Orthogonal polynomials and diffusion operators

Dominique Bakry; Stepan Orevkov; Marguerite Zani

doi:10.5802/afst.1693

Dominique Bakry ¹; Stepan Orevkov ¹; Marguerite Zani ²

¹ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
² Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, Route de Chartres, B.P. 6759, 45067, Orléans cedex 2, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 5, pp. 985-1073.

Abstract
Abstract

We study the following problem: describe the triplets $(Ω, g, μ)$ where $g = (g^{i j} (x))$ is the (co)metric associated with the symmetric second order differential operator $L (f) = \frac{1}{ρ} \sum_{i j} \partial_{i} (g^{i j} ρ \partial_{j} f)$ defined on a domain $Ω$ of $ℝ^{d}$ (that is $L$ is a diffusion operator with reversible measure $μ (d x) = ρ (x) d x$ ) and such that there exists an orthonormal basis of $ℒ^{2} (μ)$ made of polynomials which are at the same time eigenvectors of $L$ , where the polynomials are ranked according to their natural degree. We reduce this problem to a certain algebraic problem (for any $d$ ) and we find all solutions for $d = 2$ when $Ω$ is compact. Namely, in dimension $d = 2$ , and up to affine transformations, we find $10$ compact domains $Ω$ plus a one-parameter family. The proof that this list is exhaustive relies on the Plücker-like formulas for the projective dual curves applied to the complexification of $\partial Ω$ . We then describe some geometric origins for these various models. We also give some description of the non-compact cases in this dimension.

Nous considérons le problème suivant : décrire les triplets $(Ω, g, μ)$ où $g = (g^{i j} (x))$ est la (co)métrique associée à l’opérateur différentiel du second ordre symétrique $L (f) = \frac{1}{ρ} \sum_{i j} \partial_{i} (g^{i j} ρ \partial_{j} f)$ défini sur un domaine $Ω$ de $ℝ^{d}$ (i.e. $L$ est un opérateur de diffusion de mesure réversible $μ (d x) = ρ (x) d x$ ) et tels qu’il existe une base orthonormale de polynômes de $ℒ^{2} (μ)$ qui sont également vecteurs propres de $L$ , les polynômes étant classés par ordre croissant de leur degré naturel. Nous réduisons ce problème à un problème algébrique (pour tout $d$ ) et décrivons les solutions pour $d = 2$ et $Ω$ compact. Nous montrons que pour $d = 2$ , et à transformations affines près, il y a $10$ domaines compacts $Ω$ et une famille à un paramètre. La preuve de l’exhaustivité de ce classement repose sur des formules de type Plücker pour les courbes duales projectives appliquées à la complexification de $\partial Ω$ . Nous présentons alors une interprétation géométrique de ces différents modèles. Nous donnons également une description des cas non-compacts en dimension $d = 2$ .

Received: 2015-12-09
Accepted: 2020-02-13
Published online: 2022-03-11

DOI: 10.5802/afst.1693

Author's affiliations:

Dominique Bakry ¹; Stepan Orevkov ¹; Marguerite Zani ²

¹ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
² Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, Route de Chartres, B.P. 6759, 45067, Orléans cedex 2, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Dominique Bakry and Stepan Orevkov and Marguerite Zani},
     title = {Orthogonal polynomials and diffusion operators},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {985--1073},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Dominique Bakry; Stepan Orevkov; Marguerite Zani. Orthogonal polynomials and diffusion operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 5, pp. 985-1073. doi : 10.5802/afst.1693. https://afst.centre-mersenne.org/articles/10.5802/afst.1693/

References
Cited by

[1] Pavel S. Alexandrov Combinatorial Topology, Vol 1,2 and 3, Dover Publications, 1998 (reprint of 1956, 1957 and 1960)

[2] George E. Andrews; Richard Askey; Ranjan Roy Special functions, Encyclopedia of Mathematics and Its Applications, 71, Cambridge University Press, 1999, xvi+664 pages | DOI

[3] Shôrô Araki On root systems and an infinitesimal classification of irreducible symmetric spaces, Osaka J. Math., Volume 13 (1962), pp. 1-34 | MR | Zbl

[4] Dominique Bakry; Xavier Bressaud Diffusion with polynomial eigenvectors via finite subgroups of $O (3)$ , Ann. Fac. Sci. Toulouse, Math., Volume 25 (2016) no. 2-3, pp. 683-721 | DOI | MR | Zbl

