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

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.

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

Bibliographic databases:

UDC: 517.5

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