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

Abstract

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) 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.

Uncertainty principles for the weinstein transform

“…A far reaching generalization of this result has been recently proved by Bonami, Demange and Jaming [6]. They proved that a square-integrable function f on R d satisfying for an integer N ,…”

Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups

“…The above result is true even if ab = 1 / 4 under the added assumption that min(p, q) < oo. A further generalization of the Cowling-Price theorem for the case ab = 1/4 has been recently obtained by Bonami et al [4].…”

On Theorems of Hardy, Gelfand–Shilov and Beurling for Semisimple Groups

“…Recently Bonami et al [6] have established a general version of this theorem for the Fourier transform on IR n . Here we establish the following result for semisimple Lie groups.…”

“…The Euclidean analogues of Theorems 1.1 and 1.2 can be deduced from a much stronger result known as Beurling's theorem which was established in Bonami et al [6] in the most general form. This result says that if a function f on IR n and its Fourier transformf satisfy the integrability condition…”