Nonlinear Mean-Square Approximation of Finite Sets

Abstract

The properties of best approximations from a general nonlinear family with respect to the mean-square norm on a finite point set are considered. A necessary condition for an approximation to be best is obtained. Best approximations must oscillate when the tangent space contains a Haar subspace on an interval. It is shown that some approximations are best only to themselves. Special attention is paid to approximation by exponential families and polynomial rational families. It is shown that the Wittmeyer algorithm for rational approximation may converge to a non-best approximation.