Number Theory: An Introduction to Mathematics

8 Generalizations 435

For each fixeds∈G,themapˆs:χ→χ(s)is a character ofGˆ. Moreover the map
s→ˆsis one-to-one, by the theorem of Gelfand and Raikov, and every character of
Gˆis obtained in this way. In fact theduality theoremof Pontryagin and van Kampen
(1934/5) states thatGis isomorphic and homeomorphic to the dual group ofGˆ.
TheFourier transformof a functionf∈L^1 (G)is the functionfˆ:Gˆ→Cdefined
by

fˆ(χ)=

∫

G

f(s)χ(s)dμ(s),

whereμis the Haar measure onG.Iff 1 ,f 2 ∈L^1 (G)∩L^2 (G),thenfˆ 1 ,fˆ 2 ∈L^2 (Gˆ)
and, with a suitable fixed normalization of the Haar measureμˆonGˆ,

(f 1 ,f 2 )G=(fˆ 1 ,fˆ 2 )Gˆ.

Furthermore, the mapf → fˆcan be uniquely extended to a unitary map ofL^2 (G)
ontoL^2 (Gˆ). This generalizesPlancherel’s theoremfor Fourier integrals on the real
line.
Iff=g∗h,whereg,h∈L^1 (G),thenf∈L^1 (G)and

fˆ(χ)=ˆg(χ)hˆ(χ) for everyχ∈Gˆ.

If, in addition,g,h∈L^2 (G),thenfˆ∈L^1 (Gˆ)and, with the same choice as before for
the Haar measureμˆonGˆ,theFourier inversion formulaholds:

f(s)=

∫

Gˆ

fˆ(χ)χ(s)dμ(χ).ˆ

ThePoisson summation formulacan also be extended to this general setting. Let
Hbe a closed subgroup ofGand letKdenote the factor groupG/H. If the Haar
measuresμ,vˆonH,Kˆ are suitably chosen then, with appropriate hypotheses on
f∈L^1 (G),
∫

H

f(t)dμ(t)=

∫

Kˆ

fˆ(ψ)dˆv(ψ).

We now give some examples (without spelling out the topologies). IfG=Ris the
additive group of all real numbers, then its characters are the functionsχt:R→C,
witht∈R,definedby

χt(s)=eits.

In this caseGis isomorphic and homeomorphic toGˆitself under the mapt→χt.The
Haar integral off∈L^1 (G)is the ordinary Lebesgue integral

M(f)=

∫∞

−∞

f(s)ds,