On a $q$-boundary value problem with discontinuity conditions

In this paper, we studied $q$-analogue of Sturm-Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the $q$-Sturm-Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of $q$-Sturm-Liouville boundary value problem. We shown that eigenfunctions of $q$-Sturm-Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson's type.