Gaussian Quadratures with prescribed nodes via moment theory

TL;DR

This paper addresses the problem of constructing Gaussian quadrature rules with prescribed nodes using moment theory, extending previous work on minimal quadrature rules and their determinantal representations.

Contribution

It introduces a method based on moment theory to construct Gaussian quadratures with multiple prescribed nodes, advancing the theory beyond single-node prescriptions.

Findings

Provides a new construction method for quadratures with prescribed nodes

Extends previous determinantal representations to multiple nodes

Solves an open problem on prescribed nodes in quadrature rules

Abstract

Let $\mu$ be a positive Borel measure on the real line and let $L$ be the linear functional on univariate polynomials of bounded degree, defined as integration with respect to $\mu$. In 2020, Blekherman et al., the characterization of all minimal quadrature rules of $\mu$ in terms of the roots of a bivariate polynomial is given and two determinantal representations of this polynomial are established. In particular, the authors solved the question of the existence of a minimal quadrature rule with one prescribed node, leaving open the extension to more prescribed nodes. In this paper, we solve this problem using moment theory as the main tool.