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| In mathematics, the '''Denjoy–Koksma inequality''', introduced by {{harvtxt|Herman|1979|loc=p.73}} as a combination of work of [[Arnaud Denjoy]] and the [[Koksma–Hlawka inequality]] of [[Jurjen Ferdinand Koksma]], is a bound for [[Weyl sum]]s <math>\sum_{k=0}^{m-1}f(x+k\omega)</math> of functions ''f'' of [[bounded variation]].
| | Hi, everybody! <br>I'm Turkish male :D. <br>I really love Rock stacking! |
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| ==Statement==
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| Suppose that a map ''f'' from the circle ''T'' to itself has irrational rotation number ''α'', and ''p''/''q'' is a rational approximation to ''α'' with ''p'' and ''q'' [[coprime]], |''α'' – ''p''/''q''| < 1/''q''<sup>2</sup>. Suppose that ''φ'' is a function of bounded variation, and ''μ'' a [[probability measure]] on the circle invariant under ''f''. Then
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| :<math>\left|\sum_{i=0}^{q-1} \phi f^i(x) - q\int_T \phi \, d\mu \right| < \operatorname{Var}(\phi)</math>
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| {{harv|Herman|1979|loc=p.73}}
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| ==References==
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| *{{Citation | last1=Herman | first1=Michael-Robert | title=Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations | url=http://www.numdam.org/item?id=PMIHES_1979__49__5_0 | mr=538680 | year=1979 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=49 | pages=5–233}}
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| *{{Citation | last1=Kuipers | first1=L. | last2=Niederreiter | first2=H. | title=Uniform distribution of sequences | publisher=Wiley-Interscience [John Wiley & Sons] | location=New York | isbn=978-0-486-45019-3 | mr=0419394 |id= Reprinted by Dover 2006 | year=1974}}
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| {{DEFAULTSORT:Denjoy-Koksma inequality}}
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| [[Category:Mathematical analysis]]
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Latest revision as of 22:52, 25 September 2014
Hi, everybody!
I'm Turkish male :D.
I really love Rock stacking!