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

Statement

Let f(z) be an entire function of exponential type τ, with f(x) ≥ 0 for real x. Then the following are equivalent:

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

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

One has:

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

where zn are the zeros of f.
Related results

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

see Akhiezer (1948).

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 (N.S.), 63: 475–478, MR 0027333a