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Extension and Restriction Principles for the HRT Conjecture

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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.

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Notes

  1. The proof given in [8, Theorem 1.3] contains a few inaccuracies that were fixed by C. Demeter and posted on Math Arxiv as arXiv:1006.0732.

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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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  1. Department of Mathematics & Norbert Wiener Center, University of Maryland, College Park, MD, 20742, USA

    Kasso A. Okoudjou

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  1. Kasso A. Okoudjou
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Correspondence to Kasso A. Okoudjou.

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Communicated by Chris Heil.

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This work was partially supported by a grant from the Simons Foundation \(\# 319197\), and ARO grant W911NF1610008. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

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Okoudjou, K.A. Extension and Restriction Principles for the HRT Conjecture. J Fourier Anal Appl 25, 1874–1901 (2019). https://doi.org/10.1007/s00041-018-09661-x

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  • DOI: https://doi.org/10.1007/s00041-018-09661-x

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