Fejér Sums for Periodic Measures and the von Neumann Ergodic Theorem

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Fejér Sums for Periodic Measures and the von Neumann Ergodic Theorem. / Kachurovskii, A. G.; Podvigin, I. V.

In: Doklady Mathematics, Vol. 98, No. 1, 01.07.2018, p. 344-347.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{edf48935e6f6413da75274bb236cc4c3,

title = "Fej{\'e}r Sums for Periodic Measures and the von Neumann Ergodic Theorem",

abstract = "The Fej{\'e}r 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 Fej{\'e}r kernels), so this ergodic theorem is, in fact, a statement about the asymptotics of the growth of the Fej{\'e}r 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 Fej{\'e}r 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 Fej{\'e}r sums at a point can be used to obtain new estimates for the rate of convergence in this ergodic theorem.",

keywords = "CONVERGENCE",

author = "Kachurovskii, {A. G.} and Podvigin, {I. V.}",

year = "2018",

month = jul,

day = "1",

doi = "10.1134/S1064562418050149",

language = "English",

volume = "98",

pages = "344--347",

journal = "Doklady Mathematics",

issn = "1064-5624",

publisher = "Maik Nauka-Interperiodica Publishing",

number = "1",

}

RIS

TY - JOUR

T1 - Fejér Sums for Periodic Measures and the von Neumann Ergodic Theorem

AU - Kachurovskii, A. G.

AU - Podvigin, I. V.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

KW - CONVERGENCE

UR - http://www.scopus.com/inward/record.url?scp=85052864934&partnerID=8YFLogxK

U2 - 10.1134/S1064562418050149

DO - 10.1134/S1064562418050149

M3 - Article

AN - SCOPUS:85052864934

VL - 98

SP - 344

EP - 347

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - 1

ER -

ID: 16485668