Abstract and Applied Analysis
Volume 2011 (2011), Article ID 610232, 30 pages
http://dx.doi.org/10.1155/2011/610232
Research Article

Discontinuous Sturm-Liouville Problems and Associated Sampling Theories

M. M. Tharwat1,2

1Department of Mathematics, University College, Umm Al-Qura University, Makkah, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Received 23 May 2011; Revised 14 August 2011; Accepted 18 August 2011

Academic Editor: Yuming Shi

Copyright © 2011 M. M. Tharwat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the sampling analysis associated with discontinuous Sturm-Liouville problems with eigenvalue parameters in two boundary conditions and with transmission conditions at the point of discontinuity. We closely follow the analysis derived by Fulton (1977) to establish the needed relations for the derivations of the sampling theorems including the construction of Green's function as well as the eigenfunction expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green's functions. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in the work of Annaby and Tharwat (2006).