Simultaneous approximation of a set of complex-valued functions in a normed vector space

Abstract

We present a unifying characterisation theory for best simultaneous approximation of a set of complex-valued bounded functions on a compact topological space B in a normed vector space, by elements of a non-linear subset of C( B). The linear problem in the uniform norm was first considered by Diaz and McLaughlin [ J. Approximation Theory, 2 1969, 419–432] and was further developed by Blatt [ J. Approximation Theory, 8 1973, 210–248] for non-linear subsets. We now generalise their approach to an arbitrary norm using the Hahn-Banach theory.