Sampling theorem for polynomial interpolation

The classic polynomial interpolation approach is used to derive a sampling theorem for the class of signals that are the response of systems described by differential equations with constant coefficients. In particular, the polynomial that interpolates the signal between the sampling points for increasing order will give increasing accuracy for stable one-sided sequences, if the sampling rate is at least six times the highest pole frequency (Bolgiano sampling rate [7]). The convergence is ensured also for nonstable poles that lie in a certain region of the complex plane. If, instead, we use the symmetrical polynomial approach, it is enough to sample at a rate that is just two times the highest pole frequency (Nyquist sampling rate), with some constraints on the real part of the poles. A bound for the error is derived for both cases and a comparison to the Shannon-Whittaker sampling theorem is presented.