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• Distribution_(mathematics) abstract "Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function (which is historically called a "function" even though it is not considered a genuine function mathematically).According to Kolmogorov & Fomin (1999), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.The basic idea in distribution theory is to reinterpret functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the "generalized functions". There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly decreasing test functions gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the normal sense, but the converse is not true: in general the larger the space of test functions, the more restrictive the notion of distribution. On the other hand, the use of spaces of analytic test functions leads to Sato's theory of hyperfunctions; this theory has a different character from the previous ones because there are no analytic functions with non-empty compact support.".
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• Distribution_(mathematics) wikiPageID "51955".
• Distribution_(mathematics) wikiPageRevisionID "603103640".
• Distribution_(mathematics) authorLink "Vasilii Sergeevich Vladimirov".
• Distribution_(mathematics) authorlink "Sergei Lvovich Sobolev".
• Distribution_(mathematics) first "Michael".
• Distribution_(mathematics) first "Sergei".
• Distribution_(mathematics) first "V.S.".
• Distribution_(mathematics) hasPhotoCollection Distribution_(mathematics).
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• Distribution_(mathematics) id "G/g130030".
• Distribution_(mathematics) last "Oberguggenberger".
• Distribution_(mathematics) last "Sobolev".
• Distribution_(mathematics) last "Vladimirov".
• Distribution_(mathematics) title "Generalized function algebras".
• Distribution_(mathematics) title "Generalized function".
• Distribution_(mathematics) title "Generalized function, derivative of a".
• Distribution_(mathematics) title "Generalized functions, product of".
• Distribution_(mathematics) title "Generalized functions, space of".
• Distribution_(mathematics) year "1936".
• Distribution_(mathematics) year "2001".
• Distribution_(mathematics) subject Category:Functional_analysis.
• Distribution_(mathematics) subject Category:Generalized_functions.
• Distribution_(mathematics) subject Category:Smooth_functions.
• Distribution_(mathematics) type Abstraction100002137.
• Distribution_(mathematics) type Function113783816.
• Distribution_(mathematics) type GeneralizedFunctions.
• Distribution_(mathematics) type MathematicalRelation113783581.
• Distribution_(mathematics) type Relation100031921.
• Distribution_(mathematics) type SmoothFunctions.
• Distribution_(mathematics) comment "Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.".
• Distribution_(mathematics) label "Distributie (wiskunde)".
• Distribution_(mathematics) label "Distribution (Mathematik)".
• Distribution_(mathematics) label "Distribution (mathematics)".
• Distribution_(mathematics) label "Distribution (mathématiques)".
• Distribution_(mathematics) label "Distribuzione (matematica)".
• Distribution_(mathematics) label "Teoria das distribuições".
• Distribution_(mathematics) label "Teoria dystrybucji".
• Distribution_(mathematics) label "Teoría de distribuciones".
• Distribution_(mathematics) label "Обобщённая функция".
• Distribution_(mathematics) label "توزيع (رياضيات)".
• Distribution_(mathematics) label "シュワルツ超函数".
• Distribution_(mathematics) label "分布 (数学分析)".
• Distribution_(mathematics) sameAs Distribuce_(matematika).
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• Distribution_(mathematics) sameAs シュワルツ超函数.
• Distribution_(mathematics) sameAs 분포_(해석학).
• Distribution_(mathematics) sameAs Distributie_(wiskunde).
• Distribution_(mathematics) sameAs Teoria_dystrybucji.
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• Distribution_(mathematics) sameAs Distribution_(mathematics).
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• Distribution_(mathematics) isPrimaryTopicOf Distribution_(mathematics).