Completeness of the exponential system in geometric terms of width, breadth and diameter

TL;DR

This paper provides geometric criteria for the completeness of exponential systems in function spaces, linking properties like breadth, width, and diameter of sets to the distribution of exponents.

Contribution

It introduces a new geometric criterion for exponential system completeness based on the set's breadth, width, and diameter, expressed through logarithmic measures.

Findings

Criteria established for completeness in various function spaces.

Relations between geometric set properties and exponent distribution.

Main results formulated through geometric and measure-theoretic relations.

Abstract

We establish a criterion for the completeness of an exponential system in the spaces of functions continuous on a convex compact set and holomorphic in the interior of this compact set, as well as in the spaces of holomorphic functions in the convex domain in terms of the breadth of the compact set or the domain in the direction. The main results are formulated exclusively through the relations between the breadth in the direction, width or diameter of the compact set or domain on the one hand and the logarithmic submeasures or logarithmic block densities of the distribution of exponents of exponential system on the other.