On the use of green\u27s function in sampling theory

There are many papers dealing with Kramer\u27s sampling theorem associated with self-adjoint boundary-value problems with simple eigenvalues. To the best of our knowledge, Zayed was the first to introduce a theorem that deals with Kramer\u27s theorem associated with Green\u27s function of not necessarily self-adjoint problems which may have multiple eigenvalues, but no examples of sampling series associated with either non-self-adjoint problems or problems with multiple eigenvalues were given. We define two classes of not necessarily self-adjoint problems for which sampling theorems can be derived and give a sampling theorem associated with Green\u27s function of self-adjoint problems. Finally, we give some examples that illustrate our technique. © 1998 Rocky Mountain Mathematics Consortium

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\u27s 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\u27s function of the basic Sturm-Liouville systems. Examples involving basic sine and cosine transforms are given. © 2006 Elsevier B.V. All rights reserved