The algebra of interpolatory cubature formulae for generic nodes

TL;DR

This paper explores the algebraic structure of interpolatory cubature formulas for computing expected values of multivariate functions, integrating tools from algebra, polynomial theory, and probability to enhance numerical integration methods.

Contribution

It introduces a novel algebraic framework for interpolatory cubature formulas with generic nodes, combining algebraic, polynomial, and probabilistic techniques.

Findings

Develops a new algebraic approach to cubature formulas

Provides theoretical insights into the structure of interpolatory rules

Enhances understanding of numerical integration in multiple dimensions

Abstract

We consider the classical problem of computing the expected value of a real function $f$ of the $d$-variate random variable $X$ using cubature formul\ae. We use in synergy tools from Commutative Algebra for cubature rul\ae, from elementary orthogonal polynomial theory and from Probability.