Conditions for an Indefinite Sturm-Liouville Riesz Basis Property

We consider the Sturm-Liouville problem -y '' = lambda ry on [-1,1] with Dirichlet boundary conditions and with an indefinite weight function r which changes sign at 0. We discuss several conditions known to be either necessary or sufficient for the eigenfunctions to form a Riesz basis of the Hilbert space L-2 vertical bar r vertical bar(-1, 1). Assuming that the odd part of r dominates the even part in a certain sense, we show that the above conditions (and also some new ones) are in fact all equivalent to this Riesz basis property.