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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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.

Article information

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

Dates
First available in Project Euclid: 31 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1049123079

Mathematical Reviews number (MathSciNet)
MR1993414

Zentralblatt MATH identifier
1037.42010

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

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

Citation

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. https://projecteuclid.org/euclid.rmi/1049123079


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