An analog of the Jordan–Dirichlet theorem for an integral operator whose kernel has discontinuites on the diagonals

In the paper, we examine an integral operator whose kernel has first-kind discontinuites at the lines $t=x$ and $t=1-x$. For this operator, we prove an analog of the Jordan–Dirichlet theorem on the convergence of eigenfunction expansion. The convergence is studied using the method based on integration of the resolvent by the spectral parameter.