For the Sturm-Liouville eigenvalue problem −
[−1, 1] with Dirichlet boundary conditions and with an indefinite weight function
changing it’s sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space
[−1, 1]. In the nineties the sufficient so called generalized one hand Beals condition was found for this Riesz basis property. Now using a new criterion of Parfyonov we show that already the old approach gives rise to a necessary and sufficient condition for the Riesz basis property under certain additional assumptions.