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An uncertainty principle for Hankel transforms

Authors: Margit Rösler and Michael Voit
Journal: Proc. Amer. Math. Soc. 127 (1999), 183-194
MSC (1991): Primary 44A15; Secondary 43A62, 26D10, 33C45
MathSciNet review: 1459147
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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 $[0,\infty[.$

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Additional Information

Margit Rösler
Affiliation: Mathematisches Institut, Technische Universität München, Arcisstr. 21, 80333 München, Germany

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

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
Article copyright: © Copyright 1999 American Mathematical Society