Ergodic averages with generalized weights

Abstract

Two types of weighted ergodic averages are studied. It is shown that if F = {Fn} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds forF. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified. 1. Introduction. This article is inspired by the celebrated Wiener- Wintner theorem which has been instrumental in studying various proper- ties of weighted and subsequential averages as well as constructing some nontrivial classes of good sequences. It turns out that, utilizing the tools developed in (C¸, CF), one can obtain a version of this theorem in the setting of a class of superadditive processes. In turn, this superadditive version of Wiener-Wintner theorem leads to new classes of weights. In Sections 2-4 of this article, besides proving these results, we will also study convergence of sequences of weights defined by such superadditive processes. The study of the behavior of the ergodic averages modulated by means of Besicovitch's sequences was initiated by C. Ryll-Nardzewski (Ry) (for p = 1), and A. Tempelman (T) (for p > 1 and in the context of Besicovitch functions on LCA groups). Later, various generalizations of this result were obtained (BeL, JO, LO, LOT). Recently, another such attempt was made in (Li) in the non-commutative setting. The tools utilized in (Li) have some interest- ing ramifications in the commutative case as well. In Sections 5 and 6 of this article, following an observation made on the Wiener-Wintner theorem and adapting some of the techniques of (Li), we arrive at two rather distinct weighted ergodic theorems in the spirit of (Ry). First we define a generalized Besicovitch sequence associated with a subset of the group of unimodular