Theory of Distributions

see also:

• Probability Theory
• Probability Measure
• probability density function

The theory of distributions, also known as generalized functions, extends the concept of functions and provides a rigorous mathematical framework for handling objects like the Dirac delta function, which are not functions in the traditional sense but still play a crucial role in mathematical physics and engineering. Developed primarily by Laurent Schwartz in the 1940s and 1950s, the theory of distributions offers a powerful and flexible toolset for solving differential equations, especially when dealing with discontinuities, singularities, and other phenomena that are difficult to handle within the limits of classical analysis.

Basic Concepts

• Generalized Functions: Distributions generalize the notion of functions by considering them as linear Functionals acting on a space of test functions. Test functions are usually taken to be infinitely differentiable functions with compact support (they vanish outside some finite region), ensuring that distributions can act on a wide variety of functions, including those that are not smooth or well-behaved in the classical sense.

• Dirac Delta Function: One of the most famous examples of a distribution is the Dirac delta function $\delta (x)$, which is defined by its action on a test function $\varphi (x)$ as $⟨\delta ,\varphi ⟩=\varphi (0)$. Despite not being a function in the classical sense (since it is infinitely high at $x=0$ and zero elsewhere), the delta function is essential in physics for modeling point charges, impulses, and other localized phenomena.

Operations on Distributions

• Differentiation: Distributions can be differentiated an arbitrary number of times, and the derivative of a distribution is defined in such a way that it generalizes the classical notion of differentiation. If $T$ is a distribution, its derivative $T^{\prime }$ is another distribution defined by $⟨T^{\prime },\varphi ⟩=-⟨T,{\varphi }^{\prime }⟩$ for any test function $\varphi$. This definition allows for the differentiation of objects that would not be differentiable in the classical sense.

• Convolution: The convolution of distributions extends the classical convolution of functions, allowing for the combination of distributions in a way that is consistent with their interpretation as generalized functions.

Significance and Applications

• Solving Differential Equations: The theory of distributions provides a robust framework for solving differential equations, especially when solutions involve singularities or are only meaningful in a generalized sense. It allows for the formalization and solution of equations that include terms like the Dirac delta function.

• Fourier Transforms: Distributions are particularly well-suited for Fourier analysis. The Fourier Transform, which can be difficult to define for certain functions or signals, can be extended to distributions, greatly broadening the scope of problems that can be analyzed using Fourier methods.

• Quantum Mechanics: In quantum mechanics, distributions are used to describe wavefunctions, particularly in the context of the momentum space representation, where wavefunctions can be delta functions or have other singular behaviors.

• Signal Processing: The theory of distributions is applied in signal processing, where impulses and other discontinuous signals can be modeled as distributions, facilitating their analysis and manipulation.

The theory of distributions represents a profound extension of classical calculus, providing the mathematical rigor needed to work with generalized functions. It bridges the gap between abstract mathematical concepts and practical applications in physics and engineering, highlighting the versatility and power of mathematical analysis.