Distribution (mathematics)

A distribution is a continuous linear functional on a space of smooth test functions. It generalizes ordinary functions and allows derivatives to be defined even when classical derivatives do not exist.

Test function: A smooth function, often with compact support, used to probe a distribution. Distributions are defined by how they act on these functions.
Distributional derivative: The derivative defined by transferring differentiation from a distribution to its test functions with a sign change. It extends differentiation to objects that need not be ordinary differentiable functions.
Weak solution: A solution of a differential equation formulated through integrals against test functions rather than pointwise derivatives. Distributional derivatives make weak formulations meaningful for nonsmooth solutions.
Classical derivative: The pointwise derivative of a function, defined through limits of difference quotients. Distributional differentiation extends this operation to nonsmooth functions and singular objects.
Locally integrable function: A function whose absolute value has finite integral on every compact subset of its domain. Integration turns every locally integrable function into a distribution.
Dirac delta distribution: The distribution that evaluates a test function at a specified point. It demonstrates how a distribution can represent a point concentration without being a function.
Fundamental solution: A distribution that solves a linear differential operator equation with a Dirac delta source. It turns a point-source equation into a tool for constructing broader solutions.
Measure (mathematics): A countably additive function assigning sizes to sets in a measurable space. Measures also act on test functions by integration, but distributions form a broader framework.
Dual space: The vector space of continuous linear functionals on a given topological vector space. A distribution is an element of the continuous dual of a test-function space.
Convolution of distributions: An operation combining distributions when their supports or other conditions make the convolution well-defined. Convolution lets distributional methods handle smoothing and translation-invariant equations.

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