🎵Harmonic Analysis Unit 14 – Harmonic Functions: Groups & Homomorphisms

Key Concepts and Definitions

• Group a set equipped with a binary operation satisfying closure, associativity, identity, and inverse properties
• Abelian group a group where the binary operation is commutative, i.e., $a * b = b * a$ for all elements $a$ and $b$
• Homomorphism a structure-preserving map between two groups that respects the group operation
  • Formally, a function $f: G \to H$ such that $f(a * b) = f(a) \circ f(b)$ for all $a, b \in G$, where $*$ and $\circ$ are the group operations in $G$ and $H$, respectively
• Kernel of a homomorphism the set of elements in the domain that map to the identity element in the codomain
• Harmonic function a complex-valued function that satisfies the mean value property on a group
• Characters homomorphisms from a group to the multiplicative group of complex numbers of modulus 1
• Pontryagin duality the duality between a locally compact abelian group and its dual group of characters

Group Theory Foundations

• Binary operation a function that combines two elements of a set to produce another element of the same set
• Closure property states that the result of the binary operation on any two elements of the group is also an element of the group
• Associativity $(a * b) * c = a * (b * c)$ for all elements $a, b, c$ in the group
• Identity element an element $e$ such that $a * e = e * a = a$ for all elements $a$ in the group
• Inverse element for each element $a$, there exists an element $a^{-1}$ such that $a * a^{-1} = a^{-1} * a = e$
  • The inverse is unique for each element in the group
• Examples of groups include integers under addition $(\mathbb{Z}, +)$, real numbers under addition $(\mathbb{R}, +)$, and non-zero real numbers under multiplication $(\mathbb{R}^*, \times)$

Introduction to Harmonic Functions

• Harmonic functions play a crucial role in harmonic analysis and have connections to various branches of mathematics
• Mean value property a function $f$ is harmonic if the value at any point is equal to the average of its values on any sphere centered at that point
  • In the discrete setting, this translates to the average of the function values on the neighbors of a point in a group
• Harmonic functions on groups are closely related to the representation theory of the group
• Laplace operator a differential operator that appears in the definition of harmonic functions in the continuous setting
• Discrete Laplacian an analog of the Laplace operator for functions on groups or graphs
• Poisson kernel a family of functions used to construct harmonic functions and solve boundary value problems

Group Homomorphisms Explained

• Homomorphisms capture the idea of preserving the algebraic structure between groups
• Isomorphism a bijective homomorphism, i.e., a one-to-one correspondence between groups that preserves the group operation
• Automorphism an isomorphism from a group to itself
• Cayley's theorem every group is isomorphic to a subgroup of the symmetric group on the set of its elements
• First isomorphism theorem relates the kernel and image of a group homomorphism
  • Formally, if $f: G \to H$ is a homomorphism, then $G / \ker(f) \cong \operatorname{im}(f)$
• Fundamental theorem of homomorphisms connects the properties of a homomorphism with the quotient group structure

Properties of Harmonic Functions on Groups

• Uniqueness harmonic functions are uniquely determined by their values on the boundary of a group (in the appropriate sense)
• Maximum principle a non-constant harmonic function cannot attain its maximum or minimum value at an interior point of its domain
• Harnack's inequality provides a quantitative version of the maximum principle for positive harmonic functions
• Liouville's theorem a bounded harmonic function on the entire group must be constant
• Poisson integral formula expresses harmonic functions as integrals of their boundary values against the Poisson kernel
• Convolution operators translation-invariant operators that play a key role in studying harmonic functions on groups

Applications in Harmonic Analysis

• Fourier analysis the study of representing functions as superpositions of basic waves or characters
  • Harmonic functions arise naturally in the context of Fourier series and Fourier transforms
• Spectral analysis investigating the properties of operators (such as the Laplacian) using their eigenvalues and eigenfunctions
• Partial differential equations (PDEs) harmonic functions appear as solutions to Laplace's equation and other elliptic PDEs
• Potential theory the study of harmonic functions and related concepts in the context of electrostatics and gravitational fields
• Brownian motion and diffusion processes harmonic functions describe the average behavior of random walks and diffusion
• Signal processing and image analysis harmonic functions are used for smoothing, denoising, and feature extraction tasks

Problem-Solving Techniques

• Exploiting symmetries utilizing the group structure and invariance properties to simplify problems involving harmonic functions
• Fourier transform methods converting problems into the frequency domain, solving them using Fourier analytic techniques, and transforming back
• Green's functions solving boundary value problems by constructing suitable fundamental solutions (Green's functions) and integrating against boundary data
• Variational methods formulating problems in terms of energy functionals and finding harmonic functions as minimizers
• Probabilistic techniques interpreting harmonic functions as expectations of random processes and using probabilistic tools to derive their properties
• Numerical approximations discretizing the group or the function space and employing numerical linear algebra or optimization methods

Connections to Other Mathematical Areas

• Complex analysis harmonic functions are the real and imaginary parts of holomorphic functions in complex analysis
• Differential geometry Laplace-Beltrami operator and harmonic forms on Riemannian manifolds generalize the notion of harmonic functions
• Probability theory discrete and continuous harmonic functions are related to martingales and Markov processes
• Representation theory harmonic analysis on groups is closely tied to the study of unitary representations of the group
• Number theory automorphic forms and L-functions, which are important objects in number theory, have connections to harmonic analysis on certain groups
• Mathematical physics quantum mechanics, quantum field theory, and statistical mechanics heavily rely on the tools and concepts of harmonic analysis