Geodesic
On the Uniform Convergence of the Fourier Series of Functions of Bounded Variation
Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 318-326.

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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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S. A. Telyakovskii. On the Uniform Convergence of the Fourier Series of Functions of Bounded Variation. Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 318-326. http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a25/

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