A partial digestion of the HRT counterexample
This article discusses the resolution of the HRT conjecture, a long-standing mathematical problem in time-frequency analysis. The author notes that the proof was assisted by AI, marking a significant milestone in mathematical research.
Why it matters
It highlights the growing role of AI in solving complex mathematical problems and the evolving standards for academic transparency.
A function of one variable can be translated in space by a spatial shift to obtain a new function
Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid obeys the relation
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 .
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