Editing Dirac delta function (section)

==Definitions==
The Dirac delta function <math>\delta (x)</math> can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

<math display="block">\delta(x) \simeq \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>

and which is also constrained to satisfy the identity{{sfn|Gelfand|Shilov|1966–1968|loc=Volume I, §1.1, p. 1}}

<math display="block">\int_{-\infty}^\infty \delta(x) \, dx = 1.</math>

This is merely a [[heuristic]] characterization. The Dirac delta is not a function in the traditional sense as no [[extended real number]] valued function defined on the real numbers has these properties.{{sfn|Dirac|1930|p=63}}

===As a measure===
One way to rigorously capture the notion of the Dirac delta function is to define a [[Measure (mathematics)|measure]], called [[Dirac measure]], which accepts a subset {{mvar|A}} of the real line {{math|'''R'''}} as an argument, and returns {{math|1=''δ''(''A'') = 1}} if {{math|0 ∈ ''A''}}, and {{math|1=''δ''(''A'') = 0}} otherwise.<ref name="Rudin 1966 loc=§1.20">{{harvnb|Rudin|1966|loc=§1.20}}</ref>  If the delta function is conceptualized as modeling an idealized point mass at 0, then {{math|''δ''(''A'')}} represents the mass contained in the set {{mvar|A}}.  One may then define the integral against {{mvar|δ}} as the integral of a function against this mass distribution.  Formally, the [[Lebesgue integral]] provides the necessary analytic device. The Lebesgue integral with respect to the measure {{mvar|δ}} satisfies

<math display="block">\int_{-\infty}^\infty f(x) \, \delta(dx) =  f(0)</math>

for all continuous compactly supported functions {{mvar|f}}. The measure {{mvar|δ}} is not [[absolutely continuous]] with respect to the [[Lebesgue measure]]—in fact, it is a [[singular measure]]. Consequently, the delta measure has no [[Radon–Nikodym derivative]] (with respect to Lebesgue measure)—no true function for which the property

<math display="block">\int_{-\infty}^\infty f(x)\, \delta(x)\, dx = f(0)</math>

holds.{{sfn|Hewitt|Stromberg|1963|loc=§19.61}} As a result, the latter notation is a convenient [[abuse of notation]], and not a standard ([[Riemann integral|Riemann]] or [[Lebesgue integral|Lebesgue]]) integral.

As a [[probability measure]] on {{math|'''R'''}}, the delta measure is characterized by its [[cumulative distribution function]], which is the [[unit step function]].<ref>{{harvnb|Driggers|2003|p=2321}}  See also {{harvnb|Bracewell|1986|loc=Chapter 5}} for a different interpretation.  Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows.</ref>

<math display="block">H(x) = 
\begin{cases}
1 & \text{if } x\ge 0\\
0 & \text{if } x < 0.
\end{cases}</math>

This means that {{math|''H''(''x'')}} is the integral of the cumulative [[indicator function]] {{math|'''1'''<sub>(−∞, ''x'']</sub>}} with respect to the measure {{mvar|δ}}; to wit,

<math display="block">H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta(dt) = \delta\!\left((-\infty,x]\right),</math>

the latter being the measure of this interval. Thus in particular the integration of the delta function against a continuous function can be properly understood as a [[Riemann–Stieltjes integral]]:{{sfn|Hewitt|Stromberg|1963|loc=§9.19}}

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

All higher [[moment (mathematics)|moments]] of {{mvar|δ}} are zero.  In particular, [[characteristic function (probability theory)|characteristic function]] and [[moment generating function]] are both equal to one.

===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''.

===Generalizations===
The delta function can be defined in {{mvar|n}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} as the measure such that

<math display="block">\int_{\mathbf{R}^n} f(\mathbf{x})\,\delta(d\mathbf{x}) = f(\mathbf{0})</math>

for every compactly supported continuous function {{mvar|f}}.  As a measure, the {{mvar|n}}-dimensional delta function is the [[product measure]] of the 1-dimensional delta functions in each variable separately. Thus, formally, with {{math|1='''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}}, one has{{sfn|Bracewell|1986|loc=Chapter 5}}

{{NumBlk2|:|<math>\delta(\mathbf{x}) = \delta(x_1)\,\delta(x_2)\cdots\delta(x_n).</math>|2}}

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.{{sfn|Hörmander|1983|loc=§3.1}}  However, despite widespread use in engineering contexts, ({{EquationNote|2}}) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.{{sfn|Strichartz|1994|loc=§2.3}}{{sfn|Hörmander|1983|loc=§8.2}}

The notion of a '''[[Dirac measure]]''' makes sense on any set.{{sfn|Rudin |1966 |loc=§1.20}} Thus if {{mvar|X}} is a set, {{math|''x''<sub>0</sub> ∈ ''X''}} is a marked point, and {{math|Σ}} is any [[sigma algebra]] of subsets of {{mvar|X}}, then the measure defined on sets {{math|''A'' ∈ Σ}} by

<math display="block">\delta_{x_0}(A)=\begin{cases}
1 &\text{if }x_0\in A\\
0 &\text{if }x_0\notin A
\end{cases}</math>

is the delta measure or unit mass concentrated at {{math|''x''<sub>0</sub>}}.

Another common generalization of the delta function is to a [[differentiable manifold]] where most of its properties as a distribution can also be exploited because of the [[differentiable structure]].  The delta function on a manifold {{mvar|M}} centered at the point {{math|''x''<sub>0</sub> ∈ ''M''}} is defined as the following distribution:

{{NumBlk2|:|<math>\delta_{x_0}[\varphi] = \varphi(x_0)</math>|3}}

for all compactly supported smooth real-valued functions {{mvar|φ}} on {{mvar|M}}.{{sfn|Dieudonné|1972|loc=§17.3.3}} A common special case of this construction is a case in which {{mvar|M}} is an [[open set]] in the Euclidean space {{math|'''R'''<sup>''n''</sup>}}.

On a [[locally compact Hausdorff space]] {{mvar|X}}, the Dirac delta measure concentrated at a point {{mvar|x}} is the [[Radon measure]] associated with the Daniell integral ({{EquationNote|3}}) on compactly supported continuous functions {{mvar|φ}}.<ref>{{Cite book|last1=Krantz|first1=Steven G.|url={{google books |plainurl=y |id=X_BKmVphIcsC&q }}|title=Geometric Integration Theory|last2=Parks|first2=Harold R.|date=2008-12-15|publisher=Springer Science & Business Media|isbn=978-0-8176-4679-0|language=en}}</ref> At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping <math>x_0\mapsto \delta_{x_0}</math> is a continuous embedding of {{mvar|X}} into the space of finite Radon measures on {{mvar|X}}, equipped with its [[vague topology]]. Moreover, the [[convex hull]] of the image of {{mvar|X}} under this embedding is [[dense set|dense]] in the space of probability measures on {{mvar|X}}.{{sfn|Federer|1969|loc=§2.5.19}}