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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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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 $[0,\infty [.$
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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