Asymptotics for varying discrete Sobolev orthogonal polynomials
We consider a varying discrete Sobolev inner product such as $$(f,g)_S=\int f(x)g(x)d \mu+M_nf^{(j)}(c)g^{(j)}(c),$$ where $\mu$ is a finite positive Borel measure supported on an infinite subset of the real line, $c$ is adequately located on the real axis, $j \geq0,$ and $\{M_n\}_{n\geq0}$ is a...
| Main Authors: | , , |
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
| Published: |
2024
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| Subjects: | |
| Online Access: | http://hdl.handle.net/10835/15008 |
| _version_ | 1789406762192863232 |
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| author | Mañas Mañas, Juan Francisco Marcellán Español, Francisco Moreno Balcázar, Juan José |
| author_facet | Mañas Mañas, Juan Francisco Marcellán Español, Francisco Moreno Balcázar, Juan José |
| author_sort | Mañas Mañas, Juan Francisco |
| collection | DSpace |
| description | We consider a varying discrete Sobolev inner product such as
$$(f,g)_S=\int f(x)g(x)d \mu+M_nf^{(j)}(c)g^{(j)}(c),$$
where $\mu$ is a finite positive Borel measure supported on an infinite subset of the real line, $c$ is adequately located on the real axis, $j \geq0,$ and $\{M_n\}_{n\geq0}$ is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler--Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around $c,$ just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler--Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-15008 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2024 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-150082024-01-09T12:28:13Z Asymptotics for varying discrete Sobolev orthogonal polynomials Mañas Mañas, Juan Francisco Marcellán Español, Francisco Moreno Balcázar, Juan José Sobolev orthogonal polynomials Mehler–Heine formulae Asymptotics Zeros We consider a varying discrete Sobolev inner product such as $$(f,g)_S=\int f(x)g(x)d \mu+M_nf^{(j)}(c)g^{(j)}(c),$$ where $\mu$ is a finite positive Borel measure supported on an infinite subset of the real line, $c$ is adequately located on the real axis, $j \geq0,$ and $\{M_n\}_{n\geq0}$ is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler--Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around $c,$ just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler--Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones. 2024-01-09T12:28:13Z 2024-01-09T12:28:13Z 2017-12-01 info:eu-repo/semantics/article Juan F. Mañas-Mañas, Francisco Marcellán, Juan J. Moreno-Balcázar. Asymptotics for varying discrete Sobolev orthogonal polynomials , Appl. Math. Comput. 314 (2017), 65–79 0096-3003 http://hdl.handle.net/10835/15008 en https://doi.org/10.1016/j.amc.2017.06.020 Grants MTM2015-65888–C04-2-P, MTM2014-53963-P and P11-FQM-7276 Attribution-NonCommercial-NoDerivatives 4.0 Internacional http://creativecommons.org/licenses/by-nc-nd/4.0/ info:eu-repo/semantics/openAccess Juan F. Mañas-Mañas, Francisco Marcellán, Juan J. Moreno-Balcázar. Asymptotics for varying discrete Sobolev orthogonal polynomials , Appl. Math. Comput. 314 (2017), 65–79 |
| spellingShingle | Sobolev orthogonal polynomials Mehler–Heine formulae Asymptotics Zeros Mañas Mañas, Juan Francisco Marcellán Español, Francisco Moreno Balcázar, Juan José Asymptotics for varying discrete Sobolev orthogonal polynomials |
| title | Asymptotics for varying discrete Sobolev orthogonal polynomials |
| title_full | Asymptotics for varying discrete Sobolev orthogonal polynomials |
| title_fullStr | Asymptotics for varying discrete Sobolev orthogonal polynomials |
| title_full_unstemmed | Asymptotics for varying discrete Sobolev orthogonal polynomials |
| title_short | Asymptotics for varying discrete Sobolev orthogonal polynomials |
| title_sort | asymptotics for varying discrete sobolev orthogonal polynomials |
| topic | Sobolev orthogonal polynomials Mehler–Heine formulae Asymptotics Zeros |
| url | http://hdl.handle.net/10835/15008 |
| work_keys_str_mv | AT manasmanasjuanfrancisco asymptoticsforvaryingdiscretesobolevorthogonalpolynomials AT marcellanespanolfrancisco asymptoticsforvaryingdiscretesobolevorthogonalpolynomials AT morenobalcazarjuanjose asymptoticsforvaryingdiscretesobolevorthogonalpolynomials |