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

\lim _{\ell \rightarrow \infty }{\frac {1}{2\ell +1}}\sum _{j=-\ell }^{\ell }e^{ij\lambda }f(\tau ^{j}P)

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

Assani, I. (2001) [1994], "Wiener–Wintner theorem", Encyclopedia of Mathematics, EMS Press
Wiener, Norbert; Wintner, Aurel (1941), "Harmonic analysis and ergodic theory", American Journal of Mathematics, 63: 415–426, doi:10.2307/2371534, ISSN 0002-9327, JSTOR 2371534, MR 0004098

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