An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Lino G. Garza; Luis E. Garza; Edmundo J. Huertas

doi:10.3390/sym13040534

An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Lino G. Garza, Luis E. Garza, Edmundo J. Huertas

Departamento de Física y Matemáticas

Research output: Contribution to journal › Article › peer-review

Abstract

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:〈p, q〉s =∫Rp(x)q(x)dµ(x) + M0 p(0)q(0) + M1 p′ (0)q′ (0), where p, q are polynomials, M0, M1 are non-negative real numbers and µ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dµ(x) = e−x4 dx.

Original language	English
Article number	534
Journal	Symmetry
Volume	13
Issue number	4
DOIs	https://doi.org/10.3390/sym13040534
Publication status	Published - Apr 2021

Bibliographical note

Funding Information:
Funding: The work of the first author was supported by Universidad de Monterrey under grant UIN19562. The work of the second author was supported by México’s Conacyt Grant 287523. The work of the third author (EJH) was supported by Dirección General de Investigación e Innovación, Consejería de Educación e Investigación of the Comunidad de Madrid (Spain), and Universidad de Alcalá under grant CM/JIN/2019-010, (Proyectos de I+D Para Jóvenes Investigadores de la Universidad de Alcalá 2019).

Funding Information:
The work of the first author was supported by Universidad de Monterrey under grant UIN19562. The work of the second author was supported by M?xico?s Conacyt Grant 287523. The work of the third author (EJH) was supported by Direcci?n General de Investigaci?n e Innovaci?n, Consejer?a de Educaci?n e Investigaci?n of the Comunidad de Madrid (Spain), and Universidad de Alcal? under grant CM/JIN/2019-010, (Proyectos de I+D Para J?venes Investigadores de la Universidad de Alcal? 2019).

Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.

All Science Journal Classification (ASJC) codes

Computer Science (miscellaneous)
Chemistry (miscellaneous)
General Mathematics
Physics and Astronomy (miscellaneous)

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10.3390/sym13040534

Cite this

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title = "An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials",

abstract = "In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:〈p, q〉s =∫Rp(x)q(x)dµ(x) + M0 p(0)q(0) + M1 p′ (0)q′ (0), where p, q are polynomials, M0, M1 are non-negative real numbers and µ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dµ(x) = e−x4 dx.",

author = "Garza, {Lino G.} and Garza, {Luis E.} and Huertas, {Edmundo J.}",

note = "Funding Information: Funding: The work of the first author was supported by Universidad de Monterrey under grant UIN19562. The work of the second author was supported by M{\'e}xico{\textquoteright}s Conacyt Grant 287523. The work of the third author (EJH) was supported by Direcci{\'o}n General de Investigaci{\'o}n e Innovaci{\'o}n, Consejer{\'i}a de Educaci{\'o}n e Investigaci{\'o}n of the Comunidad de Madrid (Spain), and Universidad de Alcal{\'a} under grant CM/JIN/2019-010, (Proyectos de I+D Para J{\'o}venes Investigadores de la Universidad de Alcal{\'a} 2019). Funding Information: The work of the first author was supported by Universidad de Monterrey under grant UIN19562. The work of the second author was supported by M?xico?s Conacyt Grant 287523. The work of the third author (EJH) was supported by Direcci?n General de Investigaci?n e Innovaci?n, Consejer?a de Educaci?n e Investigaci?n of the Comunidad de Madrid (Spain), and Universidad de Alcal? under grant CM/JIN/2019-010, (Proyectos de I+D Para J?venes Investigadores de la Universidad de Alcal? 2019). Publisher Copyright: {\textcopyright} 2021 by the authors. Licensee MDPI, Basel, Switzerland.",

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N1 - Funding Information: Funding: The work of the first author was supported by Universidad de Monterrey under grant UIN19562. The work of the second author was supported by México’s Conacyt Grant 287523. The work of the third author (EJH) was supported by Dirección General de Investigación e Innovación, Consejería de Educación e Investigación of the Comunidad de Madrid (Spain), and Universidad de Alcalá under grant CM/JIN/2019-010, (Proyectos de I+D Para Jóvenes Investigadores de la Universidad de Alcalá 2019). Funding Information: The work of the first author was supported by Universidad de Monterrey under grant UIN19562. The work of the second author was supported by M?xico?s Conacyt Grant 287523. The work of the third author (EJH) was supported by Direcci?n General de Investigaci?n e Innovaci?n, Consejer?a de Educaci?n e Investigaci?n of the Comunidad de Madrid (Spain), and Universidad de Alcal? under grant CM/JIN/2019-010, (Proyectos de I+D Para J?venes Investigadores de la Universidad de Alcal? 2019). Publisher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

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N2 - In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:〈p, q〉s =∫Rp(x)q(x)dµ(x) + M0 p(0)q(0) + M1 p′ (0)q′ (0), where p, q are polynomials, M0, M1 are non-negative real numbers and µ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dµ(x) = e−x4 dx.

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An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Abstract

Bibliographical note

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