Project Hermite

Charles Hermite (1822-1901)

Due to oversight, the crucial role that univariate splines play in describing the error in Hermite interpolation has only recently been exploited see, e.g., `Error bounds for Lagrange interpolation' (Shadrin 1994) and `L_p-error bounds for Hermite interpolation and the associated Wirtinger inequalities' (Waldron 1994).

Because bounding the error in Hermite interpolation in terms of the derivative which kills the interpolating space is of interest in numerical analysis and in the analysis of ordinary differential equations, there is an extensive literature on the subject. See, e.g., the recent monograph `Error inequalities in polynomial interpolation and their applications' (Agarwal and Wong 1993). In view of the recent work involving splines a large part of this is now superseded. It is expected that within the next few years a much better understanding of the problem and its history will be obtained.

The purpose of this page is to efficiently communicate those changes as they occur and to coordinate the investigation into this problem.

The corresponding inequalities are of Wirtinger-Sobolev type (see Steve Finch's master page Favorite Mathematical Constants).

Important recent papers

Papers that are historically important

Relevant books

Relevant software

The following routines can be used in conjunction with the spline toolbox for MATLAB to compute the kernels associated with the error in Hermite interpolation.

The basic routine is bkfspmak.m

which makes the Birkhoff spline which represents the j-th derivative at x of the error in Hf the Hermite interpolant to f at Theta a multiset of n points. It requires the updated version of ppual.m and the function fnjmp.m which has recently been added to the spline toolbox.

Other routines include

For a quick introduction use demo.m (in preparation)

bkfmesh([0 0 0 1 1 1],2)

Work in progress and people involved

Multivariate polynomial interpolation

For those interested in the error in multivariate polynomial interpolation (and computations) see the companion page Multivariate polynomial interpolation.

This document is maintained by Shayne Waldron (
Last modified: .