Next-To-Interpolatory Approximation on Sets with Multiplicities

Abstract

It is known that given a set X of m (⩾n) distinct real numbers and a real-valued function f denned on X, there exists a unique polynomial pn-1,f,x of degree n — 1 or less which approximates best to f(x) on X, that is, which minimizes the deviation δ = δ(f, p) defined by the αth-power metric (α < 1) with positive weights, or by the positively weighted maximum of |f — p| on X; these deviations shall be denoted by δα and δβ. The polynomial pn-1,f,x has the property that f — pn-1,f,x has at least n strong sign changes; in other words, there are at least n + 1 points in X where the difference takes alternatingly positive and negative values.