Expansion theorems for a class of regular indefinite eigenvalue problems (Q1174367)

The authors consider indefinite boundary eigenvalue problems of the form \[ y^{(n)}(x)+\sum^ n_{\nu=2}f_ \nu(x)y^{(n-\nu)}(x)=\lambda r(x)y(x)\qquad (x\in[0,1]), \] \[ U_{\nu0}(y)+U_{\nu1}(y)=0\qquad (\nu=1,2,\dots,n), \] where \(n\geq 2\), the coefficients \(f_ \nu\) belong to \(L_ 1(0,1)\), \(r:[0,1]\to\mathbb{R}\backslash\{0\}\) is a step function, and the two-point boundary conditions are assumed to be normalized. In the paper reviewed above the authors have introduced the notion of regularity for such eigenvalue problems. In the present paper they study the asymptotic behaviour of the corresponding Green's function and prove the pointwise convergent of the eigenfunction expansion for functions \(f\in L_ 1(0,1)\).