Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms
Abstract
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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2001
 arXiv:
 arXiv:math/0102111
 Bibcode:
 2001math......2111B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 42B10;
 32A15;
 94A12
 EPrint:
 22 pages, submitted