In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized 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 generalized functions, but the price of such generalization...

mathematical analysis, the Dirac delta function (or δ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value...

{\displaystyle 1/p!} in § Properties of the generalized Kronecker delta below disappearing. In terms of the indices, the generalized Kronecker delta is defined as:...

vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition. A set of finitary...

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...

In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with...

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...

smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution...

known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions...

formula. Remark: Most rigorously, multiplication of the rect function with a generalized function, such as the Dirac comb, fails. This is due to undetermined...

Multiscale Green's function (MSGF) is a generalized and extended version of the classical Green's function (GF) technique for solving mathematical equations...

+1-x\right)y'+n\,y=0} are called generalized Laguerre polynomials, or associated Laguerre polynomials. One can also define the generalized Laguerre polynomials recursively...

logicians, give precise definitions for these weakly specified functions. These generalized functions may be critical in the development of a formalization of...

singularity Generalized function Distribution Minkowski's question-mark function (**) This condition depends on the references "Singular function", Encyclopedia...

paragraphs 23.8 and 23.32) Gel'fand, I. M.; Vilenkin, N. Ya (1964). Generalized Functions: Applications of Harmonic Analysis. Burlington: Elsevier Science...

function when ⁠ σ 2 = 0 {\displaystyle \sigma ^{2}=0} ⁠. However, one can define the normal distribution with zero variance as a generalized function;...

Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables...

logistic function and generalizations. In growth modeling, numerous generalizations exist, including the generalized logistic curve, the Gompertz function, the...

(2001) [1994], "Generalized function", Encyclopedia of Mathematics, EMS Press. Vladimirov, V.S. (2001) [1994], "Generalized functions, space of", Encyclopedia...

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 rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized...

Theory and Applications. 80: 175–179. doi:10.1007/BF02196600. "Generalized Rosenbrock's function". Retrieved 2008-09-16. Kok, Schalk; Sandrock, Carl (2009)...

of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as...

the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz...

a<b<c} and where b is the difficult point, at which the behavior of the function f is such that ∫ a b f ( x ) d x = ± ∞ {\displaystyle \int _{a}^{b}f(x)\...

to spaces of white noise test functions φ {\displaystyle \varphi } , and, by duality, to spaces of generalized functions Ψ {\displaystyle \Psi } of white...

sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the domain over...