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 $f:ℂ\to ℂ$ be an entire function of exponential type $\tau$, with $f(x)\ge 0$ for real $x$. Then the following are equivalent:

• There exists an entire function $F$, of exponential type $\tau /2$, having all its zeros in the (closed) upper half plane, such that

$f(z)=F(z)\overline{F(\stackrel{‾}{z})}$

• One has:

$\sum _n|\mathrm{Im}(1/z_n)|<\mathrm{\infty }$

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.

References

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