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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==
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| Let {{math|''f''(''z'')}} be an [[entire function]] of [[exponential type]] {{math|''τ''}}, with {{math|''f''(''x'') ≥ 0}} for real {{math|''x''}}. Then the following are equivalent:
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| * There exists an [[entire function]] {{math|''F''}}, of [[exponential type]] {{math|''τ''/2}}, having all its zeros in the (closed) upper half plane, such that
| | НАША КОМАНДА<br><br>КОНСТАНТИН<br>дизайнер <br><br>Я был на Вашем месте, я тоже заказывал дизайн. Обращайтесь и я Вас удивлю!<br><br>ЕКАТЕРИНА <br>режиссер <br><br>Я [http://kks.by/ САЙТЫ] арт-менеджер и мое любимое дело - это фото-режиссура.<br><br><br>РИЧАРД<br>кот <br><br>Я символ студии и я слежу за работой Кати и Кости.<br><br>http://www.kks.by<br><br>Review my web page :: [http://kks.by/?cat=27 изготовление логотипов] |
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| :: <math>f(z)=F(z)\overline{F(\overline{z})}</math>
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| * One has:
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| ::<math>
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| \sum|\operatorname{Im}(1/z_{n})|<\infty
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| </math> | |
| where {{math|''z''<sub>''n''</sub>}} are the zeros of {{math|''f''}}.
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| ==Remarks==
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| 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>
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *{{ citation
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| | last = Boas
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| | first = Jr., Ralph Philip
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| | title = Entire functions
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| | publisher = Academic Press Inc.
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| | location = New York
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| | year = 1954
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| | pages = 124–132
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| }}
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| *{{ citation
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| | last = Boas
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| | first = Jr., R. P.
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| | title = Functions of exponential type. I
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| | journal = Duke Math. J.
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| | volume = 11
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| | year = 1944
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| | pages = 9–15
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| | issn = 0012-7094
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| }}
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| *{{citation
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| | last = Akhiezer
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| | first = N. I.
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| | title = On the theory of entire functions of finite degree
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| | journal = Doklady Akad. Nauk SSSR (N.S.)
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| | volume = 63
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| | year = 1948
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| | pages = 475–478
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| | mr = 0027333
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| }}
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| [[Category:Theorems in complex analysis]]
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| {{mathanalysis-stub}}
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Revision as of 11:35, 22 February 2014
НАША КОМАНДА
КОНСТАНТИН
дизайнер
Я был на Вашем месте, я тоже заказывал дизайн. Обращайтесь и я Вас удивлю!
ЕКАТЕРИНА
режиссер
Я САЙТЫ арт-менеджер и мое любимое дело - это фото-режиссура.
РИЧАРД
кот
Я символ студии и я слежу за работой Кати и Кости.
http://www.kks.by
Review my web page :: изготовление логотипов