Poisson summation formula in weighted Lebesgue spaces

We characterize the parameters $(\alpha ,\beta ,p,q)$ for which the condition $f\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\in L^p$ and $\widehat{f}\mathrm{\mid }\xi {\mathrm{\mid }}^{\beta }\in L^q$ implies the validity of the Poisson summation formula, thus completing the study of Kahane and Lemarié-Rieusset.