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<div>In the [[mathematics|mathematical]] field of [[complex analysis]], '''Akhiezer's theorem''' is a result about [[entire function]]s proved by [[Naum Akhiezer]].<ref>see {{harvtxt|Akhiezer|1948}}.</ref><br />
<br />
==Statement==<br />
Let {{math|''f''(''z'')}} be an [[entire function]] of [[exponential type]] {{math|''&tau;''}}, with {{math|''f''(''x'')&nbsp;&ge;&nbsp;0}} for real {{math|''x''}}. Then the following are equivalent:<br />
<br />
* There exists an [[entire function]] {{math|''F''}}, of [[exponential type]] {{math|''&tau;''/2}}, having all its zeros in the (closed) upper half plane, such that<br />
<br />
:: <math>f(z)=F(z)\overline{F(\overline{z})}</math><br />
<br />
* One has:<br />
<br />
::<math><br />
\sum|\operatorname{Im}(1/z_{n})|<\infty<br />
</math><br />
where {{math|''z''<sub>''n''</sub>}} are the zeros of {{math|''f''}}.<br />
<br />
==Remarks==<br />
It is not hard to show that the [[Fejér–Riesz theorem]] is a special case.<ref>see {{harvtxt|Boas|1954}} and {{harvtxt|Boas|1944}} for references.</ref><br />
<br />
== Notes ==<br />
{{reflist}}<br />
<br />
==References==<br />
*{{ citation<br />
| last = Boas<br />
| first = Jr., Ralph Philip<br />
| title = Entire functions<br />
| publisher = Academic Press Inc.<br />
| location = New York<br />
| year = 1954<br />
| pages = 124–132<br />
}}<br />
*{{ citation<br />
| last = Boas<br />
| first = Jr., R. P.<br />
| title = Functions of exponential type. I<br />
| journal = Duke Math. J.<br />
| volume = 11<br />
| year = 1944<br />
| pages = 9–15<br />
| issn = 0012-7094<br />
}}<br />
*{{citation<br />
| last = Akhiezer<br />
| first = N. I.<br />
| title = On the theory of entire functions of finite degree<br />
| journal = Doklady Akad. Nauk SSSR (N.S.)<br />
| volume = 63<br />
| year = 1948<br />
| pages = 475–478<br />
| mr = 0027333<br />
}}<br />
<br />
[[Category:Theorems in complex analysis]]<br />
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