A Moment Matrix Approach to Multivariable Cubature

Abstract.

We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure μ, the bound is based on estimating $\rho (C): = \inf \{ {\text{rank}}(T - C):T{\text{ Toeplitz and }}T \geq C\} ,$ where C:=C^# [μ ] is a positive matrix naturally associated with the moments of μ. We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag(c₁,...,c _n ) (including the case when μ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps c _k  >c_k+1.