# Minimally supported error representations and approximation by the constants

## by Shayne Waldron

## Abstract:

Distribution theory is used to construct
*minimally supported* Peano kernel type representations for
linear functionals such as the error in multivariate Hermite interpolation.
The simplest case is that of representing the error in
approximation to $f$ by the constant polynomial $f(a)$ in terms of integrals
of the first order derivatives of $f$. This is discussed in detail.
Here it is shown that suprisingly there exist many representations which
are not minimally supported, and involve the integration of
first order derivatives over multidimensional regions.
The distance of smooth functions from the constants in the uniform norm
is estimated using our representations for the errror.

**Keywords:**
minimally supported error representation, distribution theory, intrinsic (geodesic) metric, Friedrichs' inequality, Poincare inequality, Sobolev's inequality

**Math Review Classification:**
41A10, 41A44, 41A80 (primary), 41A05, 65N30 (secondary)

**Length:**
14 pages

**Comment:**
Written in TeX with 4 encapsulated postscript figures included

**Last updated:**
26 March 1999

**Status:**
submitted

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