Revista Matemática Iberoamericana

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

Aline Bonami, Bruno Demange, and Philippe Jaming

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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.

Article information

Rev. Mat. Iberoamericana, Volume 19, Number 1 (2003), 23-55.

First available in Project Euclid: 31 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 32A15: Entire functions 94A12: Signal theory (characterization, reconstruction, filtering, etc.)

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), no. 1, 23--55.

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