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In mathematics, the '''Wiener–Wintner theorem''', named after [[Norbert Wiener]] and [[Aurel Wintner]], is a strengthening of the [[ergodic theorem]], proved by {{harvs|txt|last=Wiener|authorlink=Norbert Wiener|last2=Wintner|year=1941}}


==Statement==


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Suppose that ''&tau;'' is a measure-preserving transformation of a measure space ''S'' with finite measure. If ''f'' is a real-valued integrable function on ''S'' then the Wiener–Wintner theorem states that there is a measure 0 set ''E'' such that  the average
 
:<math> \lim_{\ell\rightarrow\infty}\frac{1}{2\ell+1}\sum_{j=-\ell}^\ell e^{ij\lambda} f(\tau^j P) </math>
 
exists for all real λ and for all ''P'' not in ''E''.
 
The special case for ''&lambda;''&nbsp;=&nbsp;0 is essentially the [[Birkhoff ergodic theorem]], from which the existence of a suitable measure 0 set ''E'' for any fixed ''&lambda;'', or any countable set of values ''&lambda;'', immediately follows. The point of the Wiener–Wintner theorem is that one can choose the measure 0 exceptional set ''E'' to be independent of&nbsp;''&lambda;''.
 
This theorem was even much more generalized by
the Return Times Theorem.
 
==References==
 
*{{eom|id=W/w130110|first=I.|last=Assani}}
*{{Citation | last1=Wiener | first1=Norbert | last2=Wintner | first2=Aurel | title=Harmonic analysis and ergodic theory | jstor=2371534 | mr=0004098 | year=1941 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=63 | pages=415–426}}
 
{{DEFAULTSORT:Wiener-Wintner theorem}}
[[Category:Ergodic theory]]

Latest revision as of 13:01, 28 December 2012

In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by Template:Harvs

Statement

Suppose that τ is a measure-preserving transformation of a measure space S with finite measure. If f is a real-valued integrable function on S then the Wiener–Wintner theorem states that there is a measure 0 set E such that the average

lim12+1j=eijλf(τjP)

exists for all real λ and for all P not in E.

The special case for λ = 0 is essentially the Birkhoff ergodic theorem, from which the existence of a suitable measure 0 set E for any fixed λ, or any countable set of values λ, immediately follows. The point of the Wiener–Wintner theorem is that one can choose the measure 0 exceptional set E to be independent of λ.

This theorem was even much more generalized by the Return Times Theorem.

References

  • Template:Eom
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