# zbMATH — the first resource for mathematics

Fejér sums and Fourier coefficients of periodic measures. (English. Russian original) Zbl 1403.37016
Dokl. Math. 98, No. 2, 464-467 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 482, No. 4, 381-384 (2018).
Summary: The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejér sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous $$2{\pi}$$-periodic function is uniquely determined by its sequence of Fejér sums at any two points whose difference is incommensurable with $${\pi}$$.

##### MSC:
 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
##### Keywords:
periodic measure; Fourier analysis
Full Text:
##### References:
 [1] R. Edwards, Fourier Series: A Modern Introduction, 2nd ed. (Springer-Verlag, New York, 1979), Vol. 1. · Zbl 0424.42001 [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 0219.60027 [4] Kachurovskii, A. G.; Knizhov, K. I., No article title, Dokl. Math, 97, 211-214, (2018) · Zbl 1400.37008 [5] Kachurovskii, A. G., No article title, Russ. Math. Surv, 51, 653-703, (1996) · Zbl 0880.60024 [6] Kachurovskii, A. G.; Sedalishchev, V. V., No article title, Sb. Math, 202, 1105-1125, (2011) · Zbl 1241.28010 [7] A. Zygmund, Trigonometric Series, 2nd ed. (Cambridge Univ. Press, New York, 1959), Vol. 1. · Zbl 0085.05601 [8] N. K. Bari, A Treatise on Trigonometric Series (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964). · Zbl 0154.06103 [9] R. Edwards, Fourier Series: A Modern Introduction, 2nd ed. (Springer-Verlag, New York, 1982), Vol. 2. · Zbl 0599.42001 [10] Kachurovskii, A. G.; Podvigin, I. V., No article title, Trans. Moscow Math. Soc, 77, 1-53, (2016) · Zbl 1370.37014 [11] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1976; Dover, New York, 1999).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.