On sampling expansions of Kramer type (Q1773519)

Summary: We treat some recent results concerning sampling expansions of Kramer type. A link of 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 \textit{W. N. Everitt} [J. Comput.Appl. Math. 148, 29--47 (2002; Zbl 1019.34031)]. For the second-order problems, we refer to the work of W. N. Everitt and G. Nasri-Roudsari in their survey paper in 1999 [Interpolation and sampling theories, and linear ordinary boundary value problems; sampling theory in Fourier and signal analysis, Vol. 5, Oxford: Oxford University Press, 96--129 (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.