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 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

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

• One has:

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

where z_n are the zeros of f.

Remarks

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 Akad. Nauk SSSR (N.S.) 63: 475–478, MR 0027333