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
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superseded.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
(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
- The functional method and its applications, E. V. Voronovskaja (1970)
- A practical guide to splines, C. de Boor (1978)
The following routines can be used in conjunction with the
spline toolbox for
to compute the kernels associated with the
error in Hermite interpolation.
The basic routine is
- 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
and the function
which has recently been added to the spline toolbox.
Other routines include
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
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
Multivariate polynomial interpolation
For those interested in the error in multivariate polynomial interpolation (and
computations) see the companion page
Multivariate polynomial interpolation.
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