Geodesic
$L_p$-convergence for expansions in terms of the eigenfunctions of a Sturm-Liouville problem
Matematičeskie zametki, Tome 3 (1968) no. 6, pp. 683-691 
V. L. Generozov. $L_p$-convergence for expansions in terms of the eigenfunctions of a Sturm-Liouville problem. Matematičeskie zametki, Tome 3 (1968) no. 6, pp. 683-691. http://geodesic.mathdoc.fr/item/MZM_1968_3_6_a7/
@article{MZM_1968_3_6_a7,
     author = {V. L. Generozov},
     title = {$L_p$-convergence for expansions in terms of the eigenfunctions of {a~Sturm-Liouville} problem},
     journal = {Matemati\v{c}eskie zametki},
     pages = {683--691},
     year = {1968},
     volume = {3},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_6_a7/}
}
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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).