Polynomial approximations to finitely oscillating functions

Abstract

The main result of this paper is an algorithm for the calculation of this polynlomial, which is first motivated by an indepenidenit proof for the existence of P. A funietion f is said to be finitely oscillating if it has at most a finite niumher of relative extrema. The following mode of approximating a continuous finiitely oscillating function f in the uniform norm so that the oscillations are preserved, is discussed: first obtain a polynomial P of minimal degree which has the same variation as f and then obtain an increasinig polynom-ial Q such that P(Q) agrees with f at all its relative extrema. Two theorems are given in the last sectioIn, to show that this method of approximation is always possible.