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 Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 1, Pages 101–136 (Mi izv1473)

Classification of periodic functions and the rate of convergence of their Fourier series

A. I. Stepanets

Abstract: The author proposes a classification for periodic functions that is based on grouping them according to the rate at which their Fourier coefficients tend to zero. The classes $L_\beta^\psi\mathfrak N$ thereby introduced coincide, for fixed values of the defining parameters, with the known classes $W^r$, $W^rH_\omega$, $W_\beta^r$, $W_\beta^rH_\omega$, and the like. Such an approach permits the classification of a wide spectrum of periodic functions, including infinitely differentiable, analytic, and entire functions. The asymptotic behavior of the deviations of the Fourier sums in these classes is studied. The assertions obtained in this direction contain results known earlier on approximation by Fourier sums of classes of differentiable functions.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1987, 28:1, 99–132

Bibliographic databases:

UDC: 517.5
MSC: 42A16, 42A10
Revised: 04.06.1985

Citation: A. I. Stepanets, “Classification of periodic functions and the rate of convergence of their Fourier series”, Izv. Akad. Nauk SSSR Ser. Mat., 50:1 (1986), 101–136; Math. USSR-Izv., 28:1 (1987), 99–132

Citation in format AMSBIB
\Bibitem{Ste86} \by A.~I.~Stepanets \paper Classification of periodic functions and the rate of convergence of their Fourier series \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1986 \vol 50 \issue 1 \pages 101--136 \mathnet{http://mi.mathnet.ru/izv1473} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=835568} \zmath{https://zbmath.org/?q=an:0615.42005} \transl \jour Math. USSR-Izv. \yr 1987 \vol 28 \issue 1 \pages 99--132 \crossref{https://doi.org/10.1070/IM1987v028n01ABEH000869} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. K. Kushpel', “Sharp estimates of the widths of convolution classes”, Math. USSR-Izv., 33:3 (1989), 631–649
2. A. I. Stepanets, “Solution of the Kolmogorov–Nikol'skii problem for the Poisson integrals of continuous functions”, Sb. Math., 192:1 (2001), 113–139
3. Feng Dai, Kunyang Wang, “Convergence rate of spherical harmonic expansions of smooth functions”, Journal of Mathematical Analysis and Applications, 348:1 (2008), 28
4. A. S. Serdyuk, Ie. Yu. Ovsii, “Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type”, ISRN Mathematical Analysis, 2014 (2014), 1
5. S. B. Vakarchuk, “Best Polynomial Approximations and Widths of Classes of Functions in the Space $L_2$”, Math. Notes, 103:2 (2018), 308–312
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