III-posed sampling theorems

There are a number of innocent appearing sampling theorems that are ill-posed, i.e., a small amount of noise superimposed on the data can render the interpolation unstable. Using Papoulis' Generalized Sampling Theorem, we show that a sufficient condition for a sampling theorem to be ill-posed is that an interpolation function has infinite energy. Specific examples include the case where (a) the signal and the
(2 n)
th derivative of the signal are both simultaneously sampled at half the Nyquist rate and (b) the signal's and the
(2 n + 1)
th derivative's samples are interlaced at Nyquist intervals.