Some results concerning best uniform coapproximation. (English) Zbl 1010.41006

Let \(G\) be a nonempty subset of a real normed linear space \(X\), a member \(g_f \in G\) is called a best coapproximation to \(f\in X\) if for every \(g\in G\), \[ \;g-g_f\;\leq \;f-g\;. \] This work is a study of the best uniform coapproximation of continuous functions on an interval. The following fundamental properties are investigated: (i) the existence and characterization of best uniform coapproximations, (ii) error estimates, (iii) a relation between interpolation and best uniform coapproximation, (iv) the existence of a selection for best coapproximations, and (v) continuity of the selection. Other basic properties are also mentioned, such as (a) the relation to best uniform approximations, and (b) general properties of best coapproximations (not necessarily uniform) used in the development here.


41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
41A50 Best approximation, Chebyshev systems
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