Gaussian Quadratures with prescribed nodes via moment theory

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.

14 pages