# Symmetries of Linear Functionals

## by Shayne Waldron

## Abstract:

It is shown that a linear functional $\gl$ on a space of functions can be described
by $G$, a group of its symmetries, together with the restriction of $\gl$ to certain $G$-invariant
functions.
This simple consequence of invariant theory has long been used, implicitly, in the construction of
numerical integration rules. It is the author's hope that, by showing that these ideas have nothing to
do with the origin of the linear functional considered, e.g., as an integral, they will be applied more
widely, and in a systematic manner.

As examples, a complete characterisation of the rules of degree (precision) 3 with 4 nodes for
integration on the square $[-1,1]^2$ is given, and a rule of degree 5 with 3 nodes for the linear
functional $f\mapsto\int_{-h}^h D^2f$ is derived.

**Keywords:**
linear functional, symmetry group, invariant theory, invariant polynomial, Poincar\'e
series, Molien series, integrity basis, numerical integration rules, numerical differentiation rules

**Math Review Classification:**
41A05, 13A50, 14D25 (primary), 65D25, 65D30, 65D32 (secondary)

**Length:**
10 pages

**Comment:**
Written in TeX, contains 1 figure

**Last updated:**
25 March 1996

**Status:**
Appeared in Approximation Theory VIII - Vol. 1, pp 541--550, (edited by C. K. Chui and L. L. Schumaker), World Scientific, 1995

## Availability:

This article is available in:
- Postscript /
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DVI
(almost exactly as it occurred in the Texas VIII conference proceedings)
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