On the Basis Properties of a System of Eigenfunctions of a Spectral Problem for a Second-Order Discontinuous Differential Operator in the Weighted Grand-Lebesgue Space with a General Weight

The question of the basis property of a system of eigenfunctions of one spectral problem for a discontinuous second-order differential operator with a spectral parameter under discontinuity conditions is considered in the weighted grand-Lebesgue spaces Lp),rho(0, 1), 1 < p < +infinity, with a general weight rho(center dot). These spaces are non-separable and therefore it is necessary to define its subspace associated with differential equation. In this paper, using the shift operator, a subspace Gp),rho(0, 1) is considered, in which the basis property of exponentials and trigonometric systems of sines and cosines is established when the weight function rho(center dot) satisfies the Muckenhoupt condition. It is proved that the system of eigenfunctions and associated functions of the discontinuous differential operator corresponding to the given problem forms a basis in the weighted space Gp),rho(0, 1) circle plus C,1 < p < +infinity with the weight rho(center dot) satisfying the Muckenhoupt condition. The question of the defect basis property of the system of eigenfunctions and associated functions of the given problem in the weighted spaces Gp),rho(0, 1),1 < p < +infinity, is considered.