Akhiezer's theorem explained

In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.^[1]

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

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

• One has:

\sum|\operatorname{Im}( 1 /z _{n} )|< inf ty where are the zeros of .

Related results

It is not hard to show that the Fejér–Riesz theorem is a special case.^[2]

Notes and References

1. see .
2. see and for references.