Pontryagin Duality - Introduction

Introduction

Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:

• Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series;

• Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and

• Complex-valued functions on a finite abelian group have discrete Fourier transforms which are functions on the dual group, which is a (non-canonically) isomorphic group. Moreover any function on a finite group can be recovered from its discrete Fourier transform.

The theory, introduced by Lev Pontryagin and combined with Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group.

It is analogous to the dual vector space of a vector space: a finite-dimensional vector space V and its dual vector space V* are not naturally isomorphic, but their endomorphism algebras (matrix algebras) are: via the transpose. Similarly, a group G and its dual group are not in general isomorphic, but their group algebras are: via the Fourier transform, though one must carefully define these algebras analytically. More categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see categorical considerations.