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Harmonic Analysis Seminar: On the HRT Conjecture May 03, 2017 (02:00 PM PDT - 03:00 PM PDT)
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Location: MSRI: Simons Auditorium
Speaker(s) Kasso Okoudjou (University of Maryland)
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Given a non-zero square integrable function $g$ and a subset  $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \R^2$,  let $$\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.$$  The Heil-Ramanathan-Topiwala (HRT) Conjecture asks whether  $\mathcal{G}(g, \Lambda)$ is linearly independent. For the last two decades, very little progress has been made in settling the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture. I will then describe a small variation of the conjecture that asks the following question: Suppose that the HRT conjecture holds for a given $g\in L^{2}(\R)$ and a given set $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \R^2$. Give a characterization of all points $(a, b)\in \R^2\setminus \Lambda$ such that the conjecture remains true for the same function $g$ and the new set of point $\Lambda_1=\Lambda\cup\{(a, b)\}$. If time permits I will illustrate this approach for the cases  $N=4$, and  $5$ and when $g$ is a real-valued function. 

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