Almost periodic extension of band-limited functions and its application to nonuniform sampling

The classical sampling theorem for band-limited functions is shown to admit generalization in the following sense: a bandlimited Fourier transform can be arbitrarily extended on the frequency axis. If this extension is performed in such a fashion that the result is an almost periodic function, the sampling theorem can be shown to hold for sampling at nonuniformly spaced instants in time. This result is proved, and the generalized sampling theorem is shown to provide a simple, constructive proof of the multichannel sampling theorem. A new sampled data Fourier transform is introduced which retains information about the sampling instants.