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

Category:Theorems in complex analysis