4 I. HECK E L-FUNCTIONS

identity in G is given by the sets of the form

{x € G : \X(g) - 1| e for all

5

€ K},

where K is a compact subset of G and e 0. It is not difficult to show that with

this topology G is a locally compact abelian group. We shall refer to G as the dual

group of G.

Observe that for each g G G, the mapping

(i) G ^ T , x^x(g),

defines a continuous homomorphism and is therefore a character of G. We shall

implicitly make frequent use of the following result:

THEOREM 4. (Pontrjagin Duality) Every character on G is of the form (1)

and the topology of uniform convergence on compact subsets of G coincides with the

original topology on G, that is to say if G is the dual of G, then G is the dual of

G.

Other useful results from duality theory are:

(1) The dual group of any compact group is discrete.

(2) The dual group of any discrete group is compact.

(3) If H is a closed subgroup of the locally compact abelian group G, we denote

by

H1-

its annihilator, i.e. the set of all characters of G which are trivial on H; H^

is clearly a closed subgroup of G. If x G

H±,

then % defines a unique character

on G/H. This establishes a topological and algebraic isomorphism between the

character group of G/H and

HL.

Thus we may identify the dual of G/H with

HL.

Abstract Fourier Transform

Since the dual of a locally compact abelian group is itself locally compact, it

also possesses a Haar measure, which is unique up to a constant factor. We denote

by

( , }:GxG^T, {g,x)=x{9)

the obvious pairing. If x £ G and $ G ^i(G), we define the Fourier transform of &

to be

*(x) = / $(9)(9,x)dg-

JG

THEOREM 5. (Fourier Inversion Formula) There is a measure d\ on G normal-

ized so that the inversion formula

$(g)= [^(x)(g,x)dx =

kg-1)

JG

holds for all continuous functions $ in L\(G) with $ G Li(G).

The measure d\ in the above theorem is called the self-dual measure. The

following result, which we do not use in the sequel is a fundamental theorem.