Lenka Motlochova

Cubature formulas of multivariate polynomials arising from symmetric orbit functions

It is shown that symmetric orbit functions, known from irreducible representations of simple Lie groups, have applications in numerical analysis. In particular, the study of remarkable properties of these functions yields cubature formulas, approximating a weighted integral of any function connected to any simple Lie group, by a weighted finite sum of function values. We summarize ideas leading to such formulas and present explicit results for simple Lie groups of rank two. We also discuss an optimal and cubature approximation of any function by multivariate polynomials arising from symmetric orbit functions and provide examples related to the Lie group $C_2$.