Number Theory: An Introduction to Mathematics

436 X A Character Study

the Fourier transform offis

fˆ(t)=

∫∞

−∞

f(s)e−itsds,

and the Fourier inversion formula has the form

f(s)=( 1 / 2 π)

∫∞

−∞

fˆ(t)eitsdt.

IfG=Zis the additive group of all integers, then its characters are the functions
χz:Z→C, withz∈Cand|z|=1, defined by

χz(n)=zn.

ThusGˆ is the multiplicative group of all complex numbers of absolute value 1. The
Haar integral off∈L^1 (G)is

M(f)=

∑∞

n=−∞

f(n),

the Fourier transform offis

fˆ(eiφ)=

∑∞

n=−∞

f(n)e−inφ,

and the Fourier inversion formula has the form

f(n)=( 1 / 2 π)

∫ 2 π

0

fˆ(eiφ)einφdφ.

Thus the classical theories of Fourier integrals and Fourier series are just special
cases. As another example, letG=Qpbe the additive group of allp-adic numbers.
The characters in this case are the functionsχt:Qp→C, witht∈Qp,definedby

χt(s)=e^2 πiλ(st),

whereλ(x) =

∑

j< 0 xjp

j if x ∈ Qpis given byx = ∑∞

j=−∞xjp

j,xj ∈

{ 0 , 1 ,...,p− 1 }andxj=0 for all largej<0. Also in this caseGis isomorphic and

homeomorphic toGˆitself under the mapt→χt. If we choose the Haar measure on
Gso that the measure of the compact setZpof allp-adic integers is 1, then the same
choice forGˆis the appropriate one for Plancherel’s theorem and the Fourier inversion
formula.
Consider next the case in which the group G is compact, but not necessarily
abelian. In this caseC 0 (G)coincides with the setC(G)of all continuous functions
f:G→C. The Haar integral is both left and right invariant, and we suppose it nor-
malized so that the integral of the constant 1 has the value 1. Then the integralM(f)
of anyf∈C(G),orL^1 (G), may be called theinvariant meanoff.