Approximation with constraints in normed linear spaces

Abstract

The purpose of this paper is to develop a unified approach to the characterization of solutions of constrained and unconstrained approximation problems. Several papers have been written on the characterization of solutions of special approximation problems with particular types of constraints or without constraints. For uniform approximation a general theory has been obtained by using generalized weight functions. Recently a new approach via optimization theory has been presented in [I]. The idea is to show, first, that the local Kolmogoroff condition is satisfied. Assuming a convexity condition, it can be shown that the local Kolmogoroff condition implies the Kolmogoroff criterion. Hence best approximations are characterized by the local Kolmogoroff condition. An essential restriction in [I] is the assumption of linear equality constraints For uniform approximation problems with nonlinear equality constraints, the local Kolmogoroff condition has been deduced in [2] under the assumption cd a regularity condition that does not seem to be practical. By deleting inequality constraints a more satisfactory regularity condition has been studied in [3]. Our aim is to treat approximation problems with nonlinear equality and inequality constraints in a normed linear space and to present a new and satisfactory regularity condition. As in [I], we consider the problem as a particular type of optimization problem. Applying new kinds of differentiability, a new approach to optimization problems has been developed in [4]. A generalization of the well-known Lagrange multiplier theorem has been obtained that can be applied to convex optimization problems as well as to differentiable optimization problems. Here we shall apply this theorem to approximation problems with constraints. In particular we obtain new characterization theorems for constrained &-approximation problems of continuous functions.