# Bases of time-frequency shifts and the uncertainty principle

# Bases of time-frequency shifts and the uncertainty principle

October 22 2018

Kasso Okoudjou - Applied Math Colloquium

Kasso Okoudjou, 2018-19 MIT MLK Visiting Professor

Professor, Department of Mathematics & Norbert Wiener Center for Harmonic Analysis and Applications, University of Maryland

Abstract

The Balian-Low Theorem (BLT) is an uncertainty principle-type result that precludes the existence of a Gabor orthonormal basis (ONB) of the form $\{e^{2\pi i kx/a }g(x-an) \}_{k, n=-\infty}^{\infty},$ where $a>0$, and $g$ is well-localized in phase space. A related ONB with a well-localized generator (hence does not obey the BLT) was numerically introduced by K. Wilson in the 80s, and formalized by Daubechies, Jaffard, and Journ\’e. The latter system is called a Wilson basis and was recently featured in the detection of the gravitational waves.

In the first part of the talk, I will review some basic structures as well as the relationship between these two systems. I will then present some recent and ongoing work on constructing Wilson-type systems from more general Gabor families.

(This is a joint work with D. Bhimani, M. Bownik, M. Jakobsen, and J. Lemvig.)