### 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.

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PDF (Русский)#### References

Babenko V.F. "The best $$$L_1$$$-approximations of $$$W^r_1$$$ classes by splines from $$$W^r_1$$$", *Ukrainian Math. J.*, 1994; 46(10): pp. 1410-1413. (in Russian) doi:10.1007/BF01066101

Babenko V.F. "Mean approximations under constraints for derivatives of approximating functions", *Vopr. analiza i priblizh.*, 1989; pp. 9-18. (in Russian)

Korneichuk N.P. *Splines in approximation theory*, Nauka, 1984. (in Russian)

Korneichuk N.P. *Exact constants in approximation theory*, Nauka, 1987. (in Russian)

Korneichuk N.P. *Extremum problems in approximation theory*, 1976. (in Russian)

DOI: https://doi.org/10.15421/249802

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Copyright (c) 1998 V.F. Babenko, I.N. Litviniuk, N.V. Parfinovych

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