The best one-sided approximation of the classes $W^rH_\omega$

In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space $L$, of the classes $W^rH_\omega$ ($r=2,4,6,\dots$) of all $2\pi$-periodic functions $f(x)$ that are continuous together with their $r$-th derivative $f^r(x)$ and such that for any points $x'$ and $x''$ we have $|f^r(x')-f^r(x'')|\le\omega(|x'-x''|)$, where $\omega(t)$ is a modulus of continuity that is convex upwards.