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This article is cited in 9 scientific papers (total in 9 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.
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
Linking options:
https://www.mathnet.ru/eng/mzm9884 https://www.mathnet.ru/eng/mzm/v12/i3/p313
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