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On certain weighted moving averages and their differentiation analogues. (English) Zbl 1130.37315
Summary: Let \((X,\Sigma,\mu,T)\) be a measure-preserving dynamical system, and \(\{I_n\}\) a sequence of intervals of nonnegative integers moving to infinity with increasing cardinality. J.M. Rosenblatt and M. Wierdl [Ergodic Theory Dyn. Syst. 12, No. 3, 509–558 (1992; Zbl 0757.28015)] constructed optimal weights \(w_n\) for the averages of the form \[ \frac{1}{w_n}\sum_{k\in I_n}f\circ T^k \] to converge a.e. in \(L_1\). In this paper, we provide modified versions of those weights to address the question of optimality for more general weighted averages and their differentiation analogues.
37A30 Ergodic theorems, spectral theory, Markov operators
28D99 Measure-theoretic ergodic theory
47A35 Ergodic theory of linear operators
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