Some aspects of best n-convex approximation

AbstractAn n-convex function is one whose nth order divided differences are nonnegative. Thus a 1-convex function is nondecreasing and a 2-convex function is convex in the classical sense. A function if ƒ is n-concave if −tf is n-convex. We consider best uniform approximation by n-convex and n-concave functions and, by considering alternation properties of the error function, we prove a variety of results, including characterizations of functions with best n-convex or n-concave approximations in Πn − 1, a sufficient condition for best n-convex approximation, and a uniqueness result. Several examples are given