zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Dunkl transform. (English) Zbl 0789.33007
In the late eighties, Dunkel found a remarkable set of commuting operators that can be associated with a finite real reflection group. The operators contain complex parameters; if these parameters are all zero, Dunkl’s operators reduce to the ordinary directional derivatives. The operators have been studied by Dunkl, who obtained (amongst others) fairly detailed information about their action on polynomials. The present paper is concerned with the spectral problem for the Dunkl operators in the case that the real parts of the parameters are all nonnegative. We obtain estimates for the simultaneous eigenfunctions of the Dunkl operators and prove an inversion theorem and Plancherel formula for the associated integral transform. Plancherel-type results were obtained earlier by Dunkl, who exhibited an orthonormal basis for the L 2 -space involved that consists of eigenfunctions for the transform with eigenvalues in {±1,±i}. Our method of proof is different and is almost exclusively based on exploiting the formal properties of the transform. The Dunkl transform contains Fourier analysis and the harmonic analysis for the Cartan motion group as a special case. This connection is explained in the paper (without proof).
Reviewer: M.F.E.de Jeu

MSC:
33C80Connections of hypergeometric functions with groups and algebras
44A15Special transforms (Legendre, Hilbert, etc.)
42B10Fourier type transforms, several variables
References:
[1][D1] Dunkl, C.F.: Reflection groups and orthogonal polyomials on the sphere. Math. Z.197, 33-60 (1988) · Zbl 0616.33005 · doi:10.1007/BF01161629
[2][D2] Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc.311, 167-183 (1989) · doi:10.1090/S0002-9947-1989-0951883-8
[3][D3] Dunkl, C.F.: Operators commuting with Coxeter groups actions on polynomials. In: Stanton, D. (ed.). Invariant Theory and Tableaux, pp. 107-117. Berlin Heidelberg New York: Springer, 1990
[4][D4] Dunkl, C.F.: Integral kernels with reflection group invariance. Can J. Math.43, 1213-1227 (1991) · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8
[5][D5] Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceedings of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications at AMS meeting in Tampa, Fa March 22-23. (Contemp. Math.138 (1992) Providence, RI: Am. Math. Soc.
[6][Hec] Heckman, G.J.: A remark on the Dunkl differential-difference operators. In: Proceedings of the Bowdoin conference on reductive groups, 1989.
[7][Hel] Helgason, S.: Groups and Geometric Analysis. New York London: Academic Press, 1984
[8][M] Macdonald, I.G.: Some conjectures for root systems. SIAM J. Math. Anal.13, 988-1007 (1982) · Zbl 0498.17006 · doi:10.1137/0513070
[9][O] Opdam, E.M.: Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group. Compos. Math.85, 333-373 (1993)
[10][R] Rudin, W.: Functional analysis. New Delhi: Tata McGraw-Hill 1973
[11][S] Sneddon, I.N.: Fourier Transforms. Toronto, London New York: McGraw-Hill 1951
[12][T] Treves, F.: Topological vector spaces, distributions and kernels. New York London: Academic Press, 1967
[13][Hec] Heckman, G.J.: A remark on the Dunkl differential-difference operators. In: Barker, W., Sally, P. (eds.) Harmonic analysis on reductive groups (Progress in Mathematics101). Boston Basel Berlin: Birkhäuser, 1991