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March, 2003 Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms
Aline Bonami, Bruno Demange, Philippe Jaming
Rev. Mat. Iberoamericana 19(1): 23-55 (March, 2003).

Abstract

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\mathbb{R}^d$ which may be written as $P(x)\exp (-\langle Ax, x\rangle)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x) \widehat{f}(y)$. We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

Citation

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Aline Bonami. Bruno Demange. Philippe Jaming. "Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms." Rev. Mat. Iberoamericana 19 (1) 23 - 55, March, 2003.

Information

Published: March, 2003
First available in Project Euclid: 31 March 2003

zbMATH: 1037.42010
MathSciNet: MR1993414

Subjects:
Primary: 32A15, 42B10, 94A12

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.19 • No. 1 • March, 2003
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