Sampling and interpolation theories associated with boundary value
problems
This talk deals with joint work with W.N. Everritt. The link of the
sampling/interpolation theorem of Shannon-Whittaker with the
original Kramer sampling theorem is considered. Also, the
connection of these two significant results with boundary value
problems associated with linear ordinary differential equations as
defined on intervals of real line is specified. The results given
in this talk are concerned with the generation from first-order
linear, ordinary boundary value problems of Kramer analytic kernels
which introduce analytic dependence of the kernel on the sampling
parameter. These kernels are represented by unbounded self-adjoint
differential operators in Hilbert function spaces. Necessary and
sufficient conditions are given to ensure that these differential
operators have a simple, discrete spectrum which then allows the
introduction of the associated Kramer analytic kernels. Finally,
the corresponding analytic interpolation functions are defined with
the required properties, to give the Lagrange interpolation series.
