, Volume 52, Issue 1, pp 85-124

A Moment Matrix Approach to Multivariable Cubature

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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(c 1,...,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.