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 (closed) 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.

Notes

References