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 especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering.
A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory.
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In mathematics, generalizedfunctions are objects extending the notion of functions. There is more than one recognized theory, for example the theory...
mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The...
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing...
(\operatorname {sgn} 0)^{2}=0} . This generalized signum allows construction of the algebra of generalizedfunctions, but the price of such generalization...
vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition. A set of finitary...
mathematical analysis, the Dirac delta function (or δ distribution), also known as the unit impulse, is a generalizedfunction on the real numbers, whose value...
smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution...
arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function f {\displaystyle f} defined on a subset...
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows...
singularity Generalizedfunction Distribution Minkowski's question-mark function (**) This condition depends on the references "Singular function", Encyclopedia...
{\displaystyle 1/p!} in § Properties of the generalized Kronecker delta below disappearing. In terms of the indices, the generalized Kronecker delta is defined as:...
potential well mathematically described by the Dirac delta function - a generalizedfunction. Qualitatively, it corresponds to a potential which is zero...
targets Generalizedfunction – Objects extending the notion of functions Homogeneous distribution Hyperfunction – Type of generalizedfunction Laplacian...
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized...
delta function yields zero. For this reason, the integration above (Fourier series coefficients determination) must be understood "in the generalized functions...
Complex functionFunction (mathematics) Generalizedfunction List of special functions and eponyms List of types of functions Rational function Special...
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...
0.2,40)} . One of the benefits of using a growth function such as the generalized logistic function in epidemiological modeling is its relatively easy...
∈ V . {\displaystyle v\in V.} This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of...
to spaces of white noise test functions φ {\displaystyle \varphi } , and, by duality, to spaces of generalizedfunctions Ψ {\displaystyle \Psi } of white...
x)+2f'(x)g'(x)+f(x)g''(x).} The formula can be generalized to the product of m differentiable functions f1,...,fm. ( f 1 f 2 ⋯ f m ) ( n ) = ∑ k 1 + k...
degenerate distribution — it is a Distribution (mathematics) in the generalizedfunction sense; but the notation treats it as if it were a continuous distribution...
In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another...
the theory of generalizedfunctions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalizedfunctions introduced by Schwartz...
In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random...
ordinary function when σ = 0 {\displaystyle \sigma =0} . However, one can define the normal distribution with zero variance as a generalizedfunction; specifically...
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