Transactions of the American Mathematical Society

Author(s): Margit Rösler
Journal: Trans. Amer. Math. Soc. 355 (2003), 2413-2438.
MSC (2000): Primary 33C52; Secondary 44A35, 35L15
Posted: January 14, 2003
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Abstract: It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

1.: M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions. Verlag Harri Deutsch, Frankfurt/Main, 1984. MR 85j:00005b
2.: R. Askey, Orthogonal Polynomials and Special Functions. SIAM, Philadelphia, 1975. MR 58:1288
3.: Y. Berest, The problem of lacunas and analysis on root systems, Trans. Amer. Math. Soc. 252 (2000), 3743-3776. MR 2001d:58030
4.: C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups. Springer-Verlag, 1975. MR 58:1204
5.: W. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter-Verlag, Berlin, 1994. MR 96a:43001
6.: C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183. MR 90k:33027
7.: C. F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213 - 1227. MR 93g:33012
8.: C. F. Dunkl, Hankel transforms associated to finite reflection groups, in ``Proceedings of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications,'' Tampa, 1991, Contemp. Math. 138 (1992), 123-138. MR 94g:33011
9.: C. F. Dunkl, Intertwining operators associated to the group , Trans. Amer. Math. Soc. 347 (1995), 3347-3374. MR 97b:22009
10.: C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables. Cambridge Univ. Press, 2001. MR 2002m:33001
11.: P. Graczyk and P. Sawyer, The product formula for the spherical functions on symmetric spaces in the complex case. Pacific J. Math. 204 (2002), 377-393.
12.: S. Helgason, Groups and Geometric Analysis. Academic Press, 1984; reprint, Amer. Math. Soc., 2000. MR 86c:22017; MR 2001h:22001
13.: L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer-Verlag, Berlin, 1983. MR 85g:35002a
14.: J. E. Humphreys, Reflection groups and Coxeter groups. Cambridge Univ. Press, 1990. MR 92h:20002
15.: M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147-162. MR 94m:22011
16.: M.F.E. de Jeu, Dunkl operators, Thesis, Leiden University, 1994.
17.: R. I. Jewett, Spaces with an abstract convolution of measures. Adv. Math. 18 (1975), 1-101. MR 52:14840
18.: J.F.C. Kingman, Random walks with spherical symmetry, Acta. Math. 109 (1965), 11-53. MR 26:7052
19.: H. Mejjaoli and K. Trimèche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transform. Spec. Funct. 12 (2001), 279-302. MR 2003a:33028
20.: E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333-373. MR 95j:33044
21.: M. Rösler, Bessel-type signed hypergroups on ${\mathbb{R} }$ , in: H. Heyer, A. Mukherjea (eds.), Probability measures on groups and related structures XI, Proc. Oberwolfach 1994. World Scientific, Singapore, 1995, 292-304. MR 97j:43004
22.: M. Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542. MR 99k:33048
23.: M. Rösler, Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463. MR 2000f:33013
24.: M. Rösler and M. Voit, Markov Processes related with Dunkl operators, Adv. Appl. Math. 21 (1998), 575-643. MR 2000j:60019
25.: M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, in: Operator Theory: Advances and Applications 73, Birkhäuser-Verlag, Basel, 1994, 369-396. MR 96b:33005
26.: G. Szegö, Orthogonal Polynomials. Amer. Math. Soc., New York, 1959. MR 21:5029
27.: K. Trimèche, Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur $(0,\infty)$ , J. Math. Pures et Appl. 60 (1981), 51-98. MR 83i:47058
28.: K. Trimèche, Paley-Wiener Theorems for the Dunkl transform and Dunkl translation operators, Integral Transform. Spec. Funct. 13 (2002), 17-38.
29.: G. N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, 1966. MR 96i:33010 (latest reprint)
30.: Y. Xu, Integration of the intertwining operator for -harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973. MR 97m:33004

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 33C52, 44A35, 35L15

Margit Rösler
Affiliation: Mathematisches Institut, Universität Göttingen, Bunsenstrasse 3--5, D-37073 Göttingen, Germany
Email: roesler@uni-math.gwdg.de

DOI: 10.1090/S0002-9947-03-03235-5
PII: S 0002-9947(03)03235-5
Keywords: Dunkl operators, Dunkl kernel, product formula, multivariable Bessel functions
Received by editor(s): October 2, 2002
Posted: January 14, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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