Seminar
| Parent Program: | |
|---|---|
| Location: | SLMath: Eisenbud Auditorium |
Given a non-zero square integrable function $g$ and a subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \R^2$, let G(g,Λ)={e2πibk⋅g(⋅−ak)}Nk=1.
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification
No Primary AMS MSC
Secondary Mathematics Subject Classification
No Secondary AMS MSC
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.