A Note on the HRT Conjecture and a New Uncertainty Principle for the Short-Time Fourier Transform
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(2020) 26:68
LETTER TO THE EDITORS
A Note on the HRT Conjecture and a New Uncertainty Principle for the Short-Time Fourier Transform Fabio Nicola1
· S. Ivan Trapasso1
Received: 4 December 2019 / Revised: 11 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this note we provide a negative answer to a question raised by Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary “bump with fat tail” profile. Keywords Short-time Fourier transform · Uncertainty principle · HRT conjecture Mathematics Subject Classification 42B10 · 42C15 · 42C30
1 Introduction A famous open problem in Gabor analysis is the so-called HRT conjecture, concerning the linear independence of finitely many time-frequency shifts of a non-trivial squareintegrable function [13]. To be precise, for x, ω ∈ Rd consider the translation and modulation operators acting on f ∈ L 2 (Rd ): Tx f (t) = f (t − x),
Mω f (t) = e2πit·ω f (t).
For z = (x, ω) ∈ R2d we say that π(z) f = Mω Tx f is a time-frequency shift of f along z. The HRT conjecture can thus be stated as follows:
Communicated by Chris Heil.
B
S. Ivan Trapasso [email protected] Fabio Nicola [email protected]
1
Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Turin, Italy 0123456789().: V,-vol
68
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Journal of Fourier Analysis and Applications
(2020) 26:68
Conjecture Given g ∈ L 2 (Rd ) \ {0} and a set of finitely many distinct points N is a linearly independent set of z 1 , . . . , z N ∈ R2d , the set G(g, ) = {π(z k )g}k=1 2 d functions in L (R ). As of today this somewhat basic question is still unanswered. Nevertheless, the conjecture has been proved for certain classes of functions or for special arrangements of points. We address the reader to the surveys [14,15], [16, Sect. 11.9] and the paper [22] for a detailed and updated state of the art on the issue. As a general remark we mention that the difficulty of the problem is witnessed by the variety of techniques involved in the known partial results, and also the surprising gap between the latter and the contexts for which nothing is known. For example, a celebrated result by Linnell [28] states that the conjecture is true for arbitrary g ∈ L 2 (Rd ) and for being a finite subset of a full-rank lattice in R2d and the proof is based on von Neumann algebras arguments. In spite of the wide range of this partial result, a solution is still lacking for smooth functions with fast decay (e.g., g ∈ S(Rd )) or for general configurations of just four points. The problem is further complicated by numerical evidence in conflict with analytic conclusions [10]. A recent contribution by Kreisel [19] proves the HRT conjecture under the assumption that the distance between points in is large compared to the decay of g. The class of funct
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