UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES Symmetries of linear functionals

It is shown that a linear functional on a space of functions can be described by G, a group of its symmetries, together with the restriction of 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] is given, and a rule of degree 5 with 3 nodes for the linear functional f 7! R h h Df is derived. AMS (MOS) Subject Classi cations: primary 41A05, (interpolation) 13A50, (invariant theory) 14D25 (geometric invariants) secondary 65D25, (numerical di erentiation 65D30, (numerical integration) 65D32 (quadrature and cubature formulas)