Category: Generalized functions

Generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especi
Laplacian of the indicator
In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can
Principal part
In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.
Unit impulse
No description available.
Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula for some given period . Here t is a real variable and the sum ext
Spaces of test functions and distributions
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usual
Homogeneous distribution
In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking, for all t > 0. More precisely, let be the scal
Singularity function
Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. Singularity functions have been heavily studied in the fi
Fourier–Bros–Iagolnitzer transform
In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer
Unit doublet
In mathematics, the unit doublet is the derivative of the Dirac delta function. It can be used to differentiate signals in electrical engineering: If u1 is the unit doublet, then where is the convolut
Algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of func
Crenel function
In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. It can be presented as a difference between two Heav
Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly sup
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transfo
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means tha
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function of n va
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order
Convolution quotient
In mathematics, a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. Convolution quotients were introduced by Mikusiński, and their theory is somet
Limit of distributions
In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the li
Multiscale Green's function
Multiscale Green's function (MSGF) is a generalized and extended version of the classical Green's function (GF) technique for solving mathematical equations. The main application of the MSGF technique
Dirac delta function
In mathematics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in
Fourier inversion theorem
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to
Microlocal analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear parti
Paley–Wiener theorem
In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymon
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differentia
White noise analysis
In probability theory, a branch of mathematics, white noise analysis, otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noi
Wave front set
In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its
Weak solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless
Green's function number
In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing solutions easier to identify, store, and
Schwartz kernel theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized func
Gelfand–Shilov space
In the mathematical field of functional analysis, a Gelfand–Shilov space is a space of test functions for the theory of generalized functions, introduced by Gelfand and Shilov .
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Gre
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero fo
In mathematics, Boehmians are objects obtained by an abstract algebraic construction of "quotients of sequences." The original construction was motivated by regular operators introduced by T. K. Boehm
Rigged Hilbert space
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional an