Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms
Bonami, Aline ; Demange, Bruno ; Jaming, Philippe
Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, p. 23-55 / Harvested from Project Euclid
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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.
Publié le : 2003-03-15
Classification:  Uncertainty principles,  short-time Fourier transform,  windowed Fourier transform,  Gabor transform,  ambiguity function,  Wigner transform,  spectrogramm,  42B10,  32A15,  94A12 
@article{1049123079,
     author = {Bonami, Aline and Demange, Bruno and Jaming, Philippe},
     title = {Hermite functions and uncertainty principles for the
 Fourier and the windowed Fourier transforms},
     journal = {Rev. Mat. Iberoamericana},
     volume = {19},
     number = {2},
     year = {2003},
     pages = { 23-55},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1049123079}
}

Bonami, Aline; Demange, Bruno; Jaming, Philippe. Hermite functions and uncertainty principles for the
 Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp.  23-55. http://gdmltest.u-ga.fr/item/1049123079/