Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type BC
HTML articles powered by AMS MathViewer
- by Margit Rösler and Michael Voit PDF
- Trans. Amer. Math. Soc. 368 (2016), 6005-6032 Request permission
Abstract:
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields $\mathbb R, \mathbb C, \mathbb H.$ These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.References
- T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), no. 1, 175–216. MR 1471336, DOI 10.1007/s002200050161
- J. F. van Diejen, Asymptotics of multivariate orthogonal polynomials with hyperoctahedral symmetry, Jack, Hall-Littlewood and Macdonald polynomials, Contemp. Math., vol. 417, Amer. Math. Soc., Providence, RI, 2006, pp. 157–169. MR 2284126, DOI 10.1090/conm/417/07920
- Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489
- Ramesh Gangolli and V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 101, Springer-Verlag, Berlin, 1988. MR 954385, DOI 10.1007/978-3-642-72956-0
- G. J. Heckman, Dunkl operators, Astérisque 245 (1997), Exp. No. 828, 4, 223–246. Séminaire Bourbaki, Vol. 1996/97. MR 1627113
- Gerrit Heckman and Henrik Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16, Academic Press, Inc., San Diego, CA, 1994. MR 1313912
- Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions; Corrected reprint of the 1984 original. MR 1790156, DOI 10.1090/surv/083
- Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991. MR 1091716, DOI 10.1017/CBO9780511840371
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
- Marcel de Jeu, Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc. 358 (2006), no. 10, 4225–4250. MR 2231377, DOI 10.1090/S0002-9947-06-03960-2
- Robert I. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. MR 394034, DOI 10.1016/0001-8708(75)90002-X
- Jyoichi Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), no. 4, 1086–1110. MR 1226865, DOI 10.1137/0524064
- Tom H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions: group theoretical aspects and applications, Math. Appl., Reidel, Dordrecht, 1984, pp. 1–85. MR 774055
- Tom H. Koornwinder, Jacobi polynomials of type $BC$, Jack polynomials, limit transitions and $O(\infty )$, Mathematical analysis, wavelets, and signal processing (Cairo, 1994) Contemp. Math., vol. 190, Amer. Math. Soc., Providence, RI, 1995, pp. 283–286. MR 1354860, DOI 10.1090/conm/190/2310
- E. K. Narayanan, A. Pasquale, and S. Pusti, Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications, Adv. Math. 252 (2014), 227–259. MR 3144230, DOI 10.1016/j.aim.2013.10.027
- E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), no. 3, 333–373. MR 1214452
- Eric M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75–121. MR 1353018, DOI 10.1007/BF02392487
- A. L. Onishchik and È. B. Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. MR 1064110, DOI 10.1007/978-3-642-74334-4
- Margit Rösler, Bessel convolutions on matrix cones, Compos. Math. 143 (2007), no. 3, 749–779. MR 2330446, DOI 10.1112/S0010437X06002594
- Margit Rösler, Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type $BC$, J. Funct. Anal. 258 (2010), no. 8, 2779–2800. MR 2593343, DOI 10.1016/j.jfa.2009.12.007
- Margit Rösler, Tom Koornwinder, and Michael Voit, Limit transition between hypergeometric functions of type BC and type A, Compos. Math. 149 (2013), no. 8, 1381–1400. MR 3103070, DOI 10.1112/S0010437X13007045
- Margit Rösler and Michael Voit, Positivity of Dunkl’s intertwining operator via the trigonometric setting, Int. Math. Res. Not. 63 (2004), 3379–3389. MR 2098643, DOI 10.1155/S1073792804141901
- Margit Rösler and Michael Voit, A limit relation for Dunkl-Bessel functions of type A and B, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 083, 9. MR 2470513, DOI 10.3842/SIGMA.2008.083
- Margit Rösler and Michael Voit, Limit theorems for radial random walks on $p\times q$-matrices as $p$ tends to infinity, Math. Nachr. 284 (2011), no. 1, 87–104. MR 2752670, DOI 10.1002/mana.200710235
- Margit Rösler and Michael Voit, A central limit theorem for random walks on the dual of a compact Grassmannian, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 013, 18. MR 3313689, DOI 10.3842/SIGMA.2015.013
- P. Sawyer, Spherical functions on $\textrm {SO}_0(p,q)/\textrm {SO}(p)\times \textrm {SO}(q)$, Canad. Math. Bull. 42 (1999), no. 4, 486–498. MR 1727346, DOI 10.4153/CMB-1999-056-5
- Bruno Schapira, Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel, Geom. Funct. Anal. 18 (2008), no. 1, 222–250. MR 2399102, DOI 10.1007/s00039-008-0658-7
- Jasper V. Stokman and Tom H. Koornwinder, Limit transitions for BC type multivariable orthogonal polynomials, Canad. J. Math. 49 (1997), no. 2, 373–404. MR 1447497, DOI 10.4153/CJM-1997-019-9
- E. C. Titchmarsh, The theory of functions, Oxford University Press, Oxford, 1958. Reprint of the second (1939) edition. MR 3155290
- Michael Voit, Limit theorems for radial random walks on homogeneous spaces with growing dimensions, Infinite dimensional harmonic analysis IV, World Sci. Publ., Hackensack, NJ, 2009, pp. 308–326. MR 2581603, DOI 10.1142/9789812832825_{0}020
- Michael Voit, Central limit theorems for hyperbolic spaces and Jacobi processes on $[0,\infty [$, Monatsh. Math. 169 (2013), no. 3-4, 441–468. MR 3019294, DOI 10.1007/s00605-012-0460-3
- Michael Voit, Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type $BC$, J. Lie Theory 25 (2015), 9–36.
- Fuzhen Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 21–57. MR 1421264, DOI 10.1016/0024-3795(95)00543-9
Additional Information
- Margit Rösler
- Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
- MR Author ID: 312683
- Email: roesler@math.upb.de
- Michael Voit
- Affiliation: Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany
- MR Author ID: 253279
- ORCID: 0000-0003-3561-2712
- Email: michael.voit@math.tu-dortmund.de
- Received by editor(s): February 27, 2014
- Received by editor(s) in revised form: January 19, 2015
- Published electronically: June 17, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6005-6032
- MSC (2010): Primary 33C67, 43A90; Secondary 33C52, 22E46
- DOI: https://doi.org/10.1090/tran6673
- MathSciNet review: 3458405