Linear dependence of time-frequency shifts of a Schwartz function

Abstract

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

arXiv commentThis version (v2) adds a separate, purely analytic proof of a qualitative counterexample, clarifying the construction mechanism; the certified numerical proof of the explicit 12-point counterexample is retained. Supplementary Python code available in the anc folder