Minimax Approximation of Optical Profiles

Abstract

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 $M\subset R$, and K is the subspace of constant functions. In this paper we investigate the minimax approximation of a given profile $[f]\in C(M)/K$ 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.