Exact order of pointwise estimates for polynomial approximation with Hermite interpolation

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 is an element of C-r (I), I = [-1, 1], at the classical pointwise rate c(k, r)rho(r)(n) (x)omega(k) (f((r)), rho(n)(x)), where rho(n)(x) = n(-1)root 1- x(2)+ 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 x(0) is an element of I, has the best possible pointwise rate of (simultaneous) approximation of f near x(0). Several applications are given. (C) 2021 Elsevier Inc. All rights reserved.