Exact order of pointwise estimates for polynomial approximation with Hermite interpolation

Abstract

We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that any algebraic polynomial of degree n approximating a function f∈Cr(I), I=[−1,1], at the classical pointwise rate c(k,r)ρnr(x)ωk(f(r),ρn(x)), where ρn(x)=n−11−x2+n−2, and c(k,r) is a constant which depends only on k and r, and is independent of f and n; and (Hermite) interpolating f and its derivatives up to the order r at a point x0∈I, has the best possible pointwise rate of (simultaneous) approximation of f near x0. Several applications are given.