Distributions (mathematics)
Introduction
In mathematics, the concept of distributions, also known as generalized functions, extends the classical notion of functions. Distributions provide a framework for analyzing objects that cannot be described by traditional functions, such as the Dirac delta function. They are particularly useful in functional analysis, partial differential equations, and Fourier analysis, where they allow the rigorous treatment of objects that arise naturally in these contexts.
Historical Background
The development of distributions is closely linked to the evolution of mathematical analysis in the 20th century. The concept was formalized by the French mathematician Laurent Schwartz in the late 1940s. Schwartz's theory provided a systematic way to handle objects like the Dirac delta function, which had been used informally in physics for many years. Prior to Schwartz, mathematicians such as Sergei Sobolev and Paul Dirac had laid the groundwork for the theory of distributions through their work on functional spaces and quantum mechanics, respectively.
Basic Definitions
A distribution is a continuous linear functional on the space of test functions, which are typically smooth functions with compact support. The space of test functions is denoted by \( \mathcal{D}(\mathbb{R}^n) \), and the space of distributions is denoted by \( \mathcal{D}'(\mathbb{R}^n) \).
Test Functions
Test functions are infinitely differentiable functions that vanish outside a compact set. They form a vector space equipped with a topology that makes it a locally convex space. The topology is defined by a family of seminorms, which measure the size of the function and its derivatives.
Distributions
A distribution \( T \) is a linear map from \( \mathcal{D}(\mathbb{R}^n) \) to the real numbers \( \mathbb{R} \) such that for every sequence of test functions \( \phi_k \) converging to zero, the sequence \( T(\phi_k) \) also converges to zero. This continuity condition ensures that distributions can be manipulated similarly to functions.
Operations on Distributions
Distributions can be added and multiplied by scalars, just like functions. More interestingly, they can also be differentiated, even if they are not functions in the traditional sense.
Differentiation
The derivative of a distribution is defined via integration by parts. If \( T \) is a distribution and \( \phi \) is a test function, the derivative \( T' \) is defined by the relation:
\[ T'(\phi) = -T(\phi') \]
This definition is consistent with the classical derivative when \( T \) is a smooth function.
Convolution
Convolution is another operation that can be extended to distributions. If \( T \) is a distribution and \( \psi \) is a test function, their convolution \( T * \psi \) is defined as a distribution by:
\[ (T * \psi)(\phi) = T(\psi * \phi) \]
where \( \psi * \phi \) is the classical convolution of two functions.
Examples of Distributions
Several important examples illustrate the utility of distributions.
Dirac Delta Function
The Dirac delta function \( \delta \) is a distribution defined by:
\[ \delta(\phi) = \phi(0) \]
for any test function \( \phi \). It is not a function in the traditional sense but acts as an identity under convolution.
Principal Value of 1/x
The principal value distribution \( \text{p.v.}(1/x) \) is defined by:
\[ \text{p.v.}\left(\frac{1}{x}\right)(\phi) = \lim_{\epsilon \to 0^+} \left( \int_{|x|>\epsilon} \frac{\phi(x)}{x} \, dx \right) \]
This distribution arises in the context of singular integrals.
Applications in Partial Differential Equations
Distributions play a crucial role in the theory of partial differential equations (PDEs). They allow for the definition of weak solutions, which are solutions that satisfy the PDE in an integral sense rather than pointwise.
Weak Solutions
A weak solution to a PDE is a distribution that satisfies the equation when integrated against test functions. This concept is essential for dealing with equations that do not have classical solutions.
Fundamental Solutions
A fundamental solution of a linear PDE is a distribution that acts as an inverse to the differential operator. For example, the fundamental solution of the Laplace equation in three dimensions is the Newtonian potential.
Fourier Transform of Distributions
The Fourier transform can be extended to distributions, providing a powerful tool for analyzing their properties. The Fourier transform of a distribution \( T \) is defined by:
\[ \hat{T}(\phi) = T(\hat{\phi}) \]
where \( \hat{\phi} \) is the Fourier transform of the test function \( \phi \).
Tempered Distributions
Tempered distributions are those that can be paired with rapidly decreasing test functions, known as Schwartz functions. The space of tempered distributions is denoted by \( \mathcal{S}'(\mathbb{R}^n) \).
Advanced Topics
Sobolev Spaces
Sobolev spaces are functional spaces that generalize the concept of differentiability and integrability. They are closely related to distributions, as they can be defined using weak derivatives.
Microlocal Analysis
Microlocal analysis is a branch of analysis that studies the singularities of distributions. It uses tools such as the wavefront set, which describes the location and direction of singularities.
Hyperfunctions
Hyperfunctions extend the concept of distributions further by allowing for the treatment of even more singular objects. They are defined using the theory of analytic functions and have applications in complex analysis.