On differentiability of the operator of best \(L_ 1\)-approximation (Q1071195)

Let X be a normed linear space and \(U_ n\) an n-dimensional subspace of X. Assume that \(U_ n\) is a unicity subspace of X; that is, each f in X possesses a unique best approximant q in \(U_ n\) for which \(\| f- q\| =\inf \{\| f-p\|:p\in U_ n\}.\) Consider the best approximation operator \(P:X\to U_ n\) mapping each f in X into its best approximant in \(U_ n\). Although this operator is bounded and continuous, it is in general nonlinear. This leads the author to consider the possibility of approximating P in a neighbourhood of f in X by a linear operator or, in other words, to the question of differentiability of the best approximation operator. In this paper, the differentiability of the operator of best \(L^ 1\)-approximation is studied. The Gatoux differentiability of P is verified for the important class of generalized convex functions. Some applications for polynomials and spline functions are given.