北京雁栖湖应用数学研究院

Let $\mathrm{\Lambda }$ be a discrete subset of the real line. We study the properties of the exponential system $\{e_{\lambda \in \mathrm{\Lambda }}^{i\lambda t}\}$ as a subset of some Banach space on the real line. The main example is the space $L^2(E)$, where $E$ is a measurable subset of the real line. The classical theory corresponds to the case when $E$ is an interval. It includes the famous Beurling-Malliavin theorem about the radius of completeness (1962), description of exponential Riesz bases given by S. Hruschev, N. Nikolski and B. Pavlov (1981), description of Fourier frames given by J. Ortega-Cerda and K. Seip (2002) and many other results. For non-connected sets $E$ the situation becomes much more complicated and we have only partial results. One of them is a construction of Riesz bases from exponentials for a finite union of intervals given by G. Kozma and S. Nitzan (2015). The second is a result about Riesz bases from exponentials for a union of two intervals given by Y. Belov and M. Mironov (2022). In addition we consider complementability problem for exponential systems.
During the lecture course we will consider classical and non-classical theorems and will provide some proofs.