Distribution theory

Distribution theory extends ordinary functions by treating them as continuous linear functionals on spaces of test functions. It gives rigorous meaning to objects such as the Dirac delta and derivatives too singular to be classical functions.

Test function: A smooth function with specified support or decay conditions, used to probe a distribution. Distributions are defined by their action on these functions.
Locally integrable function: A measurable function whose absolute value has finite integral on every compact subset of its domain. Every such function defines a distribution by integration against test functions.
Dirac delta: The distribution that maps each test function to its value at a specified point. It models a point concentration that cannot be represented by an ordinary integrable function.
Green's function: A kernel that represents the response of a linear differential operator to a point source. Point sources are modeled by delta distributions in the defining equation.
Laurent Schwartz: A French mathematician who developed the modern theory of distributions. His systematic framework established distributions as continuous functionals on test-function spaces.
Continuous linear functional: A continuous map from a topological vector space to its scalar field that preserves addition and scalar multiplication. A distribution is precisely such a functional on a test-function space.
Smooth function: A function with derivatives of every finite order. Smooth functions are the probes used to define distributions.
Heaviside step function: A function that is zero on one side of a threshold and one on the other. Its distributional derivative is a Dirac delta at the jump.
Fundamental solution: A distribution whose image under a differential operator is a Dirac delta. Its defining equation often has no classical-function solution.
Dirac delta function: An informal point-mass object introduced in physics, later given rigorous meaning as a distribution. Its use in quantum mechanics helped motivate a rigorous theory of generalized functions.

Linked from 8 pages