An uncertainty principle for Hankel transforms
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- by Margit Rösler and Michael Voit PDF
- Proc. Amer. Math. Soc. 127 (1999), 183-194 Request permission
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 \[ \qquad \quad \quad T_\alpha f(x) = f^\prime (x) +\bigl (\alpha +\frac {1}{2}\bigr ) \frac {f(x)-f(-x)}{x} ,\quad f\in C^1(\mathbb {R}).\qquad \qquad \qquad \] 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 $[0,\infty [.$References
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Additional Information
- Margit Rösler
- Affiliation: Mathematisches Institut, Technische Universität München, Arcisstr. 21, 80333 München, Germany
- MR Author ID: 312683
- 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
- MR Author ID: 253279
- ORCID: 0000-0003-3561-2712
- Email: voit@uni-tuebingen.de
- 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 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 183-194
- MSC (1991): Primary 44A15; Secondary 43A62, 26D10, 33C45
- DOI: https://doi.org/10.1090/S0002-9939-99-04553-0
- MathSciNet review: 1459147