Akhiezer's theorem

In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.

Statement
Let $f(z)$ be an entire function of exponential type $&tau;$, with $f(x) &ge; 0$ for real $x$. Then the following are equivalent:


 * There exists an entire function $F$, of exponential type $&tau;/2$, having all its zeros in the (closed) upper half plane, such that


 * $$f(z)=F(z)\overline{F(\overline{z})}$$


 * One has:



\sum|\operatorname{Im}(1/z_{n})|<\infty $$ where $z_{n}$ are the zeros of $f$.

Related results
It is not hard to show that the Fejér–Riesz theorem is a special case.