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Extension and Restriction Principles for the HRT Conjecture

  • Kasso A. OkoudjouEmail author
Article
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Abstract

The HRT (Heil–Ramanathan–Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on \(\mathbb {R}\) is linearly independent. This longstanding conjecture remains largely open even in the case when the function is assumed to be smooth. Nonetheless, the conjecture has been proved for some special families of functions and/or special sets of points. The main contribution of this paper is an inductive approach to investigate the HRT conjecture based on the following. Suppose that the HRT is true for a given set of N points and a given function. We identify the set of all new points such that the conjecture remains true for the same function and the set of \(N+1\) points obtained by adding one of these new points to the original set. To achieve this we introduce a real-valued function whose global maximizers describe when the HRT is true. To motivate this new approach we re-derive a special case of the HRT for sets of 3 points. Subsequently, we establish new results for points in (1, n) configurations, and for a family of symmetric (2, 3) configurations. Furthermore, we use these results and the refinements of other known ones to prove that the HRT holds for certain families of 4 points.

Keywords

HRT conjecture Positive definite matrix Bochner’s theorem Short-time Fourier transform Time-frequency analysis 

Mathematics Subject Classification

Primary 42C15 Secondary 42C40 

Notes

Acknowledgements

The author thanks C. Heil for introducing him to this fascinating and addictive problem, and for invaluable comments and remarks on earlier versions of this paper. He also thanks R. Balan, J. J. Benedetto, and D. Speegle for helpful discussions over the years about various versions of the results presented here. He acknowledges C. Clark’s help in generating the pictures included in the paper. Finally, he thanks W. Liu for helpful discussions, and the anonymous referees for their useful and insightful comments and remarks.

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Authors and Affiliations

  1. 1.Department of Mathematics & Norbert Wiener CenterUniversity of MarylandCollege ParkUSA

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