The uniqueness of the best non-symmetric $$$L_1$$$-approximant with a weight for continuous functions with values in KB-space

M.Ye. Tkachenko; V.M. Traktynska

doi:10.15421/241907

The uniqueness of the best non-symmetric $$$L_1$$$-approximant with a weight for continuous functions with values in KB-space

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

Abstract

We investigate the problem of uniqueness of the best non-symmetrical $$$L_1$$$-approximant with a weight for continuous functions on metric compact set $$$Q$$$ with values in strictly convex partially ordered KB-space $$$X$$$ by subspaces of space $$$C(Q, X)$$$ of continuous functions on $$$Q$$$ with values in $$$X$$$. We obtain the characterization of subspaces of uniqueness of the best $$$(\alpha, \beta)$$$-approximant in integral metric with a weight for functions of space $$$C(Q, X)$$$ in terms of classes of "test" functions.

Keywords

uniqueness of the best non-symmetric approximant with a weight; $$$(\alpha, \beta)$$$-approximation with a weight; integral metric; KB-space

MSC 2020

41A52; 41A65; 46B40

Full Text:

PDF

References

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

Pinkus A. $$$L_1$$$-Approximation, Cambridge, 1989.

Strauss H. "Unicity in $$$L_1$$$-approximation", Math. Zeitschr., 1981; pp. 63-74. (in German)

Babenko V.F., Tkachenko M.Ye. "Problems of uniqueness of an element of the best nonsymmetric $$$L_1$$$-approximation of continuous functions with values in KB-spaces", Ukrainian Math. J., 2008; 60(7): pp. 1013-1027. doi:10.1007/s11253-008-0109-5

Tkachenko M.Ye., Traktyns'ka V.M. "Criterions for the best non-symmetric $$$L_p$$$- and $$$L_1$$$-approximant for the functions with values in KB-spaces with weight", Res. Math., 2017; 22: pp. 97-102. (in Ukrainian) doi:10.15421/241712

Vulikh B.Z. Introduction to the theory of semi-ordered spaces, Fizmatgiz, 1961.