🎵Harmonic Analysis Unit 8 – Distributions and Schwartz Spaces

Key Concepts and Definitions

• Distributions generalize the concept of functions enabling the study of more irregular objects
• Schwartz space $\mathcal{S}(\mathbb{R}^n)$ consists of rapidly decreasing smooth functions
  • Functions in Schwartz space have derivatives of all orders that decay faster than any polynomial at infinity
• Tempered distributions $\mathcal{S}'(\mathbb{R}^n)$ are continuous linear functionals on the Schwartz space
  • They extend the notion of $L^p$ functions and can be differentiated infinitely many times
• Support of a distribution is the smallest closed set outside which the distribution vanishes
• Convolution of a distribution with a Schwartz function is always well-defined and produces a smooth function
• Fourier transform extends to tempered distributions providing a powerful tool for their analysis
  • Fourier transform of a tempered distribution is also a tempered distribution

Schwartz Space Fundamentals

• Schwartz space is a topological vector space with a family of seminorms $\|\cdot\|_{\alpha,\beta}$
  • These seminorms control the decay and smoothness properties of functions
• Convergence in Schwartz space is characterized by uniform convergence of all derivatives multiplied by polynomials
  • A sequence $\{\varphi_n\}$ converges to $\varphi$ in $\mathcal{S}(\mathbb{R}^n)$ if $\|\varphi_n-\varphi\|_{\alpha,\beta}\to 0$ for all multi-indices $\alpha,\beta$
• Schwartz space is dense in $L^p(\mathbb{R}^n)$ for $1\leq p<\infty$
  • This allows approximating $L^p$ functions by smooth rapidly decreasing functions
• Schwartz space is closed under multiplication, convolution, and Fourier transform
  • If $\varphi,\psi\in\mathcal{S}(\mathbb{R}^n)$, then $\varphi\psi,\varphi*\psi,\hat{\varphi}\in\mathcal{S}(\mathbb{R}^n)$
• Schwartz functions are integrable and have vanishing moments of all orders
  • $\int_{\mathbb{R}^n}x^\alpha\varphi(x)dx=0$ for all multi-indices $\alpha$ and $\varphi\in\mathcal{S}(\mathbb{R}^n)$

Types of Distributions

• Regular distributions are those that can be represented by locally integrable functions
  • If $f\in L^1_{\text{loc}}(\mathbb{R}^n)$, then $\langle T_f,\varphi\rangle=\int_{\mathbb{R}^n}f(x)\varphi(x)dx$ defines a distribution $T_f$
• Singular distributions cannot be represented by functions (Dirac delta, principal value)
  • Dirac delta distribution $\delta$ satisfies $\langle\delta,\varphi\rangle=\varphi(0)$ for all $\varphi\in\mathcal{S}(\mathbb{R}^n)$
• Distributions with compact support have their support contained in a compact set
  • They can be extended to continuous linear functionals on $C^\infty(\mathbb{R}^n)$
• Tempered distributions grow at most polynomially at infinity
  • They are continuous linear functionals on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$
• Periodic distributions are those that satisfy $\langle T,\varphi(\cdot+a)\rangle=\langle T,\varphi\rangle$ for some $a\in\mathbb{R}^n$
  • They can be identified with distributions on the torus $\mathbb{T}^n$

Properties of Distributions

• Distributions are linear: if $S,T$ are distributions and $a,b\in\mathbb{C}$, then $aS+bT$ is also a distribution
  • $\langle aS+bT,\varphi\rangle=a\langle S,\varphi\rangle+b\langle T,\varphi\rangle$ for all $\varphi\in\mathcal{S}(\mathbb{R}^n)$
• Derivatives of distributions are defined by duality: $\langle\partial^\alpha T,\varphi\rangle=(-1)^{|\alpha|}\langle T,\partial^\alpha\varphi\rangle$
  • This extends the usual notion of derivative and allows distributions to be differentiated infinitely many times
• Multiplication of a distribution by a smooth function is well-defined
  • If $T$ is a distribution and $\psi\in C^\infty(\mathbb{R}^n)$, then $\psi T$ is a distribution with $\langle\psi T,\varphi\rangle=\langle T,\psi\varphi\rangle$
• Convolution of a distribution with a Schwartz function is a smooth function
  • If $T$ is a distribution and $\varphi\in\mathcal{S}(\mathbb{R}^n)$, then $T*\varphi\in C^\infty(\mathbb{R}^n)$
• Distributions with disjoint support are always linearly independent
• Distributions can be restricted to open subsets and extended by zero outside their support
  • This allows for localization and decomposition of distributions

