Interpolation of continuous in ordered $H$-variation functions

V. V. Novikov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 418-422

Résumé

In 1972 D. Vaterman introduced a class of functions of $\Lambda$-bounded variation (in particular, a harmonic variation or an $H$-variation). Later he introduced also the class of functions of ordered $\Lambda$-bounded variation and the class of continuous in $\Lambda$-variation functions. These classes have been used by many authors in studies on the convergence and summability of the Fourier series. This paper investigates the behavior of the Lagrange interpolation of continuous in ordered $H$-variation functions. We prove a result: if $f\in C_{2\pi}$ is continuous in ordered harmonic variation on $[-\pi,\pi]$, then the Lagrange trigonometric polynomials $\{L_n(f,x)\}$ based on equidistant nodes converge to $f$ uniformly on $\mathbb{R}$.