Minimally supported error representations and approximation by the constants

by Shayne Waldron


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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