The best simultaneous uniform approximation of several continuous functions on a compact set
Let \(C(S)\) be a vector space of continuous on the compact set \(S\) functions \(f\) with the norm \(\| f\|=\max_{s\in S}| f(s)|\), and let \(V\) be an \(n\)-dimensional subspace of \(C(S)\) generated by functions \(f_{i}\in C(S), i=1,\ldots,n\). The problem of finding \(\alpha^{*}=\inf\limits_{g\in V}\max\limits_{1\leq j\leq m}\| g-\phi_{j}\|\) for given \(\phi_{j}\in C(S), j=1,\ldots,m\) is called the problem of best simultaneous uniform approximation of functions \(\phi_{j}, j=1,\ldots,m\) by elements of \(V\). The author proves existence, uniqueness and a characterization of extremal element of the considered problem.
scientific article; zbMATH DE number 1685540