Аннотация:The author considers the fourth-order differential operator Lu = pu(4)+
(ru′)′ + q u, where p is piecewise constant, r and q are smooth functions in (a,b)\{x0}; x0 in (a,b) is a fixed point. The author examines the problem of eigenvalues and eigenfunctions of the operator L in the interval (a,b) with homogeneous boundary conditions at the points a and b.
For any function f in L2(a,b), he defines a function f~ x0 (x) and proves that the expansion of f in eigenfunctions of the operator L is convergent at the point x if and only if the Fourier trigonometric series of f x 0 is convergent at x . These results generalize the results of V. A. Ilʹin [Mat. Zametki 22 (1977), no. 5, 679–698; MR0499820].