Upper bounds for the best one-sided approximation by splines of the classes $W^rL_1$

In the present note we will investigate the problem of the one-sided approximation of functions by $n$-dimensional subspaces. In particular, we will find the exact value of the best one-sided approximation of the class $W^rL_1$ ($r=1,2,\dots$) of all periodic functions $f(x)$ of period $2\pi$ for which $f^{(r-1)}(x)$ ($f^{(0)}(x)=f(x)$) is absolutely continuous and $\|f^{(r)}\|_{L_1}\le1$ by periodic spline functions $S_{2n,\mu}$ ($\mu=0,1,\dots$, $n=1,2,\dots$) of period $2\pi$, order $\mu$, and deficiency 1.