Treffer: best uniform approximation of a family of functions continuous on a compact set: The best uniform approximation of a family of functions continuous on a compact set

The problem of the best simultaneous approximation of the family of functions \(\{\phi_j\in C(S), j\in I\}\) continuous on the compact \(S\) by elements of a linear finite-dimensional subspace \(V\) generated by a linearly independent system of functions \(f_i\in C(S)\), \(i=1,\dots,n\) is considered. Here \(I\) is an arbitrary set of indexes. It is assumed that the functions \[ \Phi_1(s)=\min_{j\in I} \phi_j(s),\quad \Phi_2(s)=\max_{j\in I} \phi_j(s). \] are defined and continuous on \(S\). The existence of the extremal element \(g^{*}\in V\) such that \[ \max_{j\in I} \|g^{*}-\phi_j\|_{C(S)}= \inf_{g\in V} \max_{j\in I} \|g-\phi_j\|_{C(S)}, \] is proved and statements on characterization and uniqueness of the extremal element are obtained. These statements are extensions of certain classic results to the case of the function family \(\{\phi_j, j\in I\}\). The classic results are related to the problem of the best approximation of a function continuous on a compact set.