GoKawiilA 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 shiftto 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 towards this conjecture were known. I will mention only a few here. There is a result of Linnell that the conjecture is true if lie in a translate of a discrete subgroup of ; this (together with an argument handling the collinear case) establishes all cases where , and several partial results involving the cases are also known. The conjecture is also known if is decays at a suitably super-exponential rate, by work of Bownik and Speegle.
I was aware of this conjecture through various talks and conversations with colleagues, and even briefly tried my hand at it for a while, though not with particularly serious effort (or progress). It was thus a nice surprise to see that it has just been resolved by Faulhuber, Petersen, van Velthoven, and Voigtlaender, even in the Schwartz case:
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