On an analog of the Jordan-Dirichlet theorem for eigenfunction expansions of one differential-difference operator with an integral boundary condition (Q1040297)

The author states an analog of the Jordan-Dirichlet theorem in the theory of trigonometric series for expansions by eigenfunctions and adjoint functions of the operator \[ \beta y'(x)+ y'(1- x),\qquad x\in \langle 0,1\rangle,\;\beta^2\neq 1, \] where \[ y'(1- x)= {d\over d\xi} y(\xi)|_{\xi= 1-x}, \] with the integral boundary condition \[ \int^1_0 {k(t)\over t^\alpha(1- t)^\alpha}\, y(t)\,dt= 0,\qquad 0<\alpha< 1. \]