Finite lagrange and cauchy sampling expansions associated with regular difference equations*

Abstract

We use a discrete version of Kramer's sampling theorem to derive sampling expansions for discrete transforms whose kernels arise from regular difference equations. The kernels may be taken to be either solutions or Green's functions of second order regular self-adjoint boundary-value problems. In both cases, the sampling eexpansions obtained are written in forms of finite-Lagrange-type interpolation expansions. Cauchy-type sampling expansions for periodic bandlimited signals will be derived using first order boundary-value problems. Illustrative exampls are also given.