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
formula_1
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.