Fourier Transforms and Distributions

• Fourier transform of a tempered distribution is defined by duality
  • If $T\in\mathcal{S}'(\mathbb{R}^n)$, then $\langle\hat{T},\varphi\rangle=\langle T,\hat{\varphi}\rangle$ for all $\varphi\in\mathcal{S}(\mathbb{R}^n)$
• Fourier transform maps tempered distributions to tempered distributions
  • $\mathcal{F}:\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ is a continuous linear bijection
• Fourier transform interchanges multiplication and convolution for tempered distributions
  • If $T\in\mathcal{S}'(\mathbb{R}^n)$ and $\varphi\in\mathcal{S}(\mathbb{R}^n)$, then $\widehat{T*\varphi}=\hat{T}\hat{\varphi}$ and $\widehat{\varphi T}=\hat{\varphi}*\hat{T}$
• Fourier transform of derivatives is related to multiplication by polynomials
  • If $T\in\mathcal{S}'(\mathbb{R}^n)$, then $\widehat{\partial^\alpha T}(\xi)=(i\xi)^\alpha\hat{T}(\xi)$
• Paley-Wiener theorems characterize the Fourier transforms of distributions with compact support
  • They provide a link between the growth of a function and the size of the support of its Fourier transform

Applications in Harmonic Analysis

• Distributions are used to model singular objects like point masses, dipoles, and discontinuities
  • They provide a framework for studying partial differential equations with singular data or coefficients
• Tempered distributions are the natural setting for Fourier analysis and the study of pseudodifferential operators
  • They allow for the development of a symbolic calculus and the analysis of singular integral operators
• Distributions with compact support are used in the theory of convolution equations and the study of partial differential equations with constant coefficients
  • They provide a tool for localizing problems and reducing them to algebraic equations
• Periodic distributions arise in the study of Fourier series and the analysis of partial differential equations on the torus
  • They are used to model periodic singular objects and to study convergence properties of Fourier series
• Distributions are used in the construction of fundamental solutions of partial differential equations
  • They provide a way to represent Green's functions and to study the propagation of singularities

Theoretical Challenges and Advanced Topics

• Multiplication of distributions is not always well-defined and requires additional tools (Colombeau algebras, paraproducts)
  • This leads to the study of nonlinear theories of generalized functions
• Distributions on manifolds require the use of coordinate charts and partition of unity techniques
  • The study of distributions on manifolds is connected to the theory of pseudodifferential operators and microlocal analysis
• Wavefront set of a distribution characterizes the singularities and their directions of propagation
  • It provides a refined description of the singular support and is used in the study of propagation of singularities for partial differential equations
• Distributions with values in function spaces (vector-valued distributions) are used to study systems of partial differential equations and the regularity of their solutions
  • They require the use of tensor products and the theory of nuclear spaces
• Renormalization techniques are used to give meaning to products of distributions in quantum field theory
  • They involve the use of regularization methods and the subtraction of infinite quantities

Problem-Solving Strategies

• Identify the type of distribution (regular, singular, tempered, compactly supported, periodic) and use the appropriate tools and techniques
• Use the properties of distributions (linearity, differentiation, multiplication by smooth functions, convolution) to simplify the problem
  • Reduce the problem to a simpler one involving known distributions or functions
• Exploit the duality between distributions and test functions to transfer the problem to a more tractable setting
  • Work with the action of distributions on test functions rather than the distributions themselves
• Use the Fourier transform to convert the problem into an algebraic or analytic one
  • The Fourier transform often simplifies the problem and provides additional tools for its study
• Localize the problem by decomposing the distribution into a sum of distributions with smaller support
  • This can be done using partition of unity techniques or by restricting the distribution to open subsets
• Approximate the distribution by a sequence of smooth functions or by a sequence of distributions with simpler structure
  • Use the convergence properties of distributions to pass to the limit and obtain the desired result
• Interpret the distribution as a functional and use functional analytic techniques (Hahn-Banach theorem, closed graph theorem, Banach-Steinhaus theorem)
  • These techniques can be used to prove existence and uniqueness results or to obtain estimates for the distribution