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 Tr. Mat. Inst. Steklova, 2001, Volume 232, Pages 318–326 (Mi tm222)

On the Uniform Convergence of the Fourier Series of Functions of Bounded Variation

S. A. Telyakovskii

Abstract: It is well known that, if a function $f$ is continuous at each point of an interval $[a, b]$ and has bounded variation on the period, then the Fourier series of $f$ is uniformly convergent on $[a, b]$. This assertion is strengthened here as follows. Let $\{ n_j \}$ be an increasing sequence of positive integers that is representable as a union of a finite number of lacunary sequences. If the Fourier series of $f$ is divided into blocks consisting of the harmonics from $n_j$ to $n_{j + 1} - 1$, then the series formed by the absolute values of these blocks is uniformly convergent on $[a, b]$. Estimates for the convergence rate of the Fourier series of functions whose derivatives of prescribed order have bounded variation are strengthened likewise.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2001, 232, 310–318

Bibliographic databases:
UDC: 517.518.4

Citation: S. A. Telyakovskii, “On the Uniform Convergence of the Fourier Series of Functions of Bounded Variation”, Function spaces, harmonic analysis, and differential equations, Collected papers. Dedicated to the 95th anniversary of academician Sergei Mikhailovich Nikol'skii, Tr. Mat. Inst. Steklova, 232, Nauka, MAIK «Nauka/Inteperiodika», M., 2001, 318–326; Proc. Steklov Inst. Math., 232 (2001), 310–318

Citation in format AMSBIB
\Bibitem{Tel01} \by S.~A.~Telyakovskii \paper On the Uniform Convergence of the Fourier Series of Functions of Bounded Variation \inbook Function spaces, harmonic analysis, and differential equations \bookinfo Collected papers. Dedicated to the 95th anniversary of academician Sergei Mikhailovich Nikol'skii \serial Tr. Mat. Inst. Steklova \yr 2001 \vol 232 \pages 318--326 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm222} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1851458} \zmath{https://zbmath.org/?q=an:0999.42002} \transl \jour Proc. Steklov Inst. Math. \yr 2001 \vol 232 \pages 310--318 

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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. S. A. Telyakovskii, “On the Rate of Convergence of Fourier Series of Functions of Bounded Variation”, Math. Notes, 72:6 (2002), 872–876
2. Moricz F., “Pointwise behavior of Fourier integrals of functions of bounded variation over R”, Journal of Mathematical Analysis and Applications, 297:2 (2004), 527–539
3. S. A. Telyakovskii, “Some properties of Fourier series of functions with bounded variation. II”, Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 2, S188–S195
4. A. S. Belov, S. A. Telyakovskii, “Refinement of the Dirichlet–Jordan and Young's theorems on Fourier series of functions of bounded variation”, Sb. Math., 198:6 (2007), 777–791
5. Belov A.S., Telyakovskii S.A., “An improvement of the Dirichlet–Jordan test for Fourier series of functions of bounded variation”, Doklady Mathematics, 75:1 (2007), 101–102
6. Jenei A., “On the rate of convergence of Fourier series of functions of bounded variation in two variables”, Analysis Mathematica, 35:2 (2009), 99–118
7. S. A. Telyakovskii, “Series formed by the moduli of blocks of terms of trigonometric series. A survey”, J. Math. Sci., 209:1 (2015), 152–158
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