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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.
MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
28D99 Measure-theoretic ergodic theory
47A35 Ergodic theory of linear operators
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