Wiener–Wintner theorem
In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by Wiener and Wintner (1941).
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
- [math]\displaystyle{ \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
- Hazewinkel, Michiel, ed. (2001), "W/w130110", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=W/w130110
- Wiener, Norbert; Wintner, Aurel (1941), "Harmonic analysis and ergodic theory", American Journal of Mathematics 63: 415–426, doi:10.2307/2371534, ISSN 0002-9327
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