Orthonormal sequences in $L^2(R^d)$ and time frequency localization

TL;DR

This paper investigates the limitations of orthonormal bases in $L^2( ^d)$ concerning time-frequency localization, proving that certain boundedness conditions cannot be simultaneously satisfied, thus extending classical uncertainty principles.

Contribution

It establishes new bounds on orthonormal bases in $L^2( ^d)$ related to time-frequency localization, introducing a novel inequality with multiple applications.

Findings

No orthonormal basis in $L^2( )$ has uniformly bounded means and dispersions.

A new time-frequency localization inequality for orthonormal sequences is proven.

The results connect to and extend classical uncertainty principles.

Abstract

We study 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. In particular we prove that there is no orthonormal basis for $L^2(\R)$ for which the time and frequency means as well as the product of dispersions are uniformly bounded. The problem is related to recent results of J. Benedetto, A. Powell, and Ph. Jaming. Our main tool is a time frequency localization inequality for orthonormal sequences in $L^2(\R^d)$. It has various other applications.