Finite Lagrange and Cauchy sampling expansions associated with regular difference equations

We use a discrete version of Kramer's sampling theorem to derive sampling e xpansions for discrete transforms whose kernels arise from regular differen ce equations. The kernels may be taken to be either solutions or Green's fu nctions of second order regular selfadjoint boundary-value problems. In bot h cases, the sampling expansions obtained are written in forms of finite-La grange-type interpolation expansions. Cauchy-type sampling expansions for p eriodic bandlimited signals will be derived using first order boundary-valu e problems. Illustrative examples are also given.