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

Abstract

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 $L_{p),\rho} (0,1), 1<p<+\infty$, with a general weight $\rho(.)$. 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 $G_{p),\rho} (0,1)$ is defined, in which the basis property of exponentials and trigonometric systems of sines and cosines is established when the weight function $\rho(.)$ 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 $G_{p),\rho}(0,1)\oplus\mathbb{C}$,$1<p<+\infty$ with the weight $\rho(.)$ 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 $G_{p),\rho} (0,1)$,$1<p<+\infty$, is considered.