Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in P2. Given a valid Hilbert function Hof a zero-dimensional scheme in P2, we show how to construct a set of fat points Z⊆P2of double and reduced points such that HZ, the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function Hof a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. Fo r some families of valid Hilbert functions, we are also able to show that His the Hilbert function of only double 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.