On the best non-symmetric $$$L_1$$$-approximations under the constraints on their derivatives

Ye.V. D'yakova (Oles Honchar Dnipropetrovsk National University)
I.A. Shevchenko (Oles Honchar Dnipropetrovsk National University)


We obtained exact values of the best $$$L_1$$$-approximations of classes $$$W^r_1$$$ and $$$W^{r-1}_V$$$, non-symmetric and one-way $$$L_1$$$-approximation of classes $$$W^r_1$$$ of periodic functions by splines of order $$$r$$$ and $$$r-1$$$ with defect 1 and knots at the points $$$t_j = \frac{2\pi}{n} \left[\frac{j}{2}\right] + (1 - (-1)^j) \frac{h}{2}$$$, $$$j\in \mathbb{Z}$$$ that belong to the class $$$W^r_1$$$ and $$$W^{r-1}_V$$$.


spline; the best non-symmetric approximation; the best one-way approximation; Sobolev classes


Babenko V.F. "Approximation on the average under the restrictions on the derivatives of approximating functions", Problems of analysis and approximation, Kyiv, 1989; pp. 9-18. (in Russian)

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

Babenko V.F., Litvinyuk I.N., Parfinovich N.V. "On the best non-symmetric $$$L_1$$$-approximations by splines under the restrictions on their derivatives", Dnipr. Univ. Math. Bull., 1998; 3: pp. 11-18. (in Russian)

Babenko V.F., Azar L.E. "The best $$$L_1$$$-approximations by splines under the restrictions", Ukrainian Math. J., 1998; 11: pp. 1443-1451. (in Russian)

Babenko V.F., Azar L.E., Parfinovich N.V. "On the best non-symmetric approximations of classes of functions, defined by differential operators, by generalised splines", Dnipr. Univ. Math. Bull., 2000; 11: pp. 9-18. (in Russian)

Babenko V.F., Parfinovich N.V. "On exact values of the best approximations of classes of differentiable periodic functions by splines", Mat. zametki, 2010; 87(5): pp. 669-683. (in Russian)

Parfinovich N.V. "The best approximations of differentiable functions by splines", Proceedings of the International scientific conference "Approximation Theory and Applications", Dnipropetrovsk, 2015. (in Ukrainian)

Korneichuk N.P. Splines in approximation theory, Nauka, Moscow, 1984; 544 p. (in Russian)

Korneichuk N.P. Extremum problems of approximation theory, Nauka, Moscow, 1976; 320 p. (in Russian)

Korneichuk N.P. Exact constants in approximation theory, Nauka, Moscow, 1987; 424 p. (in Russian)

DOI: https://dx.doi.org/10.15421/241505



  • There are currently no refbacks.

Copyright (c) 2015 Ye.V. D'yakova, I.A. Shevchenko

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in

ISSN (Online): 2664-5009
ISSN (Print): 2664-4991