Codes and Expansions (CodEx) Seminar

Markus Faulhuber (University of Vienna)
Linear dependence of time-frequency shifts and the HRT conjecture

The Heil-Ramanathan-Topiwala (HRT) conjecture asserted that every finite collection of distinct time-frequency shifts of a nonzero square-integrable function is linearly independent. We begin by exploring the conjecture and some important settings in which it is known to hold. We will then follow the path to a general construction of a finite counterexample to the HRT conjecture with Schwartz-class regularity. Our results were soon followed by further work. Vignon gave a construction with the minimal number of four time-frequency shifts, while Jasper and Mixon achieved near-optimal decay of functions that can give a counterexample to the HRT conjecture.