Riesz–Haviland criterion for incomplete data

The aim of this paper is to analyze a “support-free” version of the Riesz–Haviland theorem proved recently by the present authors, which characterizes truncations of the complex moment problem via positivity condition on appropriate families of polynomials in z and z ¯ . The attention is focused on modifications of the positivity condition as well as the assumption on admissible truncations. The former results in truncations for which the corresponding “support-free” Riesz–Haviland condition locates a representing measure on the distinguished subset of the complex plane, while the latter effects a non-integral variant of the Riesz–Haviland theorem.