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arXiv · 2006.03126

Exact pointwise estimates for polynomial approximation with Hermite interpolation

Kirill A. Kopotun·Dany Leviatan·Igor A. Shevchuk
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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.

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Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk. 2020-06-04. Exact pointwise estimates for polynomial approximation with Hermite interpolation. https://arxiv.org/abs/2006.03126

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