Distribution theory

Introduction

Distribution theory, also known as the theory of distributions or generalized functions, is a branch of mathematical analysis that extends the concept of functions and derivatives. It provides a rigorous framework for dealing with objects that cannot be handled by classical analysis, such as the Dirac delta function. This theory has profound applications in various fields, including partial differential equations, quantum mechanics, signal processing, and probability theory.

Historical Background

The origins of distribution theory can be traced back to the early 20th century. The concept was formalized by the French mathematician Laurent Schwartz in the 1940s. Schwartz's work was pivotal in providing a solid mathematical foundation for the use of distributions, which had been used informally by physicists and engineers for many years. His contributions earned him the Fields Medal in 1950.

Basic Concepts

Distributions and Test Functions

A distribution is a continuous linear functional on a space of test functions. The space of test functions, denoted by \( \mathcal{D} \), consists of infinitely differentiable functions with compact support. For a distribution \( T \), the action on a test function \( \phi \) is denoted by \( \langle T, \phi \rangle \).

Examples of Distributions

1. **Dirac Delta Function**: The Dirac delta function \( \delta \) is defined by its action on a test function \( \phi \):

  \[
  \langle \delta, \phi \rangle = \phi(0)
  \]
  It is not a function in the classical sense but a distribution that represents a point mass at the origin.

2. **Principal Value of \( \frac{1}{x} \)**: The principal value distribution \( \text{p.v.} \left( \frac{1}{x} \right) \) is defined by:

  \[
  \langle \text{p.v.} \left( \frac{1}{x} \right), \phi \rangle = \lim_{\epsilon \to 0} \left( \int_{-\infty}^{-\epsilon} \frac{\phi(x)}{x} \, dx + \int_{\epsilon}^{\infty} \frac{\phi(x)}{x} \, dx \right)
  \]

Operations on Distributions

1. **Differentiation**: If \( T \) is a distribution and \( \phi \) is a test function, the derivative of \( T \), denoted \( T' \), is defined by:

  \[
  \langle T', \phi \rangle = -\langle T, \phi' \rangle
  \]

2. **Multiplication by a Smooth Function**: If \( T \) is a distribution and \( f \) is a smooth function, the product \( fT \) is defined by:

  \[
  \langle fT, \phi \rangle = \langle T, f\phi \rangle
  \]

3. **Convolution**: The convolution of a distribution \( T \) with a test function \( \phi \), denoted \( T * \phi \), is defined by:

  \[
  \langle T * \phi, \psi \rangle = \langle T, \phi * \psi \rangle
  \]

Advanced Topics

Sobolev Spaces

Sobolev spaces, denoted \( W^{k,p} \), are functional spaces that generalize the concept of differentiability and integrability. They are essential in the study of partial differential equations. A function \( u \) belongs to the Sobolev space \( W^{k,p}(\Omega) \) if it has weak derivatives up to order \( k \) that are \( L^p \)-integrable.

Fourier Transform of Distributions

The Fourier transform extends naturally to distributions. If \( T \) is a distribution, its Fourier transform \( \hat{T} \) is defined by: \[ \langle \hat{T}, \phi \rangle = \langle T, \hat{\phi} \rangle \] where \( \hat{\phi} \) is the Fourier transform of the test function \( \phi \).

Tempered Distributions

Tempered distributions are distributions that grow at most polynomially at infinity. They form the dual space of the Schwartz space \( \mathcal{S} \), which consists of rapidly decreasing functions. The Fourier transform is particularly well-behaved on tempered distributions.

Applications

Partial Differential Equations

Distribution theory is a powerful tool in the study of partial differential equations (PDEs). It allows for the definition of weak solutions, which are solutions that may not be differentiable in the classical sense but satisfy the equation in a distributional sense. This is particularly useful for equations with singularities or discontinuities.

Quantum Mechanics

In quantum mechanics, distributions are used to describe wavefunctions and observables. The Dirac delta function, for example, is used to represent point particles and to define the position operator.

Signal Processing

In signal processing, distributions are used to model and analyze signals. The Fourier transform of distributions is used to study the frequency content of signals, and the convolution operation is used to filter signals.

Probability Theory

In probability theory, distributions are used to describe random variables and their distributions. The Dirac delta function is used to represent discrete random variables, and the concept of weak convergence of probability measures is closely related to the theory of distributions.

References

Schwartz, Laurent. "Théorie des distributions." Hermann, 1950.
Rudin, Walter. "Functional Analysis." McGraw-Hill, 1991.
Hörmander, Lars. "The Analysis of Linear Partial Differential Operators I." Springer, 1983.