Complex Orthogonal Polynomials and Numerical Quadrature via Hyponormality

By interpreting a truncated moment problem in two real variables in complex terms one exploits the hyponormality of the coordinate multiplier to obtain characterizations of: the moments of a positive measure, the associated complex orthogonal polynomials and the structure of the Hessenberg matrices. All reflecting the intricate structure of multivariate positive polynomials versus sums of squares or sums of hermitian squares of polynomials. The principal step in the proposed algorithm is stated in terms of a linear matrix inequality, making our approach prone for numerical applications.