Project Hermite

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 In view of the recent work involving splines a large part of this is now superseded.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

• The Landau problem on compact intervals and optimal numerical differentiation, H. Kallioniemi, J. Approx. Theory 63 (1990) pp 72-91
• Derivative error bounds for Lagrange interpolation: an extension of Cauchy's bound for the error of Lagrange interpolation, G. W. Howell, J. Approx. Theory 67 (1991) pp 164-173
• Interpolation by Lagrange polynomials, B-splines and bounds of errors, A. Shadrin, Analysis Mathematica 20 (1994), pp 213-224
• Error bounds for Lagrange interpolation A. Shadrin, J. Approx. Theory xx (1994), to appear
• L_p-error bounds for Hermite interpolation and the associated Wirtinger inequalities, S. Waldron (1994)
• Extremising the L_p-norm of a monic polynomial with roots in a given interval and Hermite interpolation , S. Waldron (1994)
• Schmidt's inequality, S. Waldron (1996)

Papers that are historically important

• Kokai zyobibunhoteisiki ni tuite, M. Tsumura, Kansu Hoteisiki 30 (1941), pp 20-35
• On the Green's function of an N-point boundary value problem, P. R. Beesack, Pacific J. Math. 12 (1962), pp 801-812
• Hermite interpolation errors for derivatives, G. Birkhoff and A. Priver, J. Math. Phys. 46, pp 440-447
• Inequalities involving the p-norm of f and its n-th derivative for f with n zeros, J. Brink, Pacific J. Math. 42 (1972), pp 289-311

Relevant books

• The functional method and its applications, E. V. Voronovskaja (1970)
• A practical guide to splines, C. de Boor (1978)

Relevant software

The basic routine is bkfspmak.m

• function spline = bkfspmak(x,j,Theta)

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

• bkfplt.m which plots the splines constructed by bkfspmak including the points of interpolation and x. This flexible routine can be used to zoom up on interesting parts of the spline and displays information about sign changes provided by bkfchsg.m.
• bkfmesh.m and bkfsurfl.m which provide mesh and surfl (lighted surface) plots of the splines (as x varies).

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

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

Work in progress and people involved

• Shayne Waldron
• Aleksei Shadrin
• Henry Kallioniemi

Multivariate polynomial interpolation