Existence of Gaussian cubature formulas

TL;DR

This paper establishes a precise criterion for the existence of Gaussian cubature formulas by analyzing linear systems and polynomial properties, highlighting the restrictive nature of such formulas depending on variables and degree.

Contribution

It introduces a necessary and sufficient condition for Gaussian cubature formulas based on linear systems, complementing existing theorems and providing a new interpretative perspective.

Findings

Existence depends on solvability of an overdetermined linear system.

Higher variables or degree make existence conditions more restrictive.

Provides a polynomial-based interpretation of the existence condition.

Abstract

We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables the worse it gets. And for fixed number of variables, the larger the precision the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of existence of a polynomial with very specific properties.