14.1 Harmonic analysis on locally compact abelian groups

Harmonic analysis on locally compact abelian groups extends Fourier analysis to a broader setting. It combines topology and algebra, using Haar measure to define integration and the Fourier transform on these groups.

This framework allows us to study functions on groups through their Fourier transforms. Key concepts include convolution, the Plancherel theorem, and Pontryagin duality, which connect a group to its dual group of characters.

Locally Compact Abelian Groups and Haar Measure

Topological and Algebraic Structure

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• Locally compact abelian groups combine topological and algebraic structure
• Topological groups have a group operation that is continuous with respect to the topology
  • Continuity of the group operation ensures compatibility between the algebraic and topological structures
• Abelian groups have a commutative group operation, meaning the order of elements does not affect the result ($a + b = b + a$ for all $a, b$ in the group)
• Examples of locally compact abelian groups include $\mathbb{R}^n$, $\mathbb{Z}$, and the circle group $\mathbb{T}$

Haar Measure

• Haar measure is a unique (up to scaling) translation-invariant measure on a locally compact group
  • Translation-invariance means the measure of a set is equal to the measure of any translate of that set
• Existence and uniqueness of Haar measure is guaranteed for all locally compact groups
• Haar measure allows for integration on locally compact groups, which is essential for defining the Fourier transform
• Examples of Haar measure include Lebesgue measure on $\mathbb{R}^n$ and counting measure on discrete groups like $\mathbb{Z}$

Fourier Analysis on Groups

Fourier Transform

• The Fourier transform on a locally compact abelian group $G$ is a linear operator that maps functions on $G$ to functions on the dual group $\hat{G}$
  • The dual group $\hat{G}$ consists of the continuous homomorphisms from $G$ to the circle group $\mathbb{T}$
• The Fourier transform is defined using the Haar integral: $\hat{f}(\chi) = \int_G f(x) \overline{\chi(x)} dx$, where $\chi \in \hat{G}$
• The Fourier transform extends the classical Fourier transform on $\mathbb{R}^n$ to a more general setting
• Examples of Fourier transforms on groups include the discrete Fourier transform (DFT) on $\mathbb{Z}/n\mathbb{Z}$ and the Fourier series on the circle group $\mathbb{T}$

Convolution and the Plancherel Theorem

• Convolution on a locally compact abelian group $G$ is a binary operation that combines two functions $f$ and $g$ on $G$ to produce a new function $f * g$
  • Convolution is defined using the Haar integral: $(f * g)(x) = \int_G f(y) g(x-y) dy$
• The Fourier transform converts convolution into pointwise multiplication: $\widehat{f * g} = \hat{f} \cdot \hat{g}$
  • This property simplifies the analysis of convolution equations and is a key reason for the usefulness of the Fourier transform
• The Plancherel theorem states that the Fourier transform is an isometry between $L^2(G)$ and $L^2(\hat{G})$
  • Isometry means the Fourier transform preserves the $L^2$ norm: $\|f\|_{L^2(G)} = \|\hat{f}\|_{L^2(\hat{G})}$
• The Plancherel theorem allows for the study of functions on $G$ by analyzing their Fourier transforms on $\hat{G}$

Group Characters and Duality

Group Characters

• A character of a locally compact abelian group $G$ is a continuous homomorphism from $G$ to the circle group $\mathbb{T}$
  • Homomorphism means the character preserves the group operation: $\chi(x+y) = \chi(x) \chi(y)$ for all $x, y \in G$
• Characters are the building blocks of the dual group $\hat{G}$ and play a central role in Fourier analysis on groups
• Examples of characters include exponential functions $e^{2\pi i \xi x}$ on $\mathbb{R}$ and discrete characters $e^{2\pi i k/n}$ on $\mathbb{Z}/n\mathbb{Z}$

Dual Group and Duality

• The dual group $\hat{G}$ of a locally compact abelian group $G$ is the set of all continuous characters of $G$
  • The dual group has a natural group structure induced by pointwise multiplication of characters
• Pontryagin duality states that the dual of the dual group is isomorphic to the original group: $\hat{\hat{G}} \cong G$
  • This duality establishes a deep connection between a group and its dual, allowing for the study of $G$ through $\hat{G}$ and vice versa
• The Fourier transform and its inverse provide explicit isomorphisms between $G$ and $\hat{\hat{G}}$, realizing the Pontryagin duality
• Examples of dual groups include $\widehat{\mathbb{R}} \cong \mathbb{R}$, $\widehat{\mathbb{Z}} \cong \mathbb{T}$, and $\widehat{\mathbb{T}} \cong \mathbb{Z}$