Akhiezer's theorem

In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.^[1]

Statement

Let $f : ℂ \to ℂ$ 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

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

\sum_{n} | Im (1 / z_{n}) | < \infty

where $z_{n}$ are the zeros of $f$ .

It is not hard to show that the Fejér–Riesz theorem is a special case.^[2]