Paley-Wiener theorems for the dunkl transform

Transactions of the American Mathematical Society

Volumn 358, Issue 10, 2006, Pages 4225-4250

  De Jeu, Marcel ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

Author keywords

Cartan motion group; Dunkl operator; Graded Hecke algebra; Multivariable Bessel function; Paley Wiener theorem; Spherical function

Indexed keywords

EID: 33749529572     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-06-03960-2     Document Type: Article

References (42)

1. M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions, Dover Publications, New York, 1970. MR0167642 (29:4914) (review of the 1964 original edition)
2. S. Ben Saïd and B. Ørsted, Analysis on flat symmetric spaces, J. Math. Pures Appl. 84 (2005), 1393-1426. MR2170768
3. S. Ben Saïd and B. Ørsted, Bessel functions for root systems via the trigonometric setting, Int. Math. Res. Not. 2005 (9), 551-585. MR2131449 (2006d:33022)
4. I. Cherednik, A unification of Knizhnik-Zamolodchikov equations and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411-432. MR1128220 (93b:17040)
5. J.F. van Diejen and L. Vinet, Calogero-Moser-Sutherland Models, CRM Series in Mathematical Physics, Springer, 2000. MR1843558 (2002b:37003)
6. C.F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60. MR0917849 (89b:42016)
7. C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311, (1989), 167-183. MR0951883 (90k:33027)
8. C.F. Dunkl, Operators commuting with Coxeter group actions on polynomials, in "Invariant theory and tableaux," Minneapolis, 1988, IMA Vol. Math. Appl. 19, 107-177. MR1035491 (91g:20060)
9. C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227. MR1145585 (93g:33012)
10. 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, 123-138. MR1199124 (94g:33011)
11. 3, Trans. Amer. Math. Soc. 347 (1995), 3347-3374. MR1316848 (97b:22009)
12. C.F. Dunkl, Singular polynomials for the symmetric groups, Int. Math. Res. Not. 2004 (67), 3607-3635. MR2129695
13. C.F. Dunkl, Singular polynomials and modules for the symmetric groups, Int. Math. Res. Not. 2005 (39), 2409-2436. MR2181357
14. C.F. Dunkl, M.F.E. de Jeu and E.M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256. MR1273532 (96b:33012)
15. C.F. Dunkl and E.M. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70-108. MR1971464 (2004d:20040)
16. C.F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Cambridge Univ. Press, 2001. MR1827871 (2002m:33001)
17. G.J. Heckman, A remark on the Dunkl differential-difference operators, in "Harmonic analysis on reductive groups," Brunswick, 1989, Progr. Math. 101, 181-191. MR1168482 (94c:20075)
18. G.J. Heckman, Dunkl operators, Séminaire Bourbaki 828, 1996-97,
19. Astérisque 245 (1997), 223-246. MR1627113 (2000i:33021)
20. S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. MR0754767 (86c:22017)
21. S. Helgason, Geometric analysis on symmetric spaces, Amer. Math. Soc., Providence, RI, 1994. MR1280714 (96h:43009)
22. L. Hörmander, The analysis of linear partial differential operators I, Springer, 1983. MR0717035 (85g:35002a)
23. M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147-162. MR1223227 (94m:22011)
24. M.F.E. de Jeu, Dunkl operators, Thesis, Leiden University, 1994.
25. M.F.E. de Jeu, Subspaces with equal closure, Constr. Approx. 20 (2004), 93-157. MR2025416 (2005e:41083)
26. G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-685. MR0991016 (90e:16049)
27. E.M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), 1-18. MR1010152 (91h:33024)
28. E.M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Comp. Math. 85 (1993), 333-373. MR1214452 (95j:33044)
29. E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121. MR1353018 (98f:33025)
30. E.M. Opdam, Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs 8, Math. Soc. of Japan, Tokyo, 2000. MR1805058 (2003c:33001)
31. M. Rösler, Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463. MR1695797 (2000f:33013)
32. M. Rösler, Dunkl operators: theory and applications, in "Orthogonal polynomials and special functions," Leuven, 2002, Lecture Notes in Math. 1817, 93-135. MR2022853 (2004k:33024)
33. M. Rösler and M.F.E. de Jeu, Asymptotic analysis for the Dunkl kernel, J. Approx. Theory 119 (2002), 110-126. MR1934628 (2003j:33035)
34. M. Rösler and M. Voit, Markov processes related with Dunkl operators, Adv. Appl. Math. 21 (1998), 575-643. MR1652182 (2000j:60019)
35. W. Rudin, Functional analysis, Tata McGraw-Hill, New Delhi, 1973. MR0365062 (51:1315)
36. L. Schwartz, Théorie des distributions, Hermann, Paris, 1978. MR0209834 (35:730)
37. S. Thangavelu and Y. Xu, Generalized translation and convolution operator for the Dunkl transform, J. Anal. Math. 97 (2005), 25-56.
38. E.C. Titchmarsh, The theory of functions, Oxford Univ. Press, 1986.
39. C. Torossian, Une application des opérateurs de Dunkl au théorème de restriction de Chevalley, C.R. Acad. Sci. Paris 318 (1994), 895-898. MR1278147 (95h:22010)
40. K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transform. Spec. Funct. 12 (2001), 349-374. MR1872375 (2002m:33012)
41. K. Trimèche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transform. Spec. Funct. 13 (2002), 17-38. MR1914125 (2004e:44006)
42. Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973. MR1402890 (97m:33004)