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Radial orthogonality and Lebesgue constants on the disk

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Abstract

In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Universiteit Antwerpen (CMI), Middelheimlaan 1, 2020, Antwerpen, Belgium

    Annie Cuyt & Bayram Ali Ibrahimoglu

  2. Department of Mathematics, Faculty of Science, Gebze Institute of Technology, Çayırova Campus, 41400, Gebze, Kocaeli, Turkey

    Irem Yaman

  3. Department of Mathematical Engineering, Yıldız Technical University, Davutpasa Campus, 34210, Istanbul, Turkey

    Bayram Ali Ibrahimoglu

  4. Département de Mathématiques, Faculté des Sciences et Techniques, Université Hassan II, BP 146 Yasmina, 20650, Mohammadia, Morocco

    Brahim Benouahmane

Authors
  1. Annie Cuyt
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  2. Irem Yaman
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  3. Bayram Ali Ibrahimoglu
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  4. Brahim Benouahmane
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Correspondence to Irem Yaman.

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Cuyt, A., Yaman, I., Ibrahimoglu, B.A. et al. Radial orthogonality and Lebesgue constants on the disk. Numer Algor 61, 291–313 (2012). https://doi.org/10.1007/s11075-012-9615-5

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  • DOI: https://doi.org/10.1007/s11075-012-9615-5

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