Complementability of exponential systems

TL;DR

The paper demonstrates that incomplete exponential systems in certain function spaces can be extended to complete and minimal systems, with similar results for reproducing kernels in de Branges spaces.

Contribution

It establishes that any incomplete exponential system can be embedded into a complete, minimal system, extending the understanding of exponential systems and reproducing kernels.

Findings

Incomplete exponential systems can be completed to full systems.

Results apply to systems of reproducing kernels in de Branges spaces.

Provides a method for extending incomplete systems to complete ones.

Abstract

We prove that any incomplete system of complex exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-\pi,\pi)$ is a subset of some complete and minimal system of exponentials. In addition, we prove analogous statement for systems of reproducing kernels in de Branges spaces.