1.1. Periodic functions 5
If we set x = 0, this becomes the Poisson summation formula
The above argument is valid if / and / have rapid decay at ±oo.
As an example, take the function h(x) =
, which is self-dual with
respect to the Fourier transform, namely h(y) — h(y). Therefore, changing
variables, for any v 0 we have the Fourier pair f(x) =
2 — 1
. In this case (1.7) yields
= v~\ Y, e'^^einx).
Exercise. Using (1.8), derive the following formula of Dirichlet:
(1.10) ] P e f — J =
where eN = — — ,
n (mod N) ^ '
which is valid for any positive integer N. Here are some suggestions. As-
sume for simplicity that A T is odd. First prove that the sum (1.10), call it
G(N), satisfies the asymptotic formula G(N) ~ e^y/N as T V — » oo, which
is easier to do because G(N) can be smoothed before applying the Pois-
son summation. Then show by elementary arguments that
= NG(N). Hence the asymptotic formula G(N) ~
proves itself to the exact formula (1.10) by considering the sequence Nk with
k — 1,2,... (see [Dav] for a direct derivation).
The theory of periodic functions generalizes with obvious modifications
to any given dimension. The space X =
is acted on by T =
vector translations. The Poisson summation formula asserts that
f c n€Zfe
/(y) = J f(x)e(-xy)dx
and xy = x\yi + • • • + x^yu is the scalar product. To ensure convergence
it suffices that /(x) is in the Schwartz class, i.e., all its partial derivatives
/ ^ ( x ) « |x|-
and A 0 is arbitrary, the implied constant depending on A and (j) =