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