# Dissertation University of Wisconsin-Madison

$L_p$-error bounds for multivariate polynomial interpolation schemes

## by Shayne Francis Daley Waldron

## Abstract:

The $L_p(\Omega)$-distance of sufficiently smooth functions from $n$-variate polynomials of
degree $k$ is investigated.
The method, as in past approaches, is first to construct a formula for a right inverse $R$ of the
differential operator $$D^{k+1}:f\mapsto D^{k+1}f:=(D^\alpha f : |\alpha|=k+1 ),$$ and then to
manipulate the expression $$R(D^{k+1}f)$$ to obtain $L_p(\Omega)$-bounds.

New formulae for such $R$ are presented. These are based on representations for the error in the
family of polynomial interpolators which includes the maps of Kergin and Hakopian.

A multivariate form of Hardy's inequality involving the linear functional of integration against a
simplex spline is given. This inequality provides a simple way to obtain $L_p(\Omega)$-bounds from
the formulae for $R(D^{k+1}f)$ given here, and many others in the literature.

**Keywords:**
scale of mean value interpolations, Kergin interpolation, Hakopian interpolation, Lagrange interpolation, Hermite interpolation, Hermite-Genocchi formula, multivariate divided difference, plane wave, lifting, Radon transform, Hardy's inequality, B-spline, simplex spline

**Math Review Classification:**
26D10, 41A05, 41A10, 41A17, 41A63, 41A80 (primary), 41A44, 41A12 (secondary)

**Length:**
70 pages

**Comment:**
Written in TeX, contains 4 figures

** Date:**
Successfully defended 7 March 1995

** Defense Committee:**
Profs. Carl de Boor (advisor), Amos Ron (reader), John Strikwerda (reader),
Richard Askey (also read thesis), Seymour Parter

**Status:**
Copy held in University of Wisconsin Library.

## Availability:

For better, or worse, this is a postscript copy of what sits somewhere in
the basement of the University of Wisconsin memorial library
Thanks to my former flatmate Kristie for binding the copy I have :-).
I also have some reasonable photocopies

I am now located
here.