Minimax approximations subject to a constraint

Abstract:

A class of approximation problems is considered in which a continuous, positive function $\varphi (x)$ 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 $\varphi (x)$ yielding minimax logarithmic error. This approximation, in turn, is just a constant multiple of the rational approximation to $\varphi (x)$ yielding minimax relative error but not necessarily satisfying the constraint.