Minimax approximation of optical profiles

The notion of a profile is important in the theory surrounding the finishing of axisymmetric optical surfaces. Mathematically, a profile is a member of the quotient space C(M)/K, where C(M) denotes the space of continuous functions defined on a compact subset M subset of R, and K is the subspace of constant functions. In this paper we investigate the minimax approximation of a given profile [f] is an element of C(M)/K by elements of a closed convex cone in C(M)/K. We establish the existence of a minimax approximation (uniqueness does not in general hold) and prove two characterization theorems for any such best approximation. One of these theorems is then used as a basis for a "bisection" algorithm to compute a best approximation corresponding to a particular type of finishing process known as recursive operator controlled finishing.