The technique of reconstructing a higher-resolution (HR) image of size $ML\times ML$ by digitally processing $L\times L$ subpixel-shifted lower-resolution (LR) copies of it, each of size $M\times M$ , has now become well established. This particular digital superresolution problem is analyzed from the standpoint of the generalized sampling theorem. It is shown both theoretically and by computer simulation that the choice of regularly spaced subpixel shifts for the LR images tends to maximize the robustness and minimize the error of reconstruction of the HR image. In practice, since subpixel-level control of LR image shifts may be nearly impossible to achieve, however, a more likely scenario, which is also discussed, is one involving random subpixel shifts. It is shown that without reasonably tight bounds on the range of random shifts, the reconstruction is likely to fail in the presence of even small amounts of noise unless either reliable prior information or additional data are available.