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

f: C \to C be an entire function of exponential type \tau , with f(x)\geq 0 for real x . Then the following are equivalent:F , of exponential type \tau/ 2 , having all its zeros in the (closed) upper half plane, such thatf(z) = F(z)\overline{F(\overline{z})}

• One has:

\sum _{n} |\operatorname{Im}( 1 /z _{n} )|< inf ty 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.^[2]

Notes and References

1. see .
2. see and for references.