Integral Equations and Operator Theory

, Volume 52, Issue 1, pp 85–124

A Moment Matrix Approach to Multivariable Cubature

Authors

    • Department of Computer ScienceState University of New York
  • Srdjan Petrovic
    • Department of MathematicsWestern Michigan University
Original Paper

DOI: 10.1007/s00020-003-1334-9

Cite this article as:
Fialkow, L. & Petrovic, S. Integr. equ. oper. theory (2005) 52: 85. doi:10.1007/s00020-003-1334-9

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.

Mathematics Subject Classification (2000).

Primary 47A57Secondary 44A6065D32

Keywords.

Cubature rulespositive moment matrixflat extensions

Copyright information

© Birkhäuser Verlag, Basel 2005