Generalized Poisson Summation Formula for Tempered Distributions

The Poisson summation formula (PSF), which relates the sampling of an analog signal with the periodization of its Fourier transform, plays a key role in the classical sampling theory. In its current forms, the formula is only applicable to a limited class of signals in $ L _{ 1 } $ . However, this assumption on the signals is too strict for many applications in signal processing that require sampling of non-decaying signals. In this paper we generalize the PSF for functions living in weighted Sobolev spaces that do not impose any decay on the functions. The only requirement is that the signal to be sampled and its weak derivatives up to order 1∕2 + ε for arbitrarily small ε > 0, grow slower than a polynomial in the $ L _{ 2 } $ sense. The generalized PSF will be interpreted in the language of distributions