Examples of the best piecewise linear approximation with free nodes

Abstract

In 1957, E. Ya. Remez published a monograph devoted to numerical methods of Chebyshev approximations. Particularly, a problem of the best uniform approximation of a function convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A. M. Vershik, V. N. Malozemov and A. B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was established that such partition exists and generates the best piecewise polynomial approximation. Moreover, a numerical method for constructing a partition with equal deviations was proposed. In this paper, we give three examples demonstrating how the general approach works when solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.