A multivariate form of Hardy's inequality and $L_p$-error bounds for multivariate Lagrange interpolation schemes

by Shayne Waldron

Abstract:

The following multivariate generalisation of Hardy's inequality, that for $m-n/p>0$
$$\norm{\ x\mapsto\int_{[\underbrace\xs_m,\Theta]}f\ }_p \le{\norm{f}_{p}\over(m-1)!(m-n/p)_{\#\Theta}},\eqno(*)$$
valid for $f\in L_p(\Rn)$ and $\Theta$ an arbitrary finite sequence of points in $\Rn$, is discussed.

The linear functional $f\mapsto\int_\Theta f$ was introduced by Micchelli [M80] in connection with Kergin interpolation. This functional also naturally occurs in other multivariate generalisations of Lagrange interpolation, including Hakopian interpolation, and the Lagrange maps of Section 5. For each of these schemes, (*) implies $L_p$-error bounds.

We discuss why (*) plays a crucial role in obtaining $L_p$-bounds from pointwise integral error formulae for multivariate generalisations of Lagrange interpolation.

Keywords: Hardy's inequality, Lagrange interpolation, Kergin interpolation, Hakopian interpolation, B-spline, simplex spline, Hermite-Genocchi formula

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

Length: 26 pages (SIAM format), 29 pages (technical report)

Comment: Written in TeX, contains 4 figures. The following changes were made to the original CMS technical report #95-02 (these include further results on the sharpness of the inequality). This paper is the basis of Chapter 2 of Shayne Waldron's dissertation. There are a number of related papers, including Multipoint Taylor formulae.

Last updated: 28 Octover 1997

Status: Appeared in the SIAM Journal on Mathematical Analysis, 28(1), 1997, pages 233--258

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