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

This article is cited in 7 scientific papers (total in 7 papers)

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
Received in July 2000

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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  mathnet  mathscinet  zmath  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  mathnet  crossref  mathscinet  elib
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