Boundary value problems and sampling theory

This thesis is devoted to a connection between Kramer?s sampling theorem and sampling expansions generated by Lagrange interpolation. It is shown that any function that has a sampling expansion in the scope of Kramer?s theorem also has a Lagrange-type interpolation expansion provided that the kernel associated with Kramer?s theorem arises from a second-order Sturm?Liouville boundary-value problem. This new approach, which for a variety of regular and singular Sturm?Liouville problems leads to associated sampling theorems, recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions.In thesis, firstly Whittaker-Shannon-Kotel?nikov (WSK) theorem has been given and secondly some generalized theorem of WSK have been introduced. Finally, the mean theorem about sampling expansions generated by Lagrange interpolation has been proved.