Poisson's summation formula in the construction of wavelet bases

In the construction of Multiresolution Analyses for non-Hilbert spaces arising in applications, it is necessary to examine generalized sampling operators, i.e., linear operators of multiplier type from function spaces to sequence spaces. Studying these operators on the Fourier side requires Poisson's Summation Formula (PSF). It is well known that on the Schwartz space, PSF holds pointwise. We exhibit weaker conditions under which PSF holds in a more general context. To this end, we consider functions of bounded variation, functions in mixed norm spaces, and continuous vector valued functions.