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]

Notes

1. see Akhiezer (1948).
2. see Boas (1954) and Boas (1944) for references.

References

• Boas, Jr., Ralph Philip (1954), Entire functions, New York: Academic Press Inc., pp. 124–132
• 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
• Akhiezer, N. I. (1948), "On the theory of entire functions of finite degree", Doklady Akademii Nauk SSSR, New Series, 63: 475–478, MR 0027333