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,...,cn) (including the case when μ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps ck >ck+1.
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