Abstract
A generalization of the sampling theorem is presented, taking the sampled higher order derivatives of a function into account. Mathematical properties of the sampling expansion formula obtained in this paper are also discussed in some detail. Examples of sampling formulae are suggested, which include the formulae given by Lagrange, Shannon, Someya, Takizawa, Kroll, Isomiti, van der Pol, Jagerman-Fogel, and Linden-Abramson, as special cases of the present theorem. Formulae given here can be effectively applied to interpolation or extrapolation formulae, which play an important role in the fields of numerical control, such as splinefunction analysis, computer-aided design, and computer-aided manufacturing.
