# Some comments on the HRT-conjecture in L^p(R)

19.05.2021 14:00 - 15:30

Jorge Antezana (University of La Plata)

This year, in September, it will be the 25th anniversary of one of the most fascinating problems in time-frequency analysis, the HRT-conjecture. In 1996, Heil, Ramanathan, and Topiwala, motivated by the important role played by the refinement equation in wavelet theory, asked a similar question when the affine group is replaced by the Heisenberg group. More precisely, if $\{(a_k,b_k)\}_{k=1}^n$ is a finite collection of points in $\R^2$, and $f$ is a non-zero function of $f\in L^2(\mathbb{R})$, the problem is to determine whether or not the time-frequency shifts $\{e^{2\pi i b_k x}f(x-a_k)\}_{k=1}^n$ are linearly independent. In this talk, I will recall some known positive results, either assuming some structure in the set of points or some decay in the function $f$. Then, I will describe some results in $L^p(\mathbb{R})$, which have been obtained in collaboration with J. Bruna, and E. Pujals.