Introduction

We will study some mixed completeness problems, i.e. completeness problems for the union of two systems of harmonics of different nature, for a example a system from exponentials and its biorthogonal system.

We consider the spectral synthesis property for systems of exponentials and other systems of reproducing kernels of Hilbert spaces of entire functions (Paley-Wiener spaces, Fock type spaces e.t.c.).The existence of exponential systems without spectral synthesis was proved in 2013 by Yu. Belov, A. Baranov and A.Borichev. Another topic concerns a synthesis problem for differentiation operator in C^\infty(R). This problem was posed by B. Korenblum.

In particular, our technique leads us to the negative solution to the Newman-Shapiro conjecture which was stated in 1966. In addition, we will discuss the connections of these problems to operator theory and mathematical physics.

Syllabus

1. Hilbert spaces of analytic functions. Reproducing kernels.

2. Biorthogonal systems. Spectral synthesis. Examples.

3. Exponential systems on an interval. Paley-Wiener spaces. Young theorem.

4. Spectral synthesis for exponential systems. Density theorem. Systems without spectral synthesis.

5. Spectral synthesis in Fock space.

6. Carlsson-Sundberg completeness problem.

7. Derivation-invariant subspaces of C^\infty. Residual subspaces.

8. Spectral synthesis for C^\infty.

9. The weighted completeness problems in Fock space.

10. Newman-Shapiro problem. Examples.

11. Counterexamples for Newman-Shapiro problem.

12. Spectral synthesis for operators. Models.

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.