Abstract
In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.
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Kozlov, V.V. Summation of Divergent Series and Ergodic Theorems. Journal of Mathematical Sciences 114, 1473–1490 (2003). https://doi.org/10.1023/A:1022257013220
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DOI: https://doi.org/10.1023/A:1022257013220