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

Abstract

The question of the convergence of expansions in the eigenfunctions of a differential operator with discontinuous coefficients at a point x0 of discontinuity of the coefficients is studied. Given an arbitrary function f(x) in the class L2, a corresponding function\(\tilde f_{x_o } (x)\) is constructed which is such that at the point x0 the eigenfunction expansion of f(x) diverges with the expansion of\(\tilde f_{x_o } (x)\) into a Fourier trigonometric series.