Proceedings of the American Mathematical Society

Author(s): Margit Rösler; Michael Voit
Journal: Proc. Amer. Math. Soc. 127 (1999), 183-194.
MSC (1991): Primary 44A15; Secondary 43A62, 26D10, 33C45
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Abstract: There exists a generalized Hankel transform of order $\alpha\geq -1/2$ on $\mathbb{R}$ , which is based on the eigenfunctions of the Dunkl operator

$\begin{displaymath}\qquad\quad\quad T_\alpha f(x) \,=\, f^\prime(x) +\bigl(\alpha+\frac{1}{2}\bigr) \frac{f(x)-f(-x)}{x}\,,\quad f\in C^1(\mathbb{R}).\qquad\qquad\qquad\end{displaymath}$

For $\alpha=-1/2$ this transform coincides with the usual Fourier transform on $\mathbb{R}$ . In this paper the operator $T_\alpha$ replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on $\mathbb{R}$ . It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on $L^2(\mathbb{R})$ ; moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order $\alpha\geq -1/2$ on $[0,\infty[.$

1.: M. Abramowitz, I.A. Stegun: Handbook of Mathematical Functions. Dover Publ. 1990.
2.: W.R. Bloom, H. Heyer: Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter 1995. MR 96a:43001
3.: V.M. Bukhstaber, G. Felder, A.P. Veselov: Elliptic Dunkl operators, root systems, and functional equations. Duke Math. J. 76, 885-911 (1994). MR 96b:39014
4.: D.L. Donoho, P.B. Stark: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49, 906 - 931 (1989). MR 90c:42003
5.: C. F. Dunkl: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311, 167-183 (1989). MR 90k:33027
6.: C. F. Dunkl: Integral kernels with reflection group invariance. Can. J. Math. 43, 1213 - 1227 (1991). MR 93g:33012
7.: C. F. Dunkl: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123 - 138 (1992). MR 94g:33011
8.: H. Dym, H.P. McKean: Fourier Series and Integrals. Academic Press 1972. MR 56:945
9.: M.F.E. de Jeu: The Dunkl transform. Invent. Math. 113, 147 - 162 (1993). MR 94m:22011
10.: M.F.E. de Jeu: An uncertainty principle for integral operators. J. Funct. Anal. 122, 247 - 253 (1994). MR 95h:43009
11.: S. Kamefuchi, Y. Ohnuki: Quantum field theory and parastatistics. University of Tokyo Press, Springer-Verlag, 1982. MR 85b:81001
12.: Y. Ohnuki, S. Watanabe: Selfadjointness of the operators in Wigner's commutation relations. J. Math. Phys. 33 (11), 3653-3665 (1992). MR 93h:81065
13.: E.M. Opdam: Dunkl Operators, Bessel Functions and the discriminant of a finite Coxeter group. Compos. Math. 85, 333-373 (1993). MR 95j:33044
14.: M. Rösler: Convolution algebras which are not necessarily positivity-preserving. Contemp. Math. 183, 299-318 (1995). MR 96c:43006
15.: M. Rösler: On the dual of a commutative signed hypergroup. Manuscr. Math. 88, 147-163 (1995). MR 96j:43004
16.: M. Rösler: Bessel-type signed hypergroups on $\mathbb R$ . In: Probability Measures on Groups and Related Structures (Proc. Conf., Oberwolfach, 1994, Ed.: H. Heyer, A. Mukherjea), pp. 292 - 304, World Scientific 1995. MR 97j:43004
17.: M. Rösler, M. Voit: An uncertainty principle for ultraspherical expansions. J. Math. Anal. Appl. 1997, to appear. CMP 97:15
18.: M. Rosenblum: Generalized Hermite polynomials and the Bose-like oscillator calculus. In: Operator Theory: Advances and Applications, Vol. 73, pp. 369 - 396, Birkhäuser Verlag 1994. MR 96b:33005
19.: R.S. Strichartz: Uncertainty principles in harmonic analysis. J. Funct. Anal. 84, 97 - 114 (1989). MR 91a:42017
20.: M. Voit: An uncertainty principle for commutative hypergroups and Gelfand pairs. Math. Nachr. 164, 187 - 195 (1993). MR 95d:43005
21.: S. Watanabe: Sobolev type theorems for an operator with singularity. Proc. Amer. Math. Soc. 125, 129-136 (1997). MR 97c:47044
22.: E.T. Whittaker, G.N. Watson: A Course of Modern Analysis. Cambridge University Press, 1935.

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 44A15, 43A62, 26D10, 33C45

Margit Rösler
Affiliation: Mathematisches Institut, Technische Universität München, Arcisstr. 21, 80333 München, Germany
Email: roesler@mathematik.tu-muenchen.de

Michael Voit
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany, and Department of Mathematics, University of Virginia, Kerchof Hall, Charlottesville, Virginia, 22903-3199
Email: voit@uni-tuebingen.de

DOI: 10.1090/S0002-9939-99-04553-0
PII: S 0002-9939(99)04553-0
Keywords: Heisenberg-Weyl inequality, Hankel transform, Dunkl operators, hypergroups
Received by editor(s): October 14, 1996
Received by editor(s) in revised form: May 7, 1997
Additional Notes: This paper was partially written at the University of Virginia, Charlottesville, while the first author held a Forschungsstipendium of the DFG
Communicated by: J. Marshall Ash
Copyright of article: Copyright 1999, American Mathematical Society

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