Research Project of the National Plan of I+D

• Funded by the Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund (ERDF)
  Call of 2014, type 1: Projects I+D of the National Program for Development of Scientific Research and Technical Excellence, Subprogram for Generation of Knowledge.

Code and duration

• MTM2014-53171-P. Duration from January 1st, 2015 until December 31th, 2017. Extended until December 31th, 2018.

Coordinator

• Dr. Miguel A. Piñar, Department of Applied Mathematics, University of Granada.

Researchers

• Dra. Antonia M. Delgado, Dep. of Applied Mathematics, University of Granada.

• Dra. Lidia Fernández, Department of Applied Mathematics, University of Granada.

• Dra. Clotilde Martínez, Dpto. de Matemática Aplicada, Universidad de Granada.

• Dra. Teresa E. Pérez, Department of Applied Mathematics, University of Granada.

• Dr. Yuan Xu, Department of Mathematics, University of Oregon.

Abstract

In this Research Project, properties of orthogonal polynomial systems associated to standard and non-standard inner products will be studied. The non-standard case is usually called Sobolev since inner products involve multivariate derivation operators such as gradient, divergence and others. The main objectives of the projects are

1. Influence of differential properties of classical (and semi-classical) orthogonal polynomials on recurrence, common zeros and asymptotic of the multivariate orthogonal polynomials.

2. Asymptotic properties of multivariate orthogonal polynomials associated with measures obtained by rational transformations from classical measures. Connection to three terms relation.

3. Analytical and asymptotic properties of multivariate Sobolev polynomials, i.e., orthogonal polynomials associated with multivariate scalar products involving differential operators.

4. Routines for symbolic and numerical computations in multivariate orthogonal polynomials.

The study of the properties for multivariate orthogonal polynomials proposed in this project aims to be a continuation of studies begun in the previous projects. (MTM2011-28952-C02-02, MTM2008-06689-C02-02).

Publications

• Submitted

  1. A. M. Delgado, P. Iliev, L. Fernández, Darboux transformation from the Appell-Lauricella operator.
  2. F. Lizarte, T. E. Pérez, M. A. Piñar, The radial part of a class of Sobolev polynomials on the unit ball.
  3. F. Marcellán, M. E. Marriaga, T. E. Pérez, M. A. Piñar, Coherent pairs in two variables.

• Year 2019

  1. C. Martínez, M. A. Piñar, Asymptotic behaviour of the Christoffel functions on the unit ball in the presence of a mass on the sphere. Medit. J. Math. 16 (2019), no. 1, Art. 22, 18 pp.
     MR3897518

• Year 2018

  1. A. M. Delgado, L. Fernández, T. E. Pérez, Fourth order partial differential equations for Krall-type orthogonal polynomials on the triangle. Proc. Amer. Math. Soc. 146 (2018), 3961-3974.
     MR3825849
  2. F. Marcellán, M. Marriaga, T. E. Pérez, M. A. Piñar, On bivariate classical orthogonal polynomials. Appl. Math. Comput. 325 (2018), 340-357.
     MR3759149
  3. F. Marcellán, M. Marriaga, T. E. Pérez, M. A. Piñar, Matrix Pearson equations satisfied by Koornwinder weights in two variables. Acta Appl. Math. 153 (2018), 81-100.
     MR3745731
  4. M. A. Piñar, Y. Xu, Best polynomial approximation on the unit ball. IMA J. Numer. Anal. 38 (2018), 1209-1228.
     MR3829159
  5. Y. Xu, Positive definite functions on the unit sphere and integrals of Jacobi polynomials. Proc. Amer. Math. Soc. 146 (2018), 2039-2048.
     MR3767355

• Year 2017

  1. C. F. Bracciali, T. E. Pérez, Bivariate orthogonal polynomials, 2D Toda lattices and Lax-type pairs. Appl. Math. Comput. 309 (2017), 142-155.
     MR3646384
  2. M. Marriaga, T. E. Pérez, M. A. Piñar, Three term relations for a class of bivariate polynomials. Medit. J. Math. 14 (2017), no. 2, Art. 54, 26 pp.
     MR3619416
  3. Y. Xu, Minimal cubature rules and polynomial interpolation in two variables II. J. Approx. Theory 214 (2017), 49-68.
     MR3588529
  4. Y. Xu, Approximation and orthogonality in Sobolev spaces on a triangle. Constr. Approx. 46 (2017), 349-434.
     MR3691233

• Year 2016

  1. C. F. Bracciali, J. H. McCabe, T. E. Pérez, A. Sri Ranga, A class of orthogonal functions given by a three term recurrence formula. Math. Comp. 85 (2016), 1837-1859.
     MR3471110
  2. A. M. Delgado, L. Fernández, D. S. Lubinsky, T. E. Pérez, M. A. Piñar, Sobolev orthogonal polynomials on the unit ball via outward derivatives. J. Math. Anal. Appl. 440 (2016), 716-740.
     MR3484991
  3. A. M. Delgado, L. Fernández, T. E. Pérez, M. A. Piñar, Multivariate orthogonal polynomials and modified moment functionals. SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), 090, 25 pages.
     MR3545477
  4. C. Martínez, M. A. Piñar, Orthogonal polynomials on the unit ball and fourth order partial differential equations. SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), 020, 11 pages.
     MR3463057
  5. Y. Xu, Positivity and Fourier integrals over regular hexagon. J. Approx. Theory 207 (2016), 193-206.
     MR3494229

• Year 2015

  1. H. Dueñas, L. Garza, M. A. Piñar, A higher order Sobolev-type inner product for orthogonal polynomials in several variables, Numer. Algorithms 68 (2015), 35-46.
     MR3296697
  2. L. Fernández, F. Marcellán, T. E. Pérez, M. A. Piñar, Y. Xu, Sobolev orthogonal polynomials on product domains. J. Comput. Appl. Math. 284 (2015), 202-215
     MR3319504
  3. Y. Xu, Uncertainty principle on weighted spheres, balls and simplexes, J. Approx. Theory 192 (2015), 193–214
     MR3313480
  4. Y. Xu, Complex versus real orthogonal polynomials of two variables, Integral Transforms Spec. Funct. 26 (2015), no. 2, 134-151
     MR3275456

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Contents

• Code and duration
• Coordinator
• Researchers
• Abstract
• Publications
1. Submitted
2. Year 2019
3. Year 2018
4. Year 2017
5. Year 2016
6. Year 2015