14.1 Harmonic analysis on locally compact abelian groups
3 min read•august 7, 2024
Harmonic analysis on locally compact abelian groups extends Fourier analysis to a broader setting. It combines topology and algebra, using to define integration and the on these groups.
This framework allows us to study functions on groups through their Fourier transforms. Key concepts include , the , and , which connect a group to its 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
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 Rn, Z, and the circle group 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 Rn and counting measure on discrete groups like Z
Fourier Analysis on Groups
Fourier Transform
The Fourier transform on a G is a linear operator that maps functions on G to functions on the dual group G^
The dual group G^ consists of the continuous homomorphisms from G to the circle group T
The Fourier transform is defined using the Haar integral: f^(χ)=∫Gf(x)χ(x)dx, where χ∈G^
The Fourier transform extends the classical Fourier transform on Rn to a more general setting
Examples of Fourier transforms on groups include the discrete Fourier transform (DFT) on Z/nZ and the Fourier series on the circle group 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)=∫Gf(y)g(x−y)dy
The Fourier transform converts convolution into pointwise multiplication: f∗g=f^⋅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 L2(G) and L2(G^)
Isometry means the Fourier transform preserves the L2 norm: ∥f∥L2(G)=∥f^∥L2(G^)
The Plancherel theorem allows for the study of functions on G by analyzing their Fourier transforms on G^
Group Characters and Duality
Group Characters
A of a locally compact abelian group G is a continuous homomorphism from G to the circle group T
Homomorphism means the character preserves the group operation: χ(x+y)=χ(x)χ(y) for all x,y∈G
Characters are the building blocks of the dual group G^ and play a central role in Fourier analysis on groups
Examples of characters include exponential functions e2πiξx on R and discrete characters e2πik/n on Z/nZ
Dual Group and Duality
The dual group 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: G^^≅G
This duality establishes a deep connection between a group and its dual, allowing for the study of G through G^ and vice versa
The Fourier transform and its inverse provide explicit isomorphisms between G and G^^, realizing the Pontryagin duality