**Introduction**

Let $\Lambda$ be a discrete subset of the real line. We study the properties of the exponential system $\{e^{i\lambda t}_{\lambda\in\Lambda}\}$ 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.

**Syllabus**

1. Preliminaries: entire functions of finite exponential type.

2. Exponential systems on an interval. Shannon-Kotelnikov-Whittaker formula.

Paley-Wiener spaces.

3.Sampling and interpolation. Kadets 1/4 theorem.

4. Hilbert transform. Muckenhoupt condition. Description of real Riesz bases.

5. Cartwright class. First Beurling-Malliavin theorem.

6. Long system of intervals. Beurling-Malliavin density.

Radius of completeness.

7. Fourier frames. Duffin-Shaeffer problem.

8. Sampling in Hilbert spaces of entire functions.

9. Finite union of intervals. Kohlenberg theorem.

10. Kozma-Nitzan theorem. Functions with a spectral gap.

11. Riesz bases from exponentials for union of two intervals.

12. Complementability problem for exponential systems

**Reference**

[1] A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support. Acta Math., 107 (1962), 291–309.

[2] S.V. Hruscev, N.K. Nikolskii, B.S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980), pp. 214–335, Lecture Notes in Math., 864, Springer, Berlin-New York, 1981.

[3] B.S. Pavlov, The basis property of a system of exponentials and the condition of Muckenhoupt. (Russian) Dokl. Akad. Nauk SSSR 247 (1979), 37–40. English transl. in Soviet Math. Dokl. 20 (1979).

[4] J. Ortega-Cerdà, K. Seip, Fourier frames, Annals of Mathematics 155 (3), 789-806, 2002.

[5] G.Kozma, S. Nitzan, Combining Riesz bases, Inventiones Mathematicae, 199 (2014), pp. 267–285.

[6] Y. Belov, Complementability of exponential systems, C. R. Math. Acad. Sci. Paris, 353 (2015), pp.215–218;

[7] Y. Belov, M. Mironov, Exponential Riesz bases in L^2 on two intervals, Int. Math. Res. Not. IMRN, (2024), no. 7, pp. 5403–5433;

**Lecturer Intro**

Yurii Belov is a professor at St. Petersburg State University and vice-chair of educational program "Mathematics" headed by Stanislav Smirnov. He got his PhD degree in 2007 (Norwegian University of Science and Technology) and Dr.Sci. degree in 2016 (St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia). He was a postdoc at Norwegian University of Science and Technology. Yurii Belov was awarded by the St. Petersburg Mathematical Society the prize for young mathematicians and won the "Young Russian Mathematics" contest (twice). In 2016 he got the L. Euler award from the Government of St. Petersburg.