  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.
