Editing Dirac delta function (section)

===As a distribution===
In the theory of [[distribution (mathematics)|distributions]], a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them.{{sfn|Hazewinkel|2011|p=[{{google books |plainurl=y |id=_YPtCAAAQBAJ|page=41}} 41]}} In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" '''test function''' {{mvar|φ}}.  Test functions are also known as [[bump function]]s.  If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all [[smooth function]]s on {{math|'''R'''}} with [[compact support]] that have as many derivatives as required.  As a distribution, the Dirac delta is a [[linear functional]] on the space of test functions and is defined by{{sfn|Strichartz|1994|loc=§2.2}}

{{NumBlk2|:| <math>\delta[\varphi] = \varphi(0)</math>|1}}

for every test function {{mvar|φ}}.

For {{mvar|δ}} to be properly a distribution, it must be continuous in a suitable topology on the space of test functions.  In general, for a linear functional {{mvar|S}} on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer {{mvar|N}} there is an integer {{math|''M''<sub>''N''</sub>}} and a constant {{mvar|''C''<sub>''N''</sub>}} such that for every test function {{mvar|φ}}, one has the inequality{{sfn|Hörmander|1983|loc=Theorem 2.1.5}}

<math display="block">\left|S[\varphi]\right| \le C_N \sum_{k=0}^{M_N}\sup_{x\in [-N,N]} \left|\varphi^{(k)}(x)\right|</math>

where {{math|sup}} represents the [[Infimum and supremum|supremum]]. With the {{mvar|δ}} distribution, one has such an inequality (with {{math|1=''C''<sub>''N''</sub> = 1)}} with {{math|1=''M''<sub>''N''</sub> = 0}} for all {{mvar|N}}.  Thus {{mvar|δ}} is a distribution of order zero. It is, furthermore, a distribution with compact support (the [[support (mathematics)|support]] being {{math|{{brace|0}}}}).

The delta distribution can also be defined in several equivalent ways.  For instance, it is the [[distributional derivative]] of the [[Heaviside step function]]. This means that for every test function {{mvar|φ}}, one has

<math display="block">\delta[\varphi] = -\int_{-\infty}^\infty \varphi'(x)\,H(x)\,dx.</math>

Intuitively, if [[integration by parts]] were permitted, then the latter integral should simplify to

<math display="block">\int_{-\infty}^\infty \varphi(x)\,H'(x)\,dx = \int_{-\infty}^\infty \varphi(x)\,\delta(x)\,dx,</math>

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have

<math display="block">-\int_{-\infty}^\infty \varphi'(x)\,H(x)\, dx = \int_{-\infty}^\infty \varphi(x)\,dH(x).</math>

In the context of measure theory, the Dirac measure gives rise to distribution by integration.  Conversely, equation ({{EquationNote|1}}) defines a [[Daniell integral]] on the space of all compactly supported continuous functions {{mvar|φ}} which, by the [[Riesz–Markov–Kakutani representation theorem|Riesz representation theorem]], can be represented as the Lebesgue integral of {{mvar|φ}} with respect to some [[Radon measure]].

Generally, when the term ''Dirac delta function'' is used, it is in the sense of distributions rather than measures, the [[Dirac measure]] being among several terms for the corresponding notion in measure theory.  Some sources may also use the term ''Dirac delta distribution''.