$L_p$-convergence for expansions in terms of the eigenfunctions of a Sturm-Liouville problem

For the operator $Ly=-(x^{2\alpha}y')'$, $x\in[0,1]$, $y(0)=y(1)=0$ with $0\leqslant\alpha<1/2$, or $|y|<\infty$, $y(1)=0$ with $1/2\leqslant\alpha<1$ we investigate the effect which the singularity of the Sturm–Liouville operator derived from this self-adjoint expression has on $L_p$-convergence of expansions in terms of the eigenfunctions of this operator. We will prove that the orthonormalized system of eigenfunctions forms a basis in $L_p[0,1]$ for $2/(2-\alpha)<p<2/\alpha$.