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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
##### Keywords:
ergodic theory; differentiation
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