Akhiezer's theorem
# 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.

## Related results

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

