On well-posedness of the Papoulis generalized sampling expansion

K.F. Cheung and R.J. Marks (ibid., vol.CAS-32, no.5 p.481-4, 1985) have shown that if at least one of the interpolation functions used in the generalized sampling expansion of A. Papoulis (1977) is not square-integrable, then the problem is ill-posed in the sense that the variance of the reconstruction error is unbounded when noisy samples are used. It is shown that if all the interpolation functions are square-integrable, then the generalized sampling problem is well posed, essentially a converse of the Cheung-Marks result. A useful alternative sufficient condition for well-posedness that depends only on the transfer characteristics of the channels and does not require explicit calculation of the interpolation functions is developed. Several examples illustrating the results are given; in particular, the important nonuniform sampling strategy of J.L. Yen (IRE Trans. Circuits Theory, vol. CT-3, p.251-7, Dec. 1956) using bunched samples is shown to be well posed