Kachurovskii, A. G.; Podvigin, I. V. Fejér sums for periodic measures and the von Neumann ergodic theorem. (English. Russian original) Zbl 1400.37009 Dokl. Math. 98, No. 1, 344-347 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 481, No. 4, 358-361 (2018). Summary: The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculated, in fact, using the same formulas (by integrating the Fejér kernels), so this ergodic theorem is, in fact, a statement about the asymptotics of the growth of the Fejér sums at zero for the spectral measure of the corresponding dynamical system. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates of the Fejér sums at the point for periodic measures. For example, natural criteria for the polynomial growth and polynomial decrease in these sums can be obtained. On the contrary, available in the literature, numerous estimates for the deviations of Fejér sums at a point can be used to obtain new estimates for the rate of convergence in this ergodic theorem. Cited in 3 Documents MSC: 37A30 Ergodic theorems, spectral theory, Markov operators Keywords:Fejér sum; von Neumann ergodic theorem PDFBibTeX XMLCite \textit{A. G. Kachurovskii} and \textit{I. V. Podvigin}, Dokl. Math. 98, No. 1, 344--347 (2018; Zbl 1400.37009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 481, No. 4, 358--361 (2018) Full Text: DOI References: [1] N. K. Bari, A Treatise on Trigonometric Series (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964). [2] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory (Nauka, Moscow, 1980) [in Russian]. [3] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Nauka, Moscow, 1965; Wolters-Noordhoff, Groningen, 1971). · Zbl 0154.42201 [4] Kachurovskii, A. G., No article title, Russ. Math. Surv., 51, 653-703, (1996) · Zbl 0880.60024 [5] Kachurovskii, A. G.; Podvigin, I. V., No article title, Trans. Moscow Math. Soc., 77, 1-53, (2016) · Zbl 1370.37014 [6] Kachurovskii, A. G.; Sedalishchev, V. V., No article title, Sb. Math., 202, 1105-1125, (2011) · Zbl 1241.28010 [7] A. N. Shiryaev, Probability (Nauka, Moscow, 1989; Springer-Verlag, Berlin, 1994). · Zbl 0682.60001 [8] Gaposhkin, V. F., No article title, Math. Notes, 64, 316-321, (1998) · Zbl 0930.60027 [9] A. Zygmund, Trigonometric Series, 2nd ed. (Cambridge Univ. Press, Cambridge, 1959), Vol. 1. · Zbl 0085.05601 [10] I. P. Natanson, Constructive Theory of Functions (GITTL, Moscow, 1949) [in Russian]. [11] Stechkin, S. B., No article title, Tr. Mat. Inst. im. V.A. Steklova, 62, 48-60, (1961) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.