Cubature formulas of multivariate polynomials arising from symmetric orbit functions

TL;DR

This paper introduces new cubature formulas based on symmetric orbit functions from Lie group representations, enabling efficient approximation of integrals and functions in multivariate settings, especially for rank-two Lie groups.

Contribution

It develops novel cubature formulas using symmetric orbit functions, connecting Lie group theory with numerical integration techniques for multivariate functions.

Findings

Cubature formulas derived from symmetric orbit functions for simple Lie groups.

Specialized formulas for rank-two Lie groups.

Optimal polynomial approximation methods discussed.

Abstract

The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.