Revista Matemática Iberoamericana


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Volume 19, Issue 1, 2003, pp. 23–55
DOI: 10.4171/RMI/337

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

Aline Bonami[1], Bruno Demange[2] and Philippe Jaming[3]

(1) Département de Mathématiques, Université d'Orléans, B. P. 6759, 45067, ORLÉANS CEDEX 2, FRANCE
(2) Département de Mathématiques, Université d'Orléans, B. P. 6759, 45067, ORLÉANS CEDEX 2, FRANCE
(3) Département de Mathématiques, Université d'Orléans, B. P. 6759, 45067, ORLÉANS CEDEX 2, FRANCE

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.

Keywords: Uncertainty principles, short-time Fourier transform, windowed Fourier transform, Gabor transform, ambiguity function, Wigner transform, spectrogramm

Bonami Aline, Demange Bruno, Jaming Philippe: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19 (2003), 23-55. doi: 10.4171/RMI/337