In mathematics, the Wiener–Wintner theorem, named after

Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...

and

Aurel Wintner Aurel Friedrich Wintner (8 April 1903 – 15 January 1958) was a mathematician noted for his research in mathematical analysis, number theory, differential equations and probability theory. He was one of the founders of probabilistic number theor ...

, is a strengthening of the ergodic theorem, proved by .

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_\frac\sum_^\ell e^ 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 Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

, 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

* * Ergodic theory {{chaos-stub