[5] Dominique Bakry; Ivan Gentil; Michel Ledoux Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2013

[6] Dominique Bakry; Olivier Mazet Characterization of Markov semigroups on $ℝ$ associated to some families of orthogonal polynomials, Séminaire de Probabilités XXXVII (Lecture Notes in Mathematics), Volume 1832, Springer, 2003, pp. 60-80 | DOI | MR | Zbl

[7] Dominique Bakry; Marguerite Zani Dyson processes associated with associative algebras: the Clifford case, Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 2011–2013 (Lecture Notes in Mathematics), Volume 2116, Springer, 2014, pp. 1-37 | DOI | MR | Zbl

[8] Christian Berg; J. P. Reus Christensen Density questions in the classical theory of moments, Ann. Inst. Fourier, Volume 31 (1981) no. 3, pp. 99-114 | DOI | Numdam | MR | Zbl

[9] Salomon Bochner Über Sturm-Liouvillesche Polynomsysteme, Math. Z., Volume 29 (1929) no. 1, pp. 730-736 | DOI | MR | Zbl

[10] Boele L. J. Braaksma; Barend Meulenbed Jacobi polynomials as spherical harmonics, Nederl. Akad. Wet., Proc., Ser. A, Volume 71 (1968), pp. 384-389

[11] Egbert Brieskorn; Horst Knörrer Plane algebraic curves, Springer, 1986 | DOI

[12] Henri C. Brinkman; Frederik Zernike Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome, Nederl. Akad. Wet., Proc., Ser. A, Volume 38 (1935), pp. 161-170 | Zbl

[13] Élie Cartan Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl., Volume 17 (1938), pp. 177-191 | DOI | Zbl

[14] Élie Cartan Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z., Volume 45 (1939), pp. 335-367 | DOI | Zbl

[15] Élie Cartan Sur quelques familles remarquables d’hypersurfaces, C. R. Congrès Math. Liège, 1939, pp. 30-41 | Zbl

[16] Élie Cartan Sur des familles d’hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions, Rev., Ser. A, Univ. Nac. Tucumán, Volume A1 (1940), pp. 5-22 | Zbl

[17] John H. Conway; Derek A. Smith On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, 2003 | DOI

[18] Aad Dijksma; Tom Koornwinder Spherical harmonics and the product of two Jacobi polynomials, Nederl. Akad. Wet., Proc., Ser. A, Volume 74 (1971), pp. 191-196 | MR | Zbl

[19] Alexandru Dimca On the de Rham cohomology of a hypersurface complement, Am. J. Math., Volume 113 (1991) no. 4, pp. 763-771 | DOI | MR | Zbl

[20] Yan Doumerc Matrix Jacobi Process, Ph. D. Thesis, Université Toulouse 3 (2005)

[21] Charles Dunkl Reflection groups and orthogonal polynomials on the sphere, Math. Z., Volume 197 (1988) no. 1, pp. 33-60 | DOI | MR | Zbl

[22] Charles Dunkl Differential–difference operators associated to reflection groups, Trans. Am. Math. Soc., Volume 311 (1989) no. 1, pp. 167-183 | DOI | MR | Zbl

[23] Charles Dunkl; Yuan Xu Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, 81, Cambridge University Press, 2001 | DOI

[24] Anatolij Dvurčenskij; Pekka Lahti; Kari Ylinen The uniqueness question in the multidimensional moment problem with applications to phase space observables, Rep. Math. Phys., Volume 50 (2002) no. 1, pp. 55-68 | DOI | MR | Zbl

[25] Lidia Fernández; Teresa E. Pérez; Miguel A. Piñar Classical orthogonal polynomials in two variables: a matrix approach, Numer. Algorithms, Volume 39 (2005) no. 1-3, pp. 131-142 | DOI | MR | Zbl

[26] Bengt Fornberg A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics, 1, Cambridge University Press, 1998

[27] Gert-Martin Greuel; Christoph Lossen; E. Shustin Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, 2007

[28] Phillip A. Griffiths On the periods of certain rational integrals: I, II, Ann. Math., Volume 90 (1969), p. 460-495, 496–541 | DOI | MR | Zbl

