Sampling and generalized almost periodic extension of functions

The author explores the connections between nonuniform sampling of a certain function and the almost periodic extension of its Fourier transform. It is shown that the Fourier transform of a bandlimited function can be extended (as a weighted sum of translates) as a Stepanoff almost periodic function, to the whole frequency axis. This result leads to a generalized nonuniform sampling theorem which, unlike previous results, does not require the continuity of the Fourier transform of the sampled function, and is valid for finite-energy band-limited functions.