@inbook{10.2307/j.ctt130hkbh.17,
URL = {http://www.jstor.org/stable/j.ctt130hkbh.17},
abstract = {We first derive a simple classical version of the Poisson summation formula. Letfbe a function of differentiability classCvonRn. We consider the series(1)g(x)=\sum_{m_{1},\; m_{n}\; \; z}f(x_{1}+m_{1},\cdots ,x_{n}+m_{n})and assume that it and all of the series obtained by replacingfby any of its mixed partial derivatives of orders\leqq vconverge normally onRn, so that alsog∈Cv. Clearlyg(x+m=g(x) for allx∈Rn,m∈Zn; then ifνis large enough, it is known (Chapter 9, section 6) thatghas a Fourier expansion(2)g(x) = \sum_{l_{1},\; \; ,l_{n\; \; Z}}a_{l_{1}}\; _{l_{n}}e(l_{1}x_{1}+\cdots +l_{n}x_{n})converging normally onRn, where},
bookauthor = {Walter L. Baily},
booktitle = {Introductory Lectures on Automorphic Forms},
pages = {243--252},
publisher = {Princeton University Press},
title = {THETA FUNCTIONS AND AUTOMORPHIC FORMS},
year = {1973}
}