde{\lambda}_n$ , is the principal key to get the result. This value $r_0$ is related to the convergence of a series in a left--definite space. In addition, to complete the asymptotic study of this family of nonstandard polynomials we give the Mehler--Heine formulae for the corresponding orthogonal polynomials. 2024-01-18T08:16:36Z 2024-01-18T08:16:36Z 2018-06 info:eu-repo/semantics/article Lance L . Littlejohn, Juan F. Mañas Mañas, Juan J. Moreno Balcázar and Richard Wellman. Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics, J. Approx. Theory. 230 (2018), 32--49. 0021-9045 http://hdl.handle.net/10835/15245 en https://doi.org/10.1016/j.jat.2018.04.008 Grant MTM2014-53963-P and grant P11-FQM-7276 Attribution-NonCommercial-NoDerivatives 4.0 Internacional http://creativecommons.org/licenses/by-nc-nd/4.0/ info:eu-repo/semantics/openAccess Lance L . Littlejohn, Juan F. Mañas Mañas, Juan J. Moreno Balcázar and Richard Wellman. Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics, J. Approx. Theory. 230 (2018), 32--49. |
| spellingShingle | Sobolev orthogonality differential operators asymptotics Littlejohn, Lance L. Mañas Mañas, Juan Francisco Moreno Balcázar, Juan José Wellman, Richard Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics |
| title | Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics |
| title_full | Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics |
| title_fullStr | Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics |
| title_full_unstemmed | Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics |
| title_short | Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics |
| title_sort | differential operator for discrete gegenbauer--sobolev orthogonal polynomials: eigenvalues and asymptotics |
| topic | Sobolev orthogonality differential operators asymptotics |
| url | http://hdl.handle.net/10835/15245 |
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