On the Uniform Convergence of the Fourier Series of Functions of Bounded Variation

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.