14.2 Pontryagin duality and Fourier analysis on groups

Pontryagin duality connects locally compact abelian groups with their dual groups, made up of continuous homomorphisms to the circle group. This powerful concept establishes a two-way correspondence, allowing us to study groups through their characters.

Fourier analysis on groups extends classical Fourier analysis to more general settings. It introduces the Fourier transform for functions on groups, providing tools like the inversion theorem and Parseval's identity for analyzing group structures and functions.

Pontryagin Duality and Character Groups

Dual Groups and Isomorphisms

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• Pontryagin duality establishes a correspondence between a locally compact abelian group and its dual group
  • The dual group consists of all continuous homomorphisms from the original group to the circle group $\mathbb{T}$
  • These homomorphisms are called characters of the group
• The Pontryagin dual of a locally compact abelian group $G$ is denoted as $\hat{G}$
  • $\hat{G}$ is also a locally compact abelian group under pointwise multiplication and the compact-open topology
• The Pontryagin duality theorem states that the dual of the dual group $\hat{\hat{G}}$ is canonically isomorphic to the original group $G$
  • This isomorphism is a homeomorphism, preserving both the group structure and the topological properties

Character Groups and Compactification

• The character group of a locally compact abelian group $G$ is the set of all continuous homomorphisms from $G$ to the circle group $\mathbb{T}$
  • Characters are complex-valued functions $\chi: G \to \mathbb{T}$ satisfying $\chi(xy) = \chi(x)\chi(y)$ for all $x, y \in G$
  • The character group is endowed with the compact-open topology, making it a locally compact abelian group
• The Bohr compactification of a topological group $G$ is a compact Hausdorff group $bG$ together with a continuous homomorphism $b: G \to bG$
  • The Bohr compactification is characterized by the property that any continuous homomorphism from $G$ to a compact Hausdorff group factors uniquely through $b$
  • For a locally compact abelian group $G$, the Bohr compactification $bG$ is isomorphic to the dual group of the discrete group $G_d$, where $G_d$ is $G$ with the discrete topology

Fourier Analysis on Groups

Fourier Transform and Inversion Theorem

• The Fourier transform on a locally compact abelian group $G$ is a linear operator that maps functions on $G$ to functions on its dual group $\hat{G}$
  • For a function $f \in L^1(G)$, its Fourier transform $\hat{f}$ is defined as $\hat{f}(\chi) = \int_G f(x) \overline{\chi(x)} dx$ for $\chi \in \hat{G}$
  • The Fourier transform extends to a unitary operator from $L^2(G)$ to $L^2(\hat{G})$
• The Fourier inversion theorem states that under suitable conditions, a function can be recovered from its Fourier transform
  • For a function $f \in L^1(G)$ with $\hat{f} \in L^1(\hat{G})$, the inversion formula holds: $f(x) = \int_{\hat{G}} \hat{f}(\chi) \chi(x) d\chi$ for almost all $x \in G$
  • The measure $d\chi$ is the Haar measure on the dual group $\hat{G}$

Parseval's Identity and Annihilators

• Parseval's identity is a fundamental result in Fourier analysis that relates the $L^2$ norms of a function and its Fourier transform
  • For a function $f \in L^2(G)$, Parseval's identity states that $\int_G |f(x)|^2 dx = \int_{\hat{G}} |\hat{f}(\chi)|^2 d\chi$
  • This identity expresses the fact that the Fourier transform is a unitary operator on $L^2(G)$
• The annihilator of a subset $A$ of a locally compact abelian group $G$ is the set $A^{\perp} = \{\chi \in \hat{G} : \chi(x) = 1 \text{ for all } x \in A\}$
  • The annihilator $A^{\perp}$ is a closed subgroup of the dual group $\hat{G}$
  • The annihilator of a closed subgroup $H$ of $G$ is isomorphic to the dual group of the quotient group $G/H$, i.e., $(G/H)^{\wedge} \cong H^{\perp}$