Best Approximation in Euclidean Spaces

Abstract

Many classical ‘least-squares’ approximation problems are special cases of the general problem of best approximation in a Euclidean space by elements of a finite-dimensional subspace (or a convex set). We present two fundamental results on approximation by convex sets in the inner-product setting — the Kolmogorov criterion of best approximation and Phelps’s criterion for convexity of a Chebyshev set in a Euclidean space in terms of the Lipschitz continuity of the metric projection operator.