Differential properties of the best-approximation operator for complex- valued functions. I (Q1092202)

Let X be a compact metric space, \(g_ 0,g_ 1,...,g_ n\) be a Chebyshev system of complex valued functions in C(X) and P(f) be the polynomial of best uniform approximation of \(f\in C(X):\) \[ \| f- P(f)\| = \inf \{\| f-Q\|,\quad Q=\sum a_ kg_ k\}. \] Theorem. If each characteristic set of f contains \(2n+3\) points, then the operator P(f) is differentiable in each direction \(h\in C(X)\), that is for each \(h\in C(X)\) \[ \lim_{t\to +0}(P(f+th)-P(f))/t \] exists.