An alternation characterization of best uniform approximations on noncompact intervals

Abstract

We present a characterization for a best uniform approximation to a given bounded continuous function f defined on the real but not necessarily compact interval T from an n-dimensional subspace S of the bounded continuous functions on T. When S is a Haar subspace and each element of S satisfies an additional endpoint regularity condition, such a best approximation may be characterized by an appropriate generalization of the familiar alternation criterion which holds for compact T. One such best approximation that has an alternating error curve may be obtained as the uniform limit of a sequence whose vth term is the unique best uniform approximation to f on the vth member of a suitably chosen expanding sequence of compact subintervals of T. The results apply in the special case where T = [0, + ∞) and S is a family of exponential sums with real exponents.