A Moment Matrix Approach to Multivariable Cubature
نویسندگان
چکیده
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 ρ(C) := inf{rank (T − C) : T Toeplitz and T ≥ 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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