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 be an entire function of exponential type, with for real. Then the following are equivalent:

There exists an entire function, of exponential type, having all its zeros in the upper half plane, such that
One has:

where are the zeros of.

Related results

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