A problem on completeness of exponentials

TL;DR

This paper solves the classical problem of determining the exponential type of a finite measure on the real line, linking it to the measure's properties and discussing its relation to existing results.

Contribution

It provides a definitive solution to the type problem for exponential functions and explores its connections with prior research.

Findings

Established a criterion for the exponential type of a measure.

Connected the exponential type to measure properties and density conditions.

Discussed implications and relations with known results in the field.

Abstract

Let $\mu$ be a finite positive measure on the real line. For $a>0$ denote by $\EE_a$ the family of exponential functions $$\EE_a=\{e^{ist}| \ s\in[0,a]\}.$$ The exponential type of $\mu$ is the infimum of all numbers $a$ such that the finite linear combinations of the exponentials from $\EE_a$ are dense in $L^2(\mu)$. If the set of such $a$ is empty, the exponential type of $\mu$ is defined as infinity. The well-known type problem asks to find the exponential type of $\mu$ in terms of $\mu$. \ms\no In this note we present a solution to the type problem and discuss its relations with known results.