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The radial part of a class of Sobolev polynomials on the unit ball

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Abstract

In this work, a Sobolev inner product on the unit ball of \(\mathbb {R}^{d}\) involving the outward normal derivative is considered. A basis of mutually orthogonal polynomials associated with this inner product is constructed in terms of spherical harmonics and a radial part obtained from a family of univariate polynomials orthogonal with respect to a Sobolev inner product. The properties of this family constitute our main subject of study. In particular, we deduce algebraic properties, connection formulas, and some asymptotic properties. Finally, we show some numerical tests to illustrate the behavior of the roots of these univariate non-standard orthogonal polynomials.

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Acknowledgments

The authors thank the anonymous referees for their careful revision of the manuscript. Their valuable comments and suggestions have contributed to improve this work.

First author (F.L.) thanks MINECO research project MTM2017-83816-P, and grant 21.SI01.64658 from Banco de Santander.

Second and third authors (T.E.P. and M.A.P.) thank FEDER/MCIyU – AEI through the grant PGC2018-094932-B-I00, and Junta de Andalucía research group FQM-384.

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Authors and Affiliations

  1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005, Santander, Spain

    Fátima Lizarte

  2. Departamento de Matemática Aplicada & Instituto de Matemáticas (IEMath – GR), Universidad de Granada, 18071, Granada, Spain

    Teresa E. Pérez & Miguel A. Piñar

Authors
  1. Fátima Lizarte
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  2. Teresa E. Pérez
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  3. Miguel A. Piñar
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Correspondence to Miguel A. Piñar.

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Lizarte, F., Pérez, T.E. & Piñar, M.A. The radial part of a class of Sobolev polynomials on the unit ball. Numer Algor 87, 1369–1389 (2021). https://doi.org/10.1007/s11075-020-01011-7

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  • DOI: https://doi.org/10.1007/s11075-020-01011-7

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