The major result is the inequality
$$|f(x)-H_\Theta f(x)| \le {n^{1/q}\over n!} {|\omega_\Theta(x)|\over(\diam\{x,\Theta\})^{1/q}} \norm{D^nf}_q,$$
where $H_\Theta f$ is the Hermite interpolant to $f$ at the multiset of $n$
points $\Theta$,
$$\omega_\Theta(x):=\prod_{\gth\in\Theta}(x-\gth),$$
and $\diam\{x,\Theta\}$ is the diameter of $\{x,\Theta\}$. This inequality
significantly improves upon `Beesack's inequality' (cf. \ref{Be62}), on which
almost all the bounds given over the last 30 years have been based.
Keywords: Hermite interpolation, B-spline, Green's function, Beesack's inequality, Wirtinger inequality
Math Review Classification: 26D10, 41A05, 41A80 (primary), 41A10, 41A44 (secondary)
Length: 21 pages
Comment: Written in TeX, contains 7 figures. Supported by the Chebyshev Professorship of Carl de Boor. See Project Hermite for related work
Last updated: 25 March 1996
Status: To appear in Constructive Approximation