Minimax approximations subject to a constraint

Abstract

A class of approximation problems is considered in which a continuous, positive function p(x) is approximated by a rational function satisfying some identity. It is proved under certain hypotheses that there is a unique rational approximation satisfying the constraint and yielding minimax relative error and that the corresponding relative error function does not have an equal ripple graph. This approximation is, moreover, just the rational approximation to p(x) yielding minimax logarithmic error. This approximation, in turn, is just a constant multiple of the rational approximation to dx yielding minimax relative error but not necessarily satisfying the constraint. © 1971 American Mathematical Society.