Let be a separable probability space, be measure-preserving. Fix . The ergodic theorem of Wiener-Wintner asserts that for -a.e. ,
The author uses the Van der Corput inequality to show the following generalization: For -a.e. ,
for all real polynomials and all continuous periodic functions on . If, in addition, is weakly mixing and , then for -a.e. , for all and all continuous periodic ,
where the supremum is taken over all real polynomials of degree bounded by .
The proof of the Wiener-Wintner theorem is based on an equivalence which the author generalizes in the following way: Let be the set of eigenvalues of (as an operator on ) and, for , set . Suppose the linear span of is dense in , and . Then holds for all if and only if for -a.e. , for all real polynomials of degree less than or equal to and every continuous periodic ,