# Completeness of exponential systems

13.01.2021 14:00 - 15:30

Christina Neumayer (University of Vienna)

The completeness of sets of complex exponentials $\{e^{i\lambda_nt}\}_{n\geq 0}$ is one of the issues in the theory of nonharmonic Fourier series. To prove the completeness of a system of complex exponentials under certain conditions, we will introduce Carleman's formula. A consequence of this formula is that the zeros of a bounded and entire function of exponential type $f(z)$ must cluster on the real axis, and as an application we obtain a criterion for completeness of exponential systems. The presentation is based on 2nd-chapter of the book "An Introduction to Non-Harmonic Fourier Series" by Robert M. Young.