The uniqueness of the best $$$L_1$$$-approximant of continuous Banach-valued functions under interpolatory constraints

M.Ye. Tkachenko; V.M. Traktynska

doi:10.15421/242427

The uniqueness of the best $$$L_1$$$-approximant of continuous Banach-valued functions under interpolatory constraints

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 consider the best $$$L_1$$$-approximation with interpolatory constraints for continuous mapping of a metric compact set $$$Q$$$ into a Banach space $$$X$$$. The unicity set’s criterion is obtained. This result generalizes the result for real functions that was proved by A. Pinkus and H. Strauss.

Keywords

$$$L_1$$$-approximation; continuous Banach-valued functions; criterion of the best $$$L_1$$$-approximant

MSC 2020

41A52; 41A29; 41A05

Full Text:

PDF

References

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

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 Univ. Press, 1989.

Pinkus A., Strauss H. "$$$L_1$$$-approximation with Constraints", Trans. Amer. Math. Soc., 1990; 322(1): pp. 239-261. doi:10.1090/S0002-9947-1990-0986698-6

Rozema E. "Almost Chebyshev subspaces of $$$L_1(\mu, E)$$$", Pacif. J. Math, 1974; 53: pp. 585-604. doi:10.2140/pjm.1974.53.585