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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>
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==Statement==
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:
 
* 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
 
:: <math>f(z)=F(z)\overline{F(\overline{z})}</math>
 
* One has:
 
::<math>
\sum|\operatorname{Im}(1/z_{n})|<\infty
</math>
where {{math|''z''<sub>''n''</sub>}} are the zeros of {{math|''f''}}.
 
==Remarks==
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>
 
== Notes ==
{{reflist}}
 
==References==
*{{ citation
|    last  = Boas
|    first  = Jr., Ralph Philip
|    title = Entire functions
| publisher = Academic Press Inc.
|  location = New York
|      year = 1954
|    pages = 124–132
}}
*{{ citation
|    last  = Boas
|    first = Jr., R. P.
|    title = Functions of exponential type. I
|  journal = Duke Math. J.
|    volume = 11
|      year = 1944
|    pages = 9–15
|      issn = 0012-7094
}}
*{{citation
|    last  = Akhiezer
|    first  = N. I.
|    title = On the theory of entire functions of finite degree
|  journal = Doklady Akad. Nauk SSSR (N.S.)
|    volume = 63
|      year = 1948
|    pages = 475–478
|        mr = 0027333
}}
 
[[Category:Theorems in complex analysis]]
 
 
{{mathanalysis-stub}}

Latest revision as of 16:49, 10 November 2014

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