The Poisson Sum Formula and Applications

Abstract

Let f(x) be continuous for all x. We shall presently subject this function to some further restrictions. The problem is to evaluate

$$S = \sum\limits_{n = - \infty }^\infty {f(n).} $$

Poisson’s basic idea is to introduce a variable and to consider

$$S(u) = \sum\limits_{n = - \infty }^\infty {f(n + u)} $$

(35.1)

under suitable conditions of convergence. It is clear that

$$S(u + 1) = \,S(u),$$

(35.11)

since the effect of an increase of u by 1 on the right-hand member of (35.1) is the same as a change of n into n + 1, which leaves the set of integers n(— ∞ < n < ∞) unchanged.