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Mat. Zametki, 1972, Volume 12, Issue 3, Pages 313–324 (Mi mz9884)  

This article is cited in 6 scientific papers (total in 6 papers)

Generalized variation, the Banach indicatrix, and the uniform convergence of Fourier series

K. I. Oskolkov

V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR

Abstract: It is proved that if the continuous periodic function $f$ has bounded $\Phi$-variation, then the deviation of $f$ from the sum of $n$ terms of its Fourier series has the bound
$$ ||f-S_n(f)||\leqslant c\int_0^{\omega(\pi n^{-1})}\log(v_\Phi(f)/\Phi(\xi))d\xi. $$
Here $c$ is an absolute constant, $\omega$ is the modulus of continuity, $v_\Phi(f)$ is the complete $\Phi$-variation of $f$ over a period. It is established that the Salem and Garsia–Sawyer criteria for the uniform convergence of the Fourier series in terms of the $\Phi$-variation and the Banach indicatrix respectively are definitive, and it is proved that the second of these variants is a corrolary of the first.

Full text: PDF file (1206 kB)

English version:
Mathematical Notes, 1972, 12:3, 619–625

Bibliographic databases:

UDC: 517.5
Received: 27.01.1972

Citation: K. I. Oskolkov, “Generalized variation, the Banach indicatrix, and the uniform convergence of Fourier series”, Mat. Zametki, 12:3 (1972), 313–324; Math. Notes, 12:3 (1972), 619–625

Citation in format AMSBIB
\Bibitem{Osk72}
\by K.~I.~Oskolkov
\paper Generalized variation, the Banach indicatrix, and the uniform convergence of Fourier series
\jour Mat. Zametki
\yr 1972
\vol 12
\issue 3
\pages 313--324
\mathnet{http://mi.mathnet.ru/mz9884}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=316959}
\zmath{https://zbmath.org/?q=an:0239.42014}
\transl
\jour Math. Notes
\yr 1972
\vol 12
\issue 3
\pages 619--625
\crossref{https://doi.org/10.1007/BF01093998}


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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. B. I. Golubov, “On convergence of Riesz spherical means of multiple Fourier series”, Math. USSR-Sb., 25:2 (1975), 177–197  mathnet  crossref  mathscinet  zmath
    2. Z. A. Chanturiya, “On uniform convergence of Fourier series”, Math. USSR-Sb., 29:4 (1976), 475–495  mathnet  crossref  mathscinet  zmath  isi
    3. B. I. Golubov, “On the summability of Fourier integrals by Riesz spherical means”, Math. USSR-Sb., 33:4 (1977), 501–518  mathnet  crossref  mathscinet  zmath  isi
    4. V. M. Badkov, “Approximation properties of Fourier series in orthogonal polynomials”, Russian Math. Surveys, 33:4 (1978), 53–117  mathnet  crossref  mathscinet  zmath
    5. T. I. Akhobadze, “Functions of bounded generalized second variation”, Math. USSR-Sb., 37:2 (1980), 261–294  mathnet  crossref  mathscinet  zmath  isi
    6. E. A. Sevast'yanov, “On uniform approximation of functions by Fourier sums”, Math. USSR-Sb., 42:4 (1982), 515–538  mathnet  crossref  mathscinet  zmath
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