Akhiezer's theorem

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

Statement

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

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

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

• One has:

${\displaystyle \sum _{n}|\operatorname {Im} (1/z_{n})|<\infty }$

where ${\displaystyle z_{n}}$ are the zeros of ${\displaystyle f}$.

Related results

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