Dirac delta function

The Dirac delta is a generalized function concentrated at one point, with unit integral and zero values everywhere else in an idealized sense. It represents point sources, impulses, and concentrated unit mass.

Sifting property: The identity by which integrating a function against a Dirac delta evaluates the function at the delta's location. This is the delta's defining computational rule: it extracts a function's value at zero or a shifted point.
Test function: A smooth function, usually with compact support, on which a distribution acts. The delta is defined by how it acts on test functions.
Impulse response: A system's output when its input is an idealized unit impulse. For linear time-invariant systems, the delta response determines outputs for general inputs.
Kronecker delta: A discrete function equal to one when its indices match and zero otherwise. It is the discrete analogue; unlike the Dirac delta, it is an ordinary function on a discrete domain.
Distribution theory: A mathematical framework that treats generalized functions as continuous linear functionals on smooth test functions. It gives the delta a rigorous meaning despite its not being an ordinary function.
Measure (mathematics): A mathematical assignment of size or mass to sets, subject to rules such as countable additivity. A point mass is a measure-theoretic counterpart of the delta distribution.
Green's function: A function or distribution that gives a differential operator's response to a point source. The delta supplies the point-source term in the defining equation for a Green's function.
Gaussian function: A smooth bell-shaped function whose value decreases exponentially with the square of distance from its center. A normalized Gaussian can approximate the delta as its width tends to zero, but never has point support at finite width.
Approximate identity: A family of kernels that approaches the identity operator as its members concentrate around zero. Narrow normalized kernels provide a limiting route to the delta's sampling behavior.
Point mass: A measure that assigns all its mass to a single point. It captures the delta's concentrated-unit-mass interpretation.

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