Abstract and Applied Analysis
Volume 2004 (2004), Issue 5, Pages 371-385
doi:10.1155/S108533750430624X
On sampling expansions of Kramer type
Department of Mathematics, National and Capodistrian University of Athens, Panepistimioupolis, 157 84 Athens, Greece
Received 31 October 2002
Copyright © 2004 Anthippi Poulkou. 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
We treat some recent results concerning sampling
expansions of Kramer type. The linkof the sampling
theorem of Whittaker-Shannon-Kotelnikov with the Kramer sampling
theorem is considered and the connection of these theorems with
boundary value problems is specified. Essentially, this paper
surveys certain results in the field of sampling theories and
linear, ordinary, first-, and second-order boundary value problems
that generate Kramer analytic kernels. The investigation
of the first-order problems is tackled in a joint work with Everitt. For the second-order
problems, we refer to the work of Everitt and Nasri-Roudsari in
their survey paper in 1999. All these problems are represented by
unbounded selfadjoint differential operators on Hilbert function
spaces, with a discrete spectrum which allows the introduction of
the associated Kramer analytic kernel. However, for the
first-order problems, the analysis of this paper is restricted to
the specification of conditions under which the associated
operators have a discrete spectrum.