### Pompeya II

- Project funded by the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF) Coordinated with the Research Project MTM2011-28952-C02-01.

### Abstract

In this Project, we will study the properties of the multivariate orthogonal polynomial systems in the standard and non standard case, that is, orthgonal polynomial systems associated with inner products involving derivation operators in several variables. The results will be used to find the fundamental characteristics of such a systems: structure, asymptotics, and others. The main objectives of this Project can be resume as follows: 1. Development of techniques for analyzing families of multivariate orthogonal polynomials started in the previous Project. 2. Study of the analytic and structural properties for those polynomials. 3. Extension to several variables of the theory of orthogonal polynomial systems associated with inner product involving derivarion operators (the so-called Sobolev orthogonality.) 4. Aplication of the constructive approximation methods in several variables to clinic problems related with human vision as well as computered exploration techniques.### Code

- MTM2011-28952-C02-02 (subproject of MTM2011-28952-C02).

### Duration

- Begining 01/01/2012, end 12/31/2015.

### Coordinator

### Members

### Publications

### Submitted

- A. M. Delgado, L. Fernández, T. E. Pérez, M. Piñar, Multivariate orthogonal polynomials and
modified moment functionals.
- A. M. Delgado, L. Fernández, D. S. Lubinsky, T. E. Pérez, M. Piñar, Sobolev orthogonal polynomials on the unit ball via outward derivatives.
- F. Marcellán, M. Marriaga, T. E. Pérez, M. Piñar, Matrix Pearson equations satisfied by Koornwinder weights in
two variables.
- C. Martínez, M. Piñar, Orthogonal Polynomials on the unit ball and fourth order partial differential
equations.

- A. M. Delgado, L. Fernández, T. E. Pérez, M. Piñar, Multivariate orthogonal polynomials and
modified moment functionals.
### To appear

- 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.*

doi:10.1090/mcom3041

- C. F. Bracciali, J. H. McCabe, T. E. Pérez, A. Sri Ranga, A class of orthogonal functions given
by a three term recurrence formula.
### Year 2015

- L. Fernández, F. Marcellán, T. E. Pérez, M. Piñar, Y. Xu, Sobolev orthogonal polynomials on product domains.
*J. Comput. Appl. Math.***284**(2015), 202-215

MR3319504 - H. Dueñas, L. Garza, M. Piñar, A higher order Sobolev-type inner product for orthogonal
polynomials in several variables,
*Numer. Algorithms***68**(2015), 35-46.

MR3296697 - Y. Xu, Uncertainty principle on weighted spheres, balls and simplexes,
*J. Approx. Theory***192**(2015), 193–214

MR3313480 - Y. Xu, Complex versus real orthogonal polynomials of two variables,
*Integral Transforms Spec. Funct.***26**(2015), no. 2, 134-151

MR3275456

- L. Fernández, F. Marcellán, T. E. Pérez, M. Piñar, Y. Xu, Sobolev orthogonal polynomials on product domains.
### Year 2014

- M. Alfaro, A. Peña, T. E. Pérez, M. L. Rezola, On linearly related orthogonal polynomials in several variables.
*Numer. Algorithms***66**(2014), 537-553.

MR3225001 - R. K. Beatson, W. zu Castell, Y. Xu, A Pólya criterion for (strict) positive-definiteness on the
sphere.
*IMA J. Numer. Anal.***34**(2014), no. 2, 550-568.

MR3194799 - F. Dai, Y. Xu, The Hardy-Rellich inequality and uncertainty principle on the sphere.
*Constr. Approx.***40**(2014), 141-171.

MR3229998 - M. Sawa, Y. Xu, On positive cubature rules on the simplex and isometric embeddings.
*Math. Comp.***83**(2014), no. 287, 1251-1277.

MR3167458 - Y. Xu, Tight frame with Hahn and Krawtchouk polynomials of several variables.
*SIGMA Symmetry Integrability Geom. Methods Appl.***10**(2014), Paper 019, 19 pp.

MR3210616

- M. Alfaro, A. Peña, T. E. Pérez, M. L. Rezola, On linearly related orthogonal polynomials in several variables.
### Year 2013

- R. Aktaş, Y. Xu, Sobolev orthogonal polynomials on a simplex
*Int. Math. Res. Not. IMRN 2013***13**, 3087-3131.

MR3073001 - C. F. Bracciali, T. E. Pérez, M. A. Piñar, Stieltjes functions and discrete classical orthogonal
polynomials.
*Comput. Appl. Math.***32**(2013), 537-547.

MR3120139 - A. M. Delgado, T. E. Pérez, M. A. Piñar, Sobolev-type orthogonal polynomials on the unit ball.
*J. Approx. Theory***170**(2013), 94-106.

MR3044047 - T. E. Pérez, M. A. Piñar, Y. Xu, Weighted Sobolev orthogonal polynomials on the unit ball.
*J. Approx. Theory***171**(2013), 84-104.

MR3053718 - B. Rider, C. Sinclair, Y. Xu, A solvable mixed charge ensemble on the line: global results.
*Probab. Theory Related Fields***155**(2013), no. 1-2, 127–164.

MR3010395

- R. Aktaş, Y. Xu, Sobolev orthogonal polynomials on a simplex
### Year 2012

- A. M. Delgado, L. Fernández, T. E. Pérez, M. A. Piñar, On the Uvarov modification of two variable
orthogonal polynomials on the disk, Complex Anal. Oper. Theory 6 (2012), 665-676.

MR2944078 - L. Fernández, T. E. Pérez, M. A. Piñar, On Koornwinder classical orthogonal polynomials in two
variables, J. Comp. Appl. Math. 236 (2012), 3817-3826.

MR2923514 - K. Ivanov, P. Petrushev, Y. Xu, Decomposition of spaces of distributions induced by tensor
product bases, J. Funct. Anal. 263 (2012), no. 5, 1147-1197.

MR2943727 - H. Li, J. Sun, Y. Xu, Discrete Fourier analysis and Chebyshev polynomials with G2 group. SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), Paper 067, 29 pp.
- Y. Xu, Orthogonal polynomials and expansions for a family of weight functions in two variables,
Constr. Approx. 36 (2012), no. 2, 161-190.

MR2957307 - Y. Xu, Minimal cubature rules and polynomial interpolation in two variables, J. Approx. Theory
164 (2012), no. 1, 6-30.

MR2855767

- A. M. Delgado, L. Fernández, T. E. Pérez, M. A. Piñar, On the Uvarov modification of two variable
orthogonal polynomials on the disk, Complex Anal. Oper. Theory 6 (2012), 665-676.