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Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics
dc.contributor.author | Littlejohn, Lance Lee | |
dc.contributor.author | Mañas Mañas, Juan Francisco | |
dc.contributor.author | Moreno Balcázar, Juan José | |
dc.contributor.author | Wellman, Richard | |
dc.date.accessioned | 2024-01-18T08:16:36Z | |
dc.date.available | 2024-01-18T08:16:36Z | |
dc.date.issued | 2018-06 | |
dc.identifier.citation | 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. | es_ES |
dc.identifier.issn | 0021-9045 | |
dc.identifier.uri | http://hdl.handle.net/10835/15245 | |
dc.description.abstract | 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. | es_ES |
dc.language.iso | en | es_ES |
dc.rights | Attribution-NonCommercial-NoDerivatives 4.0 Internacional | * |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | * |
dc.source | 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. | es_ES |
dc.subject | Sobolev orthogonality | es_ES |
dc.subject | differential operators | es_ES |
dc.subject | asymptotics | es_ES |
dc.title | Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics | es_ES |
dc.type | info:eu-repo/semantics/article | es_ES |
dc.relation.publisherversion | https://doi.org/10.1016/j.jat.2018.04.008 | es_ES |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | es_ES |
dc.relation.projectID | Grant MTM2014-53963-P and grant P11-FQM-7276 | es_ES |