On sampling theory and basic Sturm-Liouville systems

We investigate the sampling theory associated with basic Sturm-Liouville eigenvalue problems. We derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson's type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic Green's function of the basic Sturm-Liouville systems. Examples involving basic sine and cosine transforms are given.