Pontryagin dual - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact Hausdorff group that is abelian. Inside the complex numbers, let

\mathbb T=\{z\in\mathbb C:\lvert z\rvert=1\}

be the circle group. The Pontryagin dual of $G$ is

\widehat G=\operatorname{Hom}_{\mathrm{cont}}(G,\mathbb T),

the set of continuous group homomorphisms $\chi:G\to\mathbb T$ , called characters. Pointwise multiplication, $(\chi\psi)(x)=\chi(x)\psi(x)$ , makes $\widehat G$ an abelian group. It carries the compact-open topology, generated by uniform control of characters on compact subsets of $G$ ; with this topology, $\widehat G$ is again locally compact and Hausdorff.

Biduality

Each $x\in G$ defines a character on $\widehat G$ by evaluation, $\chi\mapsto\chi(x)$ . The Pontryagin duality theorem says that

G\longrightarrow\widehat{\widehat G},\qquad x\longmapsto(\chi\mapsto\chi(x)),

is an isomorphism of topological groups. The compact-open topology is essential: the statement is stronger than an abstract group isomorphism Rudin, Chapter 1.

Basic examples

With the usual normalizations,

\widehat{\mathbb R}\cong\mathbb R,\qquad t\longmapsto(x\longmapsto e^{2\pi itx}),

while $\widehat{\mathbb Z}\cong\mathbb T$ and $\widehat{\mathbb T}\cong\mathbb Z$ . More generally, the dual of a compact abelian group is discrete, and the dual of a discrete abelian group is compact. Finite abelian groups are isomorphic to their duals, though generally not by a canonical isomorphism.

Fourier pairing and conventions

The evaluation $\langle x,\chi\rangle=\chi(x)$ is the canonical pairing between $G$ and $\widehat G$ . After choosing a Haar measure, it defines the Fourier transform

\widehat f(\chi)=\int_G f(x)\overline{\chi(x)}\,dx.

Changing the exponential convention changes signs or factors of $2\pi$ , but not the dual group itself.

References

W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. Wiley DOI record. Relevant: Chapter 1, locally compact abelian groups, dual groups, and the Pontryagin duality theorem.