[29] Benyu Guo; Jie Shen; Li-Lian Wang Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. Sci. Comput., Volume 27 (2006) no. 1-3, pp. 305-322 | MR | Zbl

[30] Benyu Guo; Jie Shen; Li-Lian Wang Generalized Jacobi polynomials/functions and their applications, Appl. Numer. Math., Volume 59 (2009) no. 5, pp. 1011-1028 | MR | Zbl

[31] Benyu Guo; Zhengsu Wan; Zhongqing Wang Jacobi pseudospectral method for fourth order problems, J. Comput. Math., Volume 24 (2006) no. 4, pp. 481-500 | MR | Zbl

[32] Wolfgang Hahn Über Orthogonalpolynome, die $q$ -Differenzengleichungen genügen, Math. Nachr., Volume 2 (1949), pp. 4-34 | DOI | MR | Zbl

[33] Peter Hall The Bootstrap and Edgeworth expansion, Springer Series in Statistics, Springer, 1992 | DOI

[34] Harish-Chandra Spherical Functions on a Semisimple Lie Group I, Am. J. Math., Volume 80 (1958), pp. 241-310 | DOI

[35] Harish-Chandra Spherical Functions on a Semisimple Lie Group II, Am. J. Math., Volume 80 (1958), pp. 553-613 | DOI | MR | Zbl

[36] Gert J. Heckman Root systems and hypergeometric functions II, Compos. Math., Volume 64 (1987), pp. 353-373 | Numdam | MR | Zbl

[37] Gert J. Heckman An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math., Volume 103 (1991) no. 2, pp. 341-350 | DOI | MR | Zbl

[38] Gert J. Heckman Dunkl operators, Séminaire Bourbaki. Volume 1996/97. Exposés 820–834 (Astérisque), Volume 245, Société Mathématique de France, 1997, pp. 223-246 | Numdam | Zbl

[39] Gert J. Heckman; Eric M. Opdam Root sytems and hypergeometric functions I, Compos. Math., Volume 64 (1987), pp. 329-352

[40] Gert J. Heckman; Henrik Schlichtkrull Harmonic Analysis and Special Functions on Symmetric Spaces, Perspectives in Mathematics, 16, Academic Press Inc., 1994

[41] Sigurdur Helgason Groups and Geometric Analysis. Integral geometry, invariant differential operators, and spherical functions, Pure and Applied Mathematics, 113, Academic Press Inc., 1984

[42] Tom Koornwinder The addition formula for Jacobi polynomials and spherical harmonics, SIAM J. Appl. Math., Volume 25 (1973), pp. 236-246 Lie algebras: applications and computational methods (Conf., Drexel Univ., Philadelphia, Pa., 1972) | DOI | MR | Zbl

[43] Tom Koornwinder Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I, Nederl. Akad. Wet., Proc., Ser. A, Volume 77 (1974), pp. 48-58 | Zbl

[44] Tom Koornwinder Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. II, Nederl. Akad. Wet., Proc., Ser. A, Volume 77 (1974), pp. 59-66 | Zbl

[45] Tom Koornwinder Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III, Nederl. Akad. Wet., Proc., Ser. A, Volume 77 (1974), pp. 357-369 | Zbl

[46] Tom Koornwinder Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. IV, Nederl. Akad. Wet., Proc., Ser. A, Volume 77 (1974), pp. 370-381 | Zbl

[47] Tom Koornwinder Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions. Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Volume 35 (1975) | MR | Zbl

[48] Tom Koornwinder; Alan L. Schwartz Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle, Constr. Approx., Volume 13 (1997) no. 4, pp. 537-567 | DOI | MR | Zbl

[49] Harry L. Krall; Isador M. Sheffer Orthogonal polynomials in two variables, Ann. Mat. Pura Appl., Volume 76 (1967), pp. 325-376 | DOI | MR

[50] Vik. S. Kulikov A remark on classical Pluecker’s formulae, Ann. Fac. Sci. Toulouse, Math., Volume 25 (2016) no. 5, pp. 959-967 | DOI | Numdam | MR | Zbl

[51] Solomon Lefschetz Algebraic Topology, Colloquium Publications, 27, American Mathematical Society, 1942

[52] Ian G. Macdonald Symmetric functions and orthogonal polynomials, University Lecture Series, 12, American Mathematical Society, 1998

