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

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

2001-02-14

arxiv.org/abs/math/0102111v1

Mathematics

Classical Analysis and ODEs

22 pages, submitted

Scientific paper

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be written as $P(x)\exp (Ax,x)$, with $A$ a real symmetric definite positive matrix, are characterized by integrability conditions on the product $f(x)\hat{f}(y)$. We also give the best constant in uncertainty principles of Gelf'and Shilov type. 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.

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