Upper bounds of best one-sided approximations of the classes $W^rL_\psi$ in the metric of $L$

The lowest upper bound is obtained for best one-sided approximations of classes $W^rL_\psi$ ($r=1,2,\dots$) by trigonometric polynomials and splines of minimum deficiency with equidistant knots, in the metric of space $L$, where $W^rL_\psi=\{f:f(x+2\pi)=f(x)$, $f^{(r-1)}(x)$ is absolutely continuous, $\|f^{(r)}\|_{L_\psi}\le1\}$ and $L_\psi$ is an Orlicz space.