Non-symmetric approximations of functional classes by splines on the real line

Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.