Extensions of Gauss Quadrature Via Linear Programming
Gauss quadrature is a well-known method for estimating the integral of a continuous function with respect to a given measure as a weighted sum of the function evaluated at a set of node points. Gauss quadrature is traditionally developed using orthogonal polynomials. We show that Gauss quadrature can also be obtained as the solution to an infinite-dimensional linear program (LP): minimize the \(n\)th moment among all nonnegative measures that match the \(0\) through \(n-1\) moments of the given measure. While this infinite-dimensional LP provides no computational advantage in the traditional setting of integration on the real line, it can be used to construct Gauss-like quadratures in more general settings, including arbitrary domains in multiple dimensions.
gauss quadrature semi-infinite programming convex optimization 65d32 90c34 90c48