Best one-sided approximation of certain classes of functions

This considers the question of the best one-sided approximation of certain classes of continuous periodic functions by means of trigonometric polynomials of order $\leqslant n-1$ in the metric $L_{2\pi}^p$ ($1\leqslant p<\infty$). Precise upper bounds are obtained for the best one-sided approximation of classes of $2\pi/n$-periodic functions $H_{\omega,n}$ [having arbitrary prescribed modulus of continuity $\omega(t)$] in the metric $L_{2\pi}^p$, as well as of classes of $2\pi$-periodic functions $H_\omega$ [having prescribed modulus of continuity $\omega(t)$ with definite limits] in the metric $L_{2\pi}^1$.