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: As per the time-frequency uncertainty principle, the two shifts do not quite commute with each other. For instance, we have As such, is not a representation of the abelian group , but rather a portion of the Weyl representation of the Heisenberg group, ...