Uniform non-linear constrained approximation in normed linear spaces

Abstract

The theory of H-sets as originally defined by L. Collatz has been developed in recent papers and shown to be a unifying concept for linear approximation problems. We here extend the theory of H-sets for non-linear uniform approximation with functional constraints where the approximating functions have a compact domain and range in a Banach space. With the theory of H-sets as here developed the characterization of global and local best approximations follow, and conditions for local to be global approximations are given. A convergence theorem for a descent algorithm is given.