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

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive m...

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Bibliographic Details
Published inSymmetry (Basel) Vol. 13; no. 4; p. 534
Main Authors Garza, Lino G., Garza, Luis E., Huertas, Edmundo J.
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.04.2021
Subjects
Approximation
Asymptotic properties
Coefficients
Difference equations
Mathematical analysis
Norms
orthogonal polynomials
Polynomials
Real numbers
sobolev type orthogonal polynomials
symmetric weights
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Summary:In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(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−x4dx.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13040534