[53] Ian G. Macdonald Orthogonal polynomials associated with root systems, Sémin. Lothar. Comb., Volume 45 (2000), B45a, 40 pages | MR | Zbl

[54] Olivier Mazet Classification des semi–groupes de diffusion sur $ℝ$ associés à une famille de polynômes orthogonaux, Séminaire de probabilités XXXI (Lecture Notes in Mathematics), Volume 1655, Springer, 1997, pp. 40-53 | DOI | MR | Zbl

[55] Burnett Meyer On the symmetries of spherical harmonics, Can. J. Math., Volume 6 (1954), pp. 135-157 | DOI | MR | Zbl

[56] John Milnor Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press, 1968

[57] María Álvarez de Morales; Lidia Fernández; Teresa E. Pérez; Miguel A. Piñar A matrix Rodrigues formula for classical orthogonal polynomials in two variables, J. Approx. Theory, Volume 157 (2009) no. 1, pp. 32-52 | DOI | MR | Zbl

[58] Arnold F. Nikiforov; Sergei K. Suslov; Vasili. B. Uvarov Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer, 1991 | DOI

[59] Eric M. Opdam Root systems and hypergeometric functions, III, IV, Compos. Math., Volume 67 (1988), p. 21-49, 191–209 | Numdam | Zbl

[60] Lev Pontrjagin The general topological theorem of duality for closed sets, Ann. Math., Volume 35 (1934) no. 4, pp. 904-914 | DOI | MR | Zbl

[61] Nicolas Privault Random Hermite polynomials and Girsanov identities on the Wiener space, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 13 (2010) no. 4, pp. 663-675 | DOI | MR | Zbl

[62] Margit Rösler Generalized Hermite polynomials and the heat equation for Dunkl operators, Commun. Math. Phys., Volume 192 (1998) no. 3, pp. 519-542 | MR | Zbl

[63] Margit Rösler Dunkl operators: Theory and applications, Orthogonal polynomials and special functions (Leuven, 2002) (Lecture Notes in Mathematics), Volume 1817, Springer, 2003 | DOI | MR | Zbl

[64] Thomas O. Sherman The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one, Acta Math., Volume 164 (1990) no. 1-2, pp. 73-144 | DOI | MR | Zbl

[65] Lev Soukhanov On the phenomena of constant curvature in the diffusion-orthogonal polynomials (2014) (https://arxiv.org/abs/1409.5332v1)

[66] Lev Soukhanov Diffusion-orthogonal polynomial systems of maximal weighted degree, Ann. Fac. Sci. Toulouse, Math., Volume 26 (2017) no. 2, pp. 511-518 | DOI | Numdam | MR | Zbl

[67] Ida G. Sprinkhuizen-Kuyper Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola, SIAM J. Math. Anal., Volume 7 (1976), pp. 501-518 | DOI | MR

[68] Elias M. Stein; Guido Weiss Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton University Press, 1971

[69] Pavel K. Suetin Orthogonal polynomials in two variables (Translated from the 1988 Russian original by E. V. Pankratiev), Analytical Methods and Special Functions, 3, Gordon and Breach Science Publishers, 1999

[70] Gabor Szegő Orthogonal polynomials, Colloquium Publications, 23, American Mathematical Society, 1975, xiii+432 pages | MR

[71] François Trèves Topological vector spaces, distributions and kernels, Academic Press Inc., 1967

[72] W. T. M. Verkley A spectral model for two-dimensional incompressible fluid flow in a circular basin. I, J. Comput. Phys., Volume 136 (1997) no. 1, pp. 110-114 | MR | Zbl

[73] W. T. M. Verkley A spectral model for two-dimensional incompressible fluid flow in a circular basin. II, J. Comput. Phys., Volume 136 (1997) no. 1, pp. 115-131 | DOI | MR | Zbl

[74] Luc Vinet; Alexei Zhedanov Generalized Bochner theorem: characterization of the Askey–Wilson polynomials, J. Comput. Appl. Math., Volume 211 (2008) no. 1, pp. 45-56 | DOI | MR | Zbl

[75] Robert J. Walker Algebraic curves, Princeton Mathematical Series, 13, Princeton University Press, 1950 | MR

[76] Paul Williams Jacobi Pseudospectral Method for Solving Optimal Control Problems, J. Guid. Control Dyn., Volume 27 (2004) no. 2, pp. 293-297 | DOI

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