Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in the projective plane. Given a valid Hilbert function H of a zero-dimensional scheme inthe projective plane, we show how to construct a set of fat points Z of double and reduced points such that the Hilbert function of Z is the same as H. In other words, we show that any valid Hilbert function H of a zero-dimensional scheme is the Hilbert function of some set of double and reduced points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines