Geodesic
Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a~differential operator
Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 679-698

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The question of the convergence of expansions in the eigenfunctions of a differential operator with discontinuous coefficients at a point $x_0$ of discontinuity of the coefficients is studied. Given an arbitrary function $f(x)$ in the class $L_2$, a corresponding function $\widetilde f_{x_0}(x)$ is constructed which is such that at the point $x_0$ the eigenfunction expansion of $f(x)$ diverges with the expansion of $\widetilde f_{x_0}(x)$ into a Fourier trigonometric series. 
@article{MZM_1977_22_5_a8,
     author = {V. A. Il'in},
     title = {Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a~differential operator},
     journal = {Matemati\v{c}eskie zametki},
     pages = {679--698},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a8/}
}
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V. A. Il'in. Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a~differential operator. Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 679-698. http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a8/