# 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

## Availability:

This article is available in:
- Postscript (as it will be formatted in the SIAM Journal of Mathematical Analysis)

Posted with permission from the SIAM Journal on Mathematical Analysis,
Vol. 28, No. 1, pp. 233-258. © Copyright 1997 by the Society for
Industrial and Applied Mathematics, Philadelphia, PA. All rights reserved.
- Postscript
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