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.
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