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 $τ$ , with $f (x) \geq 0$ for real $x$ . Then the following are equivalent:

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

where $z n$ are the zeros of $f$ .

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