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), https://orcid.org/0000-0002-9242-194X
V.M. Traktynska (Oles Honchar Dnipro National University)

Abstract


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 $$$\mathbb{R}^m_p, p\in (1;+\infty )$$$.

Keywords


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

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References


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DOI: https://doi.org/10.15421/242104

  

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