Gaussian Quadratures with prescribed nodes via moment theory

TL;DR

The paper tackles constructing Gaussian quadrature rules with finitely many prescribed nodes for a positive Borel measure on $\mathbb{R}$. It adopts a moment-theoretic approach: the prescribed nodes are encoded via a monic polynomial with those zeros, generating a univariate moment sequence that extends the original data, after which the classical $\mathbb{R}$-TMP framework is applied. The main results provide explicit necessary and sufficient conditions for the existence of a (generalized) GQR with prescribed nodes, realized through an extension of the moment sequence and resulting in constructive recovery of the remaining nodes and weights. By solving the multi-prescribed-node case, the work connects prior determinantal characterizations to a practical moment-extension method, advancing both theory and computation of minimal quadrature rules.

Abstract

Let $μ$ 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 $μ$. In 2020, Blekherman et al., the characterization of all minimal quadrature rules of $μ$ 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.