A Generalization of the Sampling Theorem

The sampling theorem for bandlimited functions allows one to reconstruct exactly a function containing no frequencies higher than W cps, given the values of the function at equispaced sampling points ( R + 1)/ W sec apart. This theorem is generalized to allow reconstruction, given the values of the function and its first R derivatives at equispaced sampling points, ( R + 1)/2 W sec apart. For large R , the R -derivative expansion approaches a Taylor's series weighted by a Gaussian density about each sample point.