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

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