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) ≥ 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

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

• One has:

$\sum|\operatorname{Im}(1/z_{n})|<\infty$

where z_n are the zeros of f.

Remarks

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