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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 a distribution, not a function as its historical name might suggest.
The practical use of distributions can be traced back to the use of Green functions in the s to solve ordinary differential equations, but was not formalized until much later.
Use of the space of smooth, rapidly faster than any polynomial increases decreasing test functions these functions are called Schwartz functions gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform.
Jump to navigation Jump to search This article is about generalized functions in mathematical analysis.
In particular, any locally integrable function has a distributional derivative. 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".
Distributions or generalized functions are objects that generalize the classical notion of functions in mathematical analysis. For the concept of distributions in probability theory, see Probability distribution.
Distribution theory reinterprets functions as linear functionals acting on a space of test functions.
The basic space of test function consists of smooth functions with compact supportleading to standard distributions. Every tempered distribution is a distribution in the normal sense, but the converse is not true: For artificial landscapes, see Test functions for optimization.
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Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. Distributions are widely used in the theory of partial differential equationswhere it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist.
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. There are different possible choices for the space of test functions, leading to different spaces of distributions.