Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory
[Polynômes orthogonaux matriciels du type de Jacobi : le rôle de la théorie de représentations des groupes]
Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2051-2068.

Le résultat principal du présent article est la construction d’une nouvelle famille de polynômes orthogonaux matriciels, du type de Jacobi. Ces polynômes proviennent du groupe sous-jacent SU(n) et de ses représentations : ce sont des fonctions propres d’un opérateur différentiel symétrique du second ordre, hypergéométrique et à valeurs matricielles. Le résultat final est valable pour des valeurs arbitraires des paramètres α,β>-1, mais est dérivé uniquement pour les valeurs provenant de la théorie des groupes.

The main purpose of this paper is to present new families of Jacobi type matrix valued orthogonal polynomials obtained from the underlying group SU(n) and its representations. These polynomials are eigenfunctions of some symmetric second order hypergeometric differential operator with matrix coefficients. The final result holds for arbitrary values of the parameters α,β>-1, but it is derived only for those values that come from the group theoretical setup.

DOI : 10.5802/aif.2151
Classification : 33C45, 22E45
Keywords: Matrix valued orthogonal polynomials, Jacobi polynomials
Mot clés : polynômes orthogonaux matriciels, polynômes de Jacobi
Grünbaum, F. Alberto 1 ; Pacharoni, Inés  ; Alfredo Tirao, Juan 

1 University of California, department of mathematics, Berkeley CA 94705 (USA), Universidad Nacional de Córdoba, CIEM-FaMAF, Córdoba 5000 (Argentine)
@article{AIF_2005__55_6_2051_0,
     author = {Gr\"unbaum, F. Alberto and Pacharoni, In\'es and Alfredo Tirao, Juan},
     title = {Matrix valued orthogonal polynomials of {Jacobi} type: the role of group representation theory},
     journal = {Annales de l'Institut Fourier},
     pages = {2051--2068},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     doi = {10.5802/aif.2151},
     mrnumber = {2187945},
     zbl = {1082.33006},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2151/}
}
TY  - JOUR
AU  - Grünbaum, F. Alberto
AU  - Pacharoni, Inés
AU  - Alfredo Tirao, Juan
TI  - Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2051
EP  - 2068
VL  - 55
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2151/
DO  - 10.5802/aif.2151
LA  - en
ID  - AIF_2005__55_6_2051_0
ER  - 
%0 Journal Article
%A Grünbaum, F. Alberto
%A Pacharoni, Inés
%A Alfredo Tirao, Juan
%T Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory
%J Annales de l'Institut Fourier
%D 2005
%P 2051-2068
%V 55
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2151/
%R 10.5802/aif.2151
%G en
%F AIF_2005__55_6_2051_0
Grünbaum, F. Alberto; Pacharoni, Inés; Alfredo Tirao, Juan. Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2051-2068. doi : 10.5802/aif.2151. http://www.numdam.org/articles/10.5802/aif.2151/

[D1] A. Duran Matrix inner product having a matrix symmetric second order differential operators, Rocky Mountain Journal of Mathematics, Volume 27 (1997) no. 2, pp. 585-600 | DOI | MR | Zbl

[D2] A. Duran On orthogonal polynomials with respect to a positive definite matrix of measures, Canadian J. Math., Volume 47 (1995), pp. 88-112 | DOI | MR | Zbl

[D3] A. Duran Markov Theorem for orthogonal matrix polynomials, Canadian J. Math., Volume 48 (1996) no. 4i, pp. 1180-1195 | MR | Zbl

[D4] A. Duran Ratio Asymptotics for orthogonal matrix polynomials, J. Approx. Th., Volume 100 (1999), pp. 304-344 | DOI | MR | Zbl

[DG] J. J. Duistermaat; F. A. Grünbaum Differential equations in the spectral parameter, Comm. Math. Phys., Volume 103 (1986), pp. 177-240 | DOI | MR | Zbl

[DG1] A. J. Duran; F. A. Grünbaum Orthogonal matrix polynomials satisfying second order differential equations, International Math. Research Notices, Volume 10 (2004), pp. 461-484 | MR | Zbl

[DG2] A. J. Duran; F. A. Grünbaum A characterization for a class of weight matrices with orthogonal matrix polynomials satisfying second order differential equations (To appear in Math. Research Notices) | MR | Zbl

[DP] A. Duran; B. Polo Gaussian quadrature formulae for matrix weights, Linear Algebra and Appl., Volume 355 (2002), pp. 119-146 | DOI | MR | Zbl

[DvA] A. Duran; van W. Assche Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra and Its Applications, Volume 219 (1995), pp. 261-280 | DOI | MR | Zbl

[G] F. A. Grünbaum Matrix valued Jacobi polynomials, Bull. Sciences Math., Volume 127 (2003) no. 3, pp. 207-214 | DOI | MR | Zbl

[Ge] J. S. Geronimo Scattering theory and matrix orthogonal polynomials on the real line, Circuits Systems Signal Process, Volume 1 (1982), pp. 471-495 | DOI | MR | Zbl

[GPT1] F. A. Grünbaum; I. Pacharoni; J. Tirao Matrix valued spherical functions associated to the complex projective plane, J. Functional Analysis, Volume 188 (2002), pp. 350-441 | DOI | MR | Zbl

[GPT2] F. A. Grünbaum; I. Pacharoni; J. Tirao A matrix valued solution to Bochner's problem, J. Physics A: Math. Gen., Volume 34 (2001), pp. 10647-10656 | DOI | MR | Zbl

[GPT3] F. A. Grünbaum; I. Pacharoni; J. Tirao Matrix valued spherical functions associated to the three dimensional hyperbolic space, Internat. J. of Mathematics, Volume 13 (2002), pp. 727-784 | DOI | MR | Zbl

[GPT4] F. A. Grünbaum; I. Pacharoni; J. Tirao; D. Healyand D. Rockmore An invitation to matrix valued spherical functions: Linearization of products in the case of the complex projective space P 2 (), Modern Signal Processing (MSRI publication), Volume 46 (2003), pp. 147-160 | Zbl

[GPT5] F. A. Grünbaum; I. Pacharoni; J. Tirao Matrix valued orthogonal polynomials of the Jacobi type, Indag. Mathem., Volume 14 (2003) no. 3,4, pp. 353-366 | MR | Zbl

[GV] R. Gangolli; V. S. Varadarajan Harmonic analysis of spherical functions on real reductive groups, Series title: Ergebnisse der Mathematik und ihrer Grenzgebiete, 101, Springer-Verlag, Berlin, New York, 1988 | MR | Zbl

[K1] M. G. Krein Fundamental aspects of the representation theory of Hermitian operators with deficiency index (m,m) (AMS Translations, Series 2), Volume 97 (1971), pp. 75-143 | Zbl

[K2] M. G. Krein Infinite J-matrices and a matrix moment problem, Dokl. Akad. Nauk SSSR, Volume 69 (1949) no. 2, pp. 125-128 | MR

[PT] I. Pacharoni; J. Tirao Matrix valued spherical functions associated to the complex projective space (in preparation)

[SvA] A. Sinap; van W. Assche Orthogonal matrix polynomials and applications, J. Comput. and Applied Math., Volume 66 (1996), pp. 27-52 | DOI | MR | Zbl

[T1] J. Tirao Spherical functions, Rev. de la Unión Matem. Argentina, Volume 28 (1977), pp. 75-98 | MR | Zbl

[T2] J. Tirao The matrix valued hypergeometric equation, Proc. Nat. Acad. Sci. U.S.A, Volume 100 (2003) no. 14, pp. 8138-8141 | DOI | MR | Zbl

[VK] N. Vilenkin; A. Klimyk Representation of Lie Groups and Special functions, Kluwer Academic, Dordrecht, MA, Volume 3 (1992) | Zbl

Cité par Sources :