Positive Functionals and Hessenberg Matrices

TL;DR

This paper explores conditions under which positive functionals on bivariate polynomials can be represented by positive measures, using matrix analysis techniques involving Hessenberg matrices and harmonic polynomials.

Contribution

It introduces new criteria involving harmonic polynomials and defect indicators to characterize when positive functionals admit measure representations, with constructive cubature formulas.

Findings

Criteria involving harmonic polynomials for measure representation

Use of Hessenberg matrices in constructing cubature formulas

Conditions for positive functionals to be represented by positive measures

Abstract

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.