Characterization theorems for constrained approximation problems via optimization theory

Abstract

There have been several attempts to develop a unified approach to the characterization of solutions of L p approximation problems, for example [5] and [21]. However, the approaches developed in these papers do not readily lend themselves to handling problems where the satisfaction of additional constraints, such as interpolation or convexity conditions on the approximating function, is required. On the other hand, there have been many papers which have individually dealt with the characterization of solutions of special approximation problems with particular types of constraints, especially in the area of Chebyshev approximation. Examples of such special problems include interpolation by the approximating function [4], approximation by a monotone function [16], approximation from one side of the function to be approximated [14], [2, 6], approximation with a vector-valued norm [1, 11], simultaneous approximation of a function and its derivatives [18], and a series of papers by Taylor: [23–25]. The purpose of this paper is to develop a unified approach to the characterization of solutions of Chebyshev and L 1 approximation problems with the various types of constraints mentioned above. In addition, it is recognized that many approximation problems with non- L p norms can easily be handled in the same manner. In Section 1 the necessary results from optimization theory are outlined. The remaining sections of the paper are devoted to applications of these results to various approximation problems: Section 2 to constrained linear Chebyshev approximation, Section 3 to rational Chebyshev approximation, Section 4 to Chebyshev approximation with a vector-valued norm, Section 5 to Chebyshev approximation with nonstandard norms, and Section 6 to constrained L 1 approximation.