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 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 (weak solutions) than classical solutions, or where 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 singular, such as the Dirac delta function. A function is normally thought of as on the in the function domain by "sending" a point in the domain to the point Instead of acting on points, distribution theory reinterprets functions such as as acting on in a certain way. In applications to physics and engineering, are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset . (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by or Most commonly encountered functions, including all continuous maps if using can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that such a function "acts on" a test function by "sending" it to the number which is often denoted by This new action of defines a scalar-valued map whose domain is the space of test functions This functional turns out to have the two defining properties of what is known as a : it is linear, and it is also continuous when is given a certain topology called . The action (the integration ) of this distribution on a test function can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined.