# L_{p}-error bounds for Hermite interpolation and the
associated Wirtinger inequalities

## by Shayne Waldron

## Abstract:

The B-spline representation for divided differences is used, for the first time,
to provide $L_p$-bounds for the error in Hermite interpolation, and its
derivatives, thereby simplifying and improving the results to be found in the
extensive literature on the problem. These bounds are equivalent to certain
Wirtinger inequalities (cf. \ref{FMP91:p66}).
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

## Availability:

This article is available in:
- Postscript
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- Contact the author if you need a copy mailed to you.