Abstract:

The subject of the talk is uncertainty principles for orthonormal bases and sequences in L^{2}(R^{d}). As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform.

We will prove that there is no orthonormal basis for L^{2}(R) for which the time and frequency means as well as the products of dispersions are uniformly bounded. The problem is related to recent results of J. Benedetto, A. Powell, and Ph. Jaming.

A higher dimensional version of the above result will be presented. It involves p-dispersions with p > d (here d is the dimension of the space). Our main tool is a time-frequency localization inequality for orthonormal sequences.

On the other hand, for p ≤ d we show that the situation is different. We use the construction from a theorem of Bourgain to prove, in particular, that for d > 1 there exists an orthonormal basis for L^{2}(R^{d}) with uniformly bounded time and frequency means and uniformly bounded products of variations.