Volume 12, Issue 3
Sobolev Orthogonal Polynomials: Asymptotics and Symbolic Computation

Juan F. Mañas-Mañas & Juan J. Moreno-Balcázar

East Asian J. Appl. Math., 12 (2022), pp. 535-563.

Published online: 2022-04

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  • Abstract

The Sobolev polynomials, which are orthogonal with respect to an inner product involving derivatives, are considered. The theory about these nonstandard polynomials has been developed along the last 40 years. The local asymptotics of these polynomials can be described by the Mehler-Heine formulae, which connect the polynomials with the Bessel functions of the first kind. In recent years, the formulae have been computed for discrete Sobolev orthogonal polynomials in several particular cases. We improve various known results by unifying them. Besides, an algorithm to compute these formulae effectively is presented. The algorithm allows to construct a computer program based on Mathematica$^®$ language, where the corresponding Mehler-Heine formulae are automatically obtained. Applications and examples show the efficiency of the approach developed.

  • Keywords

Sobolev orthogonal polynomials, asymptotics, algorithm, computer program.

  • AMS Subject Headings

33F10, 33C47, 42C05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-12-535, author = {}, title = {Sobolev Orthogonal Polynomials: Asymptotics and Symbolic Computation}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {3}, pages = {535--563}, abstract = {

The Sobolev polynomials, which are orthogonal with respect to an inner product involving derivatives, are considered. The theory about these nonstandard polynomials has been developed along the last 40 years. The local asymptotics of these polynomials can be described by the Mehler-Heine formulae, which connect the polynomials with the Bessel functions of the first kind. In recent years, the formulae have been computed for discrete Sobolev orthogonal polynomials in several particular cases. We improve various known results by unifying them. Besides, an algorithm to compute these formulae effectively is presented. The algorithm allows to construct a computer program based on Mathematica$^®$ language, where the corresponding Mehler-Heine formulae are automatically obtained. Applications and examples show the efficiency of the approach developed.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.240221.130921}, url = {http://global-sci.org/intro/article_detail/eajam/20406.html} }
TY - JOUR T1 - Sobolev Orthogonal Polynomials: Asymptotics and Symbolic Computation JO - East Asian Journal on Applied Mathematics VL - 3 SP - 535 EP - 563 PY - 2022 DA - 2022/04 SN - 12 DO - http://doi.org/10.4208/eajam.240221.130921 UR - https://global-sci.org/intro/article_detail/eajam/20406.html KW - Sobolev orthogonal polynomials, asymptotics, algorithm, computer program. AB -

The Sobolev polynomials, which are orthogonal with respect to an inner product involving derivatives, are considered. The theory about these nonstandard polynomials has been developed along the last 40 years. The local asymptotics of these polynomials can be described by the Mehler-Heine formulae, which connect the polynomials with the Bessel functions of the first kind. In recent years, the formulae have been computed for discrete Sobolev orthogonal polynomials in several particular cases. We improve various known results by unifying them. Besides, an algorithm to compute these formulae effectively is presented. The algorithm allows to construct a computer program based on Mathematica$^®$ language, where the corresponding Mehler-Heine formulae are automatically obtained. Applications and examples show the efficiency of the approach developed.

Juan F. Mañas-Mañas & Juan J. Moreno-Balcázar. (2022). Sobolev Orthogonal Polynomials: Asymptotics and Symbolic Computation. East Asian Journal on Applied Mathematics. 12 (3). 535-563. doi:10.4208/eajam.240221.130921
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