Consider a regular Sturm-Liouville problem \(-f'' = \lambda r f\) with a weight function \(r \in L^1[-1,1]\) , equipped with Neumann boundary conditions. If r is positive then \([f,g]_r:= \int f \overline{g} r \, dx\) defines a Hilbert space inner product and with the normed orthogonal eigenfunctions \(f_n\) the Parseval equation \([f,f]_r = \Sigma \, |[f,f_n]_r|^2\) is well known. If r changes its sign, say \(x r(x) > 0\) , then \([\cdot , \cdot ]_r\) induces a Krein space. In this case a Parseval type equation is obtained if and only if the eigenfunctions \(f_n\) form a Riesz basis. The setting from Fleige, A.: Positive and negative eigenfunction expansion results for indefinite Sturm-Liouville problems. Integr. Equ. Oper. Theory 95, 5 (2023) is used in order to present an example for its failure. A similar result is obtained for the eigenvalue problem \(-(\frac{u'}{r})' = \lambda u\) with Dirichlet boundary conditions.
