A partial digestion of the HRT counterexample

A function of one variable can be translated in space by a spatial shift to obtain a new function and also modulated in frequency by a frequency shift to obtain a new function One can compose these two operations to obtain a time-frequency shift: (This can be viewed of as a portion of the Weyl representation of the Heisenberg group, but we will not adopt a representation-theoretic perspective here.) Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid obeys the relation However, the Heil-Ramanathan-Topiwala (HRT) conjecture states that once one imposes some reasonable decay condition on , no such relations exist: Conjecture 1 (HRT conjecture) If is non-zero, then there is no relation of the form for some distinct points and some coefficients , not all zero. A special case of the HRT conjecture, which was also open, makes the additional assumption that was Schwartz. Many positive results of this conjecture were known. I will mention…