Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a~differential operator

V. A. Il'in

Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 679-698.

Résumé

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.