Editing Dirac delta function (section)

===Nascent delta function===
The delta function can be viewed as the limit of a sequence of functions

<math display="block">\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), </math>

where {{math|''η<sub>ε</sub>''(''x'')}} is sometimes called a '''nascent delta function'''{{anchor|nascent delta function}}. This limit is meant in a weak sense: either that

{{NumBlk2|:|<math> \lim_{\varepsilon\to 0^+} \int_{-\infty}^\infty \eta_\varepsilon(x)f(x) \, dx = f(0) </math>|5}}

for all [[continuous function|continuous]] functions {{mvar|f}} having [[compact support]], or that this limit holds for all [[smooth function|smooth]] functions {{mvar|f}} with compact support.  The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the [[vague topology]] of measures, and the latter is convergence in the sense of [[distribution (mathematics)|distributions]].

====Approximations to the identity====
Typically a nascent delta function {{mvar|η<sub>ε</sub>}} can be constructed in the following manner.  Let {{mvar|η}} be an absolutely integrable function on {{math|'''R'''}} of total integral {{math|1}}, and define
<math display="block">\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). </math>

In {{mvar|n}} dimensions, one uses instead the scaling
<math display="block">\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). </math>

Then a simple change of variables shows that {{mvar|η<sub>ε</sub>}} also has integral {{math|1}}.  One may show that ({{EquationNote|5}}) holds for all continuous compactly supported functions {{mvar|f}},{{sfn|Stein|Weiss|1971|loc=Theorem 1.18}} and so {{mvar|η<sub>ε</sub>}} converges weakly to {{mvar|δ}} in the sense of measures.

The {{mvar|η<sub>ε</sub>}} constructed in this way are known as an '''approximation to the identity'''.{{sfn|Rudin|1991|loc=§II.6.31}} This terminology is because the space {{math|''L''<sup>1</sup>('''R''')}} of absolutely integrable functions is closed under the operation of [[convolution]] of functions: {{math|''f'' ∗ ''g'' ∈ ''L''<sup>1</sup>('''R''')}} whenever {{mvar|f}} and {{mvar|g}} are in {{math|''L''<sup>1</sup>('''R''')}}.  However, there is no identity in {{math|''L''<sup>1</sup>('''R''')}} for the convolution product: no element {{mvar|h}} such that {{math|1=''f'' ∗ ''h'' = ''f''}} for all {{mvar|f}}.  Nevertheless, the sequence {{mvar|η<sub>ε</sub>}} does approximate such an identity in the sense that

<math display="block">f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0.</math>

This limit holds in the sense of [[mean convergence]] (convergence in {{math|''L''<sup>1</sup>}}).  Further conditions on the {{mvar|η<sub>ε</sub>}}, for instance that it be a mollifier associated to a compactly supported function,<ref>More generally, one only needs {{math|1=''η'' = ''η''<sub>1</sub>}} to have an integrable radially symmetric decreasing rearrangement.</ref> are needed to ensure pointwise convergence [[almost everywhere]].

If the initial {{math|1=''η'' = ''η''<sub>1</sub>}} is itself smooth and compactly supported then the sequence is called a [[mollifier]].  The standard mollifier is obtained by choosing {{mvar|η}} to be a suitably normalized [[bump function]], for instance

<math display="block">\eta(x) = \begin{cases}
\frac{1}{I_n} \exp\Big( -\frac{1}{1-|x|^2} \Big) & \text{if } |x| < 1\\
0                     & \text{if } |x|\geq 1.
\end{cases}</math>
(<math>I_n</math> ensuring that the total integral is 1).

In some situations such as [[numerical analysis]], a [[piecewise linear function|piecewise linear]] approximation to the identity is desirable.  This can be obtained by taking {{math|''η''<sub>1</sub>}} to be a [[hat function]].  With this choice of {{math|''η''<sub>1</sub>}}, one has

<math display="block"> \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) </math>

which are all continuous and compactly supported, although not smooth and so not a mollifier.

====Probabilistic considerations====
In the context of [[probability theory]], it is natural to impose the additional condition that the initial {{math|''η''<sub>1</sub>}} in an approximation to the identity should be positive, as such a function then represents a [[probability distribution]].  Convolution with a probability distribution is sometimes favorable