Completeness of exponential systems in spaces of functions on a compact or domain and their area

TL;DR

This paper establishes new criteria for the completeness of exponential systems in various function spaces, linking geometric properties of compact sets and indicator distributions to functional completeness.

Contribution

It introduces novel completeness conditions based on Euclidean area and indicator distribution, applicable to spaces of continuous and holomorphic functions.

Findings

New geometric completeness criteria involving Euclidean area.

Conditions applicable to both compact and simply connected domains.

Enhanced understanding of exponential system distribution in function spaces.

Abstract

We present new completeness conditions for exponential systems on the complex plane in Banach algebras of continuous functions on a compact with a connected complement that are simultaneously holomorphic in the interior of this compact if it is nonempty, as well as in spaces of holomorphic functions on a simply connected bounded domain with a topology of uniform convergence on compact subsets. These conditions are formulated in terms of the Euclidean area of the convex hull of a compact or a region on the one hand, and new characteristics of the distribution of exponential system indicators on the other.