文档介绍：CHAPTER II
FOURIER ANALYSIS ON GROUPS
In this chapter we describe how the familiar theory of Fourier analysis for L2
functions on the circle (as used in Theorem ) may be extended to pact
abelian group. We shall also state formally the properties of Haar measure.
Introduction
Definition . A topological abelian group is a Hausdorff topological space
G which is also an abelian group, provided that the map (g, h) 7→ g − h is a
continuous map from G × G (product topology) to G. If the topology of G is also
pact (every point has an open neighbourhood whose closure pact),
then G is a pact abelian group, or an LCA group.
Example . Almost every group you know is a topological group in some
natural topology.
(1) R, usual topology.
(2) R, with the discrete topology.
(3) Q, with the discrete topology.
(4) S1, the multiplicative circle; T = R/Z, the additive circle.
(5) Qp, the field of p–adic numbers, in the p–adic topology.
(6) Z[x], in the discrete topology.
(7) any finite or countable group with the discrete topology.
(8) If k is a pact field, then GL(n, k) is a non–abelian pact
2
group with the topology obtained from the product topology on kn .
A homomorphism φ: G → H of LCA groups is a homomorphism of the groups
which is also continuous. Most of the ways of constructing new groups from old
preserve pactness.
(1) If {Gi}i∈I are LCA group