Geodesic
Von Neumann's ergodic theorem and Fejer sums for signed measures on the circle
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1313-1321

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The Fejer sums for measures on the circle and the norms of the deviations from the limit in von Neumann's ergodic theorem are calculated, in fact, using the same formulas (by integrating the Fejer kernels) — and so, this ergodic theorem is a statement about the asymptotics of the Fejer sums at zero for the spectral measure of the corresponding dynamical system. It made it possible, having considered the integral Holder condition for signed measures, to prove a theorem that unifies both following well-known results: classical S.N. Bernstein's theorem on polynomial deviations of the Fejer sums for Holder functions — and theorem about polynomial rates of convergence in von Neumann's ergodic theorem.
Keywords: deviations of Fejer sums, rates of convergence in von Neumann's ergodic theorem, integral Holder condition. 
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     author = {A. G. Kachurovskii and M. N. Lapshtaev and A. J. Khakimbaev},
     title = {Von {Neumann's} ergodic theorem and {Fejer} sums for signed measures on the circle},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1313--1321},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a139/}
}
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A. G. Kachurovskii; M. N. Lapshtaev; A. J. Khakimbaev. Von Neumann's ergodic theorem and Fejer sums for signed measures on the circle. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1313-1321. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a139/