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 $f:\mathbb {C} \to \mathbb {C}$ be an entire function of exponential type $\tau$, with $f(x)\geq 0$ 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

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

• One has:

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

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

• Boas, Jr., Ralph Philip (1954), Entire functions, New York: Academic Press Inc., pp. 124–132{{citation}}: CS1 maint: multiple names: authors list (link)
• Boas, Jr., R. P. (1944), "Functions of exponential type. I", Duke Math. J., 11: 9–15, doi:10.1215/s0012-7094-44-01102-6, ISSN 0012-7094{{citation}}: CS1 maint: multiple names: authors list (link)
• Akhiezer, N. I. (1948), "On the theory of entire functions of finite degree", Doklady Akademii Nauk SSSR, New Series, 63: 475–478, MR 0027333