On the best non-symmetric $$$L_1$$$-approximations by splines under constraints for their derivatives

Abstract

We find exact values of non-symmetric $$$L_1$$$-approximations of classes $$$W_1^r$$$ of periodic functions by splines $$$s \in S_{2n,r-1}$$$ and $$$s \in S_{2n,r}$$$ ($$$S_{2n,r}$$$ is the set of $$$2\pi$$$-periodic polynomial splines of degree $$$r$$$, of defect 1, with knots in the points $$$k \pi / n$$$, $$$k \in \mathbb{Z}$$$) such that $$$\bigvee\limits_0^{2\pi} s^{(r-1)} \leqslant 1$$$ and $$$\| s^{(r)} \| \leqslant 1$$$ respectively when $$$r$$$ is even, and, as a corollary, we obtain exact values for the corresponding best one-side approximations.