Other meanings of Generalized function
Mathematics
In mathematics, a generalized function is an object that extends the classical notion of a function, allowing for operations such as differentiation that are not possible for ordinary functions. The most widely used generalized functions are distributions, introduced by Laurent Schwartz in the 1940s, which formalize entities like the Dirac delta function. Generalized functions are essential in partial differential equations, physics, and engineering, where they model point sources, impulses, and singularities.
A generalized function is a continuous linear functional on a space of test functions, typically the space of infinitely differentiable functions with compact support. The action of a generalized function f on a test function φ is denoted ⟨f, φ⟩. This duality allows ordinary locally integrable functions to be embedded as generalized functions via integration: ⟨f, φ⟩ = ∫ f(x)φ(x) dx. Distributions can be differentiated arbitrarily many times, and differentiation is defined by transferring the derivative to the test function: ⟨f′, φ⟩ = −⟨f, φ′⟩. This property makes distributions a natural setting for solving linear partial differential equations, as every distribution has derivatives that are also distributions.1
The concept of generalized functions emerged from the need to formalize the Dirac delta, introduced by Paul Dirac in the 1920s for quantum mechanics. Earlier, Sergei Sobolev had used similar ideas in the 1930s for solving hyperbolic equations. Laurent Schwartz published his comprehensive theory in 1945, earning a Fields Medal in 1950. His work unified and extended previous approaches, providing a rigorous framework that became standard in analysis. Later developments include the theory of hyperfunctions by Mikio Sato and the Colombeau algebras, which allow multiplication of distributions, a problem in Schwartz's theory.2
Generalized functions are indispensable in physics and engineering. In electromagnetism, the Dirac delta models point charges and dipoles, while the Heaviside step function represents sudden switches. In control theory, impulses are used to analyze system responses. In signal processing, distributions underpin the theory of Fourier transforms of non-integrable functions, such as the constant function. In partial differential equations, fundamental solutions are distributions, enabling the solution of Poisson's equation and wave equations. The theory also extends to stochastic processes, where white noise is treated as a generalized random process.
Beyond Schwartz distributions, there are other types of generalized functions. Hyperfunctions, introduced by Mikio Sato in 1958, are defined via boundary values of holomorphic functions and are more general than distributions. Colombeau algebras, developed in the 1980s, allow multiplication of distributions, which is impossible in Schwartz's theory. In nonstandard analysis, generalized functions can be represented as equivalence classes of internal functions. Also, the theory of generalized functions has been extended to manifolds, leading to the concept of distributional sections of vector bundles, used in differential geometry and mathematical physics. A notable edge case is the derivative of the Heaviside step function, which is the Dirac delta, a result that is rigorously justified only within distribution theory.
This article focuses on the mathematical concept of generalized functions, primarily distributions.
Help improve the encyclopedia. Reports go straight to the site manager.