Constants in estimates for the rates of convergence in von Neumann’s and Birkhoff’s ergodic theorems. (English. Russian original) Zbl 1241.28010

Kachurovskii, A. G.; Sedalishchev, V. V.

Sb. Math. 202, No. 8, 1105-1125 (2011); translation from Mat. Sb. 202, No. 8, 21-40 (2011).

The authors study rate questions arising from the pointwise (Birkhoff) ergodic theorem and the mean (von Neumann) ergodic theorem, relating the power type of convergence in the mean ergodic theorem to the power type singularity at zero of the spectral measure associated to the transformation and the function averaged. Constants are also identified for a measure of rate of convergence in the pointwise ergodic theorem. This makes more explicit earlier results of the first author with A. V. Reshetenko [Sb. Math. 201, No. 4, 493–500 (2010); translation from Mat. Sb. 201, No. 4, 25–32 (2010; Zbl 1200.28018)] and the first author with V. V. Sedalishchev [Sb. Math. 202, No. 8, 1105–1125 (2011); translation from Mat. Sb. 202, No. 8, 21–40 (2011; Zbl 1241.28010)].

Reviewer: Thomas B. Ward (Norwich)

MSC:

28D05	Measure-preserving transformations
37A30	Ergodic theorems, spectral theory, Markov operators
37A50	Dynamical systems and their relations with probability theory and stochastic processes
47A35	Ergodic theory of linear operators
60G10	Stationary stochastic processes

Keywords:

ergodic theorems; rate of convergence; spectral measures

Citations:

Zbl 1200.28018; Zbl 1241.28010