Extremal Solutions of the Two-dimensional L-problem of Moments, Ii

All extremal solutions of the truncated L-problem of moments in two real variables , with support contained in a given compact set, are described as characteristic functions of semi-algebraic sets given by a single polynomial inequality. An exponential kernel, arising as the determinantal function of a naturally associated hyponormal operator with rank-one self-commutator, provides a natural defining function for these semi-algebraic sets. We find an intrinsic characterization of this kernel and we describe a series of analytic continuation properties of it which are closely related to the behaviour of the Schwarz reflection function in portions of the boundary of the extremal supporting set.