The uniqueness of the best non-symmetric $$$L_1$$$-approximant for continuous functions with values in $$$\mathbb{R}^m_p$$$

M.Ye. Tkachenko (Oles Honchar Dnipro National University),
V.M. Traktynska (Oles Honchar Dnipro National University)


The article considers the questions of the uniqueness of the best non-symmetric $$$L_1$$$-approximations of continuous functions with values in $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$ by elements of the two-dimensional subspace  $$$H_2= \mathrm{span} \{1, g_{a,b}\}$$$, where
g_{a,b}(x)=\left\{ \begin{matrix}
-b\cdot (x-1)^2, & x\in [0;1), & \\
0, & x\in [1;a-1), & (a\geq 2, b>0),\\
(x-a+1)^2,& x\in [a-1,a],&
\end{matrix} \right.
It is obtained that when $$$b\in (0;1)\cup (1;+\infty), a\geq 2$$$, the subspace $$$H_2$$$ is a unicity space of the best $$$(\alpha ,\beta )$$$-approximations for continuous on the $$$[0;a]$$$ functions with values in the space  $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$. In case $$$b=1$$$, $$$a\geq 4$$$ it is proved that the subspace  $$$H_2$$$ is not a unicity subspace of the best non-symmetric approximations for these functions.
Received results summarize the previously obtained Strauss results for the real functions in the case $$$\alpha = \beta = 1$$$, as well as the results of Babenko and Glushko for the the best $$$(\alpha ,\beta )$$$-approximation for continuous functions on a segment with values in the space $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$.


non-symmetric approximation; unicity space of the best non-symmetric approximations; vector-valued functions; integral

Full Text:



Kroo A. "A general approach to the study of Chebyshev subspaces in $$$L_1$$$-approximation of continuous functions", J. Approx. Theory, 1987; 51: pp. 98-111. doi:10.1016/0021-9045(87)90024-4

Strauss H. "Unicity in $$$L_1$$$-approximation", Math. Zeitschr., 1981; 176: pp. 63-74. (in German) doi:10.1007/BF01258905

Babenko V.F., Tkachenko M.Ye. "Problems of unicity of the best non-symmetric $$$L_1$$$-approximant of continuous functions with values in KB-spaces", Ukrainian Math. J., 2008; 60(7): pp. 867-878. (in Russian) doi:10.1007/s11253-008-0109-5

Vulikh B.Z. Introduction to the theory of semi-ordered spaces, Moscow, Fizmatgiz, 1961; 407 p. (in Russian)

Pinkus A. $$$L_1$$$-Approximation, Cambridge Univ. Press, 1989; 239 p.

Babenko V.F., Glushko V.N. "On unicity of the best approximant in metric of $$$L_1$$$ space", Ukrainian Math. J., 1994; 46(5): pp. 475-483. (in Russian) doi:10.1007/BF01058514

Babenko V.F., Pichugov S.A. "Approximation of continuous vector functions", Ukrainian Math. J., 1994; 46(11): pp. 1435-1448. (in Russian) doi:10.1007/BF01058878

Babenko V.F., Gorbenko M.Ye. "On the uniqueness of the best $$$L_1$$$-approximant for functions with values in a Banach space", Ukrainian Math. J., 2000; 52: pp. 29-34. doi:10.1007/BF02514134




  • There are currently no refbacks.

Copyright (c) 2021 M.Ye. Tkachenko, V.M. Traktynska

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