QLunch: Aleksei Kulikov

Speaker: Aleksei Kulikov

Title: Time-frequency localization operator and its singular values

Abstract: For a set A of finite measure on the real line we consider the time-frequency localization operator S = PFP where P is a restriction to A and F is the Fourier transform. It turns out that S is a compact operator, and as such it has a sequence of singular values decaying to zero. The simplest case is when A is an interval of length L. In this situation the singular values exhibit a phase transition: first ~L^2 of them are very close to 1, then there are only ~log L intermediate singular values and after that singular values tend to zero extremely fast. In this talk we will discuss known quantitative estimates for these singular values in all three regimes, with a special focus on their behaviour before the phase transition, that is how close the first L^2 singular values are to 1.

The talk is based on a joint work with Fedor Nazarov.