Exact pointwise estimates for polynomial approximation with Hermite interpolation

TL;DR

This paper derives optimal pointwise estimates for polynomial approximation with Hermite interpolation on a finite interval, showing these bounds are sharp and cannot be improved, with various applications demonstrated.

Contribution

It establishes the best possible pointwise approximation estimates for polynomials satisfying Hermite interpolation conditions, extending classical approximation theory results.

Findings

Optimal pointwise approximation estimates are proven to be sharp.

Hermite interpolation conditions do not improve the approximation rate beyond established bounds.

Applications demonstrate the practical relevance of these estimates.

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 {\bf any} algebraic polynomial of degree $n$ approximating a function $f\in C^r(I)$, $I=[-1,1]$, at the classical pointwise rate $\rho_n^r(x) \omega_k(f^{(r)}, \rho_n(x))$, where $\rho_n(x)=n^{-1}\sqrt{1-x^2}+n^{-2}$, and (Hermite) interpolating $f$ and its derivatives up to the order $r$ at a point $x_0\in I$, has the best possible pointwise rate of (simultaneous) approximation of $f$ near $x_0$. Several applications are given.