en>OrenBochman: Cleaned up using AutoEd

2012-12-28T12:01:44Z

Cleaned up using AutoEd

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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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			:<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 ''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==

			*{{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]]

en>ChrisGualtieri: /* Development of Batman */Typo fixing, typos fixed: , → , using AWB

2012-08-10T05:07:28Z

Development of Batman: Typo fixing, typos fixed: , → , using AWB

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en>OrenBochman: Cleaned up using AutoEd

en>ChrisGualtieri: /* Development of Batman */Typo fixing, typos fixed: , → , using AWB