Chebyshev Approximation by Families with the Betweeness Property
Charles B. Dunham · Transactions of the American Mathematical Society · 1969
Introduction.In this note a theory of Chebyshev approximation is obtained for approximating families with a property which is a generalization of convexity, the betweeness property.This theory is of interest for several reasons.Most of the approximating families for which a tractable theory exists characterize best approximations by the extrema of their error curve.The betweeness property is the weakest easily verifiable condition giving such a characterization of best approximations.The development of the theory sheds considerable light on the well-known linear theory [2], [5] and rational theory [1], [2], [3].A necessary and sufficient condition for the uniqueness of best approximations is obtained; it is the most general known necessary and sufficient condition for any theory.Let A'be a compact space and for a function g define \\g\\ = sup {\g(x)\ : x e X}.Let <3 be a family of real continuous functions with elements F, G, H,....The Chebyshev problem is: given a continuous function/, to find an element G* of 'S to minimize e(G)=\\E(G, ) [| where E(G, x)=f(x) -G(x).Such an element G* is called a best approximation in 'S to / on X.It will be assumed throughout the discussion that / is fixed, and mention of / is suppressed in the notation e(G) and E(G, ■).