1.1. Periodic functions 5
If we set x = 0, this becomes the Poisson summation formula
(i-8)
£/(ra)
=
E/(n)-
The above argument is valid if / and / have rapid decay at ±oo.
As an example, take the function h(x) =
e~nx
, 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) =
e_7nE v
1
f(y) =
_ 1
2 1
v~2e~wy v
. In this case (1.7) yields
(1.9) Yl
e~x^2v
= v~\ Y, e'^^einx).
Exercise. Using (1.8), derive the following formula of Dirichlet:
(1.10) ] P e f J =
SNVN,
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
G(N2)
= N
and
G(N3)
= NG(N). Hence the asymptotic formula G(N) ~
SNVN
im-
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 =
Rfc
is acted on by T =
Zk
as integral
vector translations. The Poisson summation formula asserts that
(i.ii) E
/(n)
= E
/(n)
neZ
f c n€Zfe
where
/(y) = J f(x)e(-xy)dx
Rk
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
satisfy
/ ^ ( x ) « |x|-
A
where
|x|
=
(s?
+
.••
+
*!)*
and A 0 is arbitrary, the implied constant depending on A and (j) =
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