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From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals

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Orthogonal Polynomials (AIMSVSW 2018)

Abstract

In this contribution, we present an overview of standard orthogonal polynomials by using an algebraic approach. Discrete Darboux transformations of Jacobi matrices are studied. Next, we emphasize the role of semiclassical orthogonal polynomials as a basic background to analyze sequences of polynomials orthogonal with respect to a Sobolev inner product. Some illustrative examples are discussed. Finally, we summarize some results in multivariate Sobolev orthogonal polynomials.

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Acknowledgements

The authors acknowledge the careful revision of the manuscript by the referees. Their comments and suggestions have contributed to improve its presentation.

The work of J. C. García-Ardila, F. Marcellán and M. Marriaga has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía, Industria y Competitividad of España, research project [MTM2015-65888-C4-2-P].

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García-Ardila, J.C., Marcellán, F., Marriaga, M.E. (2020). From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_13

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