Hereditary completeness for systems of exponentials and reproducing kernels

TL;DR

This paper proves that exponential systems in L^2 spaces are almost hereditarily complete, with at most a one-dimensional defect, and constructs nonhereditarily complete systems of reproducing kernels in de Branges spaces.

Contribution

It establishes the hereditary completeness of exponential systems up to a one-dimensional defect and constructs nonhereditarily complete systems in de Branges spaces, answering a longstanding question.

Findings

Exponential systems are hereditarily complete up to a one-dimensional defect.

Existence of nonhereditarily complete exponential systems.

Construction of nonhereditarily complete systems of reproducing kernels in de Branges spaces.

Abstract

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a one-dimensional defect. This means that there is at most one (up to a constant factor) function $f$ which is orthogonal to all the summands in its formal Fourier series $\sum_n (f,\tilde e_n) e^{i\lambda_n t}$, where $\{\tilde e_n\}$ is the system biorthogonal to $\{e^{i\lambda_n t}\}$. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.