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

S. A. Telyakovskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 318-326

Résumé

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.