Necessary Conditions for a Minimax Approximation

$φ(x,λ1*,λ2*,...,λn*)$ is a minimax approximation to f(x) if the values $λi=λi*$, i = 1 to n of the parameters are such that the maximum value (over x) of |f (x) − Φ(x, λ)| is minimized. Rice (1960) has established a number of conditions characterizing minimax approximations, but these apply only to a limited class of approximating functions, Φ(x, λ). A simple example serves to illustrate that more general conditions are required—some are derived in this paper.