# Dissertation University of Wisconsin-Madison $L_p$-error bounds for multivariate polynomial interpolation schemes

## 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