Constructing a Basis from Systems of Eigenfunctions of one not Strengthened Regular Boundary Value Problem

We investigate a nonlocal boundary value spectral problem for an ordinary di ff erential equation in an interval. Such problems arise in solving the nonlocal boundary value problem for partial equations by the Fourier method of variable separation. For example, they arise in solving nonstationary problems of di ff usion with boundary conditions of Samarskii-Ionkin type. Or they arise in solving problems with stationary di ff usion with opposite flows on a part of the interval. The boundary conditions of this problem are regular but not strengthened regular. The principal di ff erence of this problem is: the system of eigenfunctions is comlplete but not forming a basis. Therefore the direct applying of the Fourier method is impossible. Based on these eigenfunctions there is constructed a special system of functions that already forms the basis. However the obtained system is not already the system of the eigenfunctions of the problem. We demonstrate how this new system of functions can be used for solving a nonlocal boundary value problem on the example of the Laplace equation.