A problem on completeness of exponentials


Let $\mu$ be a finite positive measure on the real line. For $a>0$, denote by $\mathcal{E}_a$ the family of exponential functions $$\mathcal{E}_a=\{e^{ist}| s\in[0,a]\}.$$ The exponential type of $\mu$ is the infimum of all numbers $a$ such that the finite linear combinations of the exponentials from $\mathcal{E}_a$ 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 $\mu$. In this note we present a solution to the type problem and discuss its relations with known results.


Alexei Poltoratski

Texas A&M University, College Station, TX