Generalization of intersecting planes method to the case of the problem of the best simultaneous uniform approximation of several continuous on the compact functions by finite-dimensional subspace (Q2768816)

Let \(C(S)\) be a vector space of continuous functions \(f\) on the compact \(S\) 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 \(\alpha^{*}= \inf_{g\in V} \max_{1\leq j\leq m} \|g-\varphi_{j}\|\) for given \(\varphi_{j}\in C(S)\), \(j=1,\ldots,m\) is called the problem of the best simultaneous uniform approximation of functions \(\varphi_{j}\), \(j=1,\ldots,m\) by elements of \(V\). The authors propose the generalization of intersecting planes algorithms for approximation of optimal points and proves convergence of this algorithm.