Possion Summation for LCA Groups
November 1, 2013
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Possion Summations
Possion Summation on R. Let f ∈ L1 (R). Then for all x ∈ [0, 1] define
X
F (x) =
f (x + n)
(1)
n∈Z
If F is well-defined then F ∈ L1 (0, 1) since f ∈ L1 (R). In this case, let cm = hF, en i. Then
Z
cm =
1
−2πimx
F (x)e
Z
dx =
0
1X
0
f (x + n)e
−2πimx
Z
dx =
f (x)e−2πimx dx = fˆ(m)
(2)
R
n
From the other side we have
F =
X
cm e−2πimx
(3)
m
Therefore by (1), (2) and (3) we obtain the following.
X
n
f (x + n) =
X
cm e−2πimx =
m
X
fˆ(m)e−2πimx
m
for all x ∈ [0, 1]. If we let x = 0, then we obtain the Possion summation:
X
f (n) =
n
X
fˆ(m).
m
Observations: Notice that in above we used the fact that the real line is tiled by the
interval [0, 1] and for any y ∈ R there are unique n ∈ Z and x ∈ [0, 1] such that y = x + n.
Dual space. Let G be a locally compact abelian group. A map α : G → S 1 from
G into the circle is called a character of G if it is continuous and homomorphism, i.e.,
α(g1 g2 ) = α(g1 )α(g2 ), where g1 g2 is the product in the group G.
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Let Ĝ denote the set of all such characters α. We call Ĝ the dual space of G. The dual
space Ĝ is a multiplicative group and
(α1 α2 )(g) = α1 (g)α2 (g)
α(g)α(g −1 ) = α(e) = 0
Let Γ be a disceret countable subgroup of G such that the quotient group G/Γ is
compact. And, let F ⊂ G relatively compact domain such that for any g ∈ G there is γ ∈ Γ
and f ∈ F such that g can be uniquely written as g = γf . In this case G = ΓF = ∪γ∈Γ γF .
The set F is called fundamental domain.
Example 1. Let G = R. The character space of R is identified by the group of map
ξ → eξ for all ξ ∈ R and eξ (x) = e−2πixξ for all x ∈ R. Therefore one can identify the
character space R̂ by the space R, i.e., R ≡ R̂.
Let Γ = Z. Then R/Z is compact and is identified by the unit circle S 1 . In this case,
F = [0, 1] and for any x ∈ R one has x = n + τ for a unit n ∈ Z and τ ∈ [0, 1].
Example 2. The characters of Zn , the group of n-dimensional integer numbers, is
identified by the group of n-dimensional torus Tn by ξ → eξ . And, the dual space of Tn is
identified by Zn by n → en .
Definition Let Γ be a subgroup of G and α ∈ Ĝ be a character of G. We say α
annihilates Γ if α(γ) = 1 for all γ ∈ Γ. The set of such α’s is called the annihilator of Γ
and is denoted by Γ⊥ . This set a closed subgroup of Ĝ.
Example: Let G = R. Then Γ = Z as a set in R is the annihilator of Z as a subgroup
of R.
Lemma 1.0.1. Let G equipped
with Haar measure. Let Γ be a discrete countable subgroup
P
of G. Define F (gΓ) = γ∈Γ f (gγ). If f ∈ L1 (G, dg), then F is well-defined and belongs
to L1 (G/H). Moreover, F̂ = fˆ| ⊥ where (G/Γb) can be identified by Γ⊥ .
Γ
Theorem 1.0.2 (Theorem (4.42) [1]). Let Γ be a discrete countable subgroup of G and
f ∈ L1 (G, dg). Then for any g ∈ G/Γ
Z
X
f (gγ) =
fˆ(ξ)ξ(g)dξ
ξ∈Γ⊥
γ∈Γ
Corollary 1.0.3 (Poisson summation). With the hypothesis of the Theorem, for g = e we
have
Z
X
f (γ) =
fˆ(ξ)dξ
ξ∈Γ⊥
γ∈Γ
If Γ⊥ is discrete, then we can write
X
f (γ) =
X
ξ∈Γ⊥
γ∈Γ
2
fˆ(ξ)
The classical Poisson summation is the case when G = R, Γ = Z, and Γ⊥ = Z.
References
[1] G. Folland, A course in Abstract Harmonic Analysis, CRC Press, 1995.
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