  Not every positive functional defined on bi-variate polynomials of a
prescribed degree bound is represented by the integration against a positive
measure. We isolate a couple of conditions filling this gap, either by
restricting the class of polynomials to harmonic ones, or imposing the
vanishing of a defect indicator. Both criteria offer constructive cubature
formulas and they are obtained via well known matrix analysis techniques
involving either the dilation of a contractive matrix to a unitary one or the
specific structure of the Hessenberg matrix associated to the multiplier by the
underlying complex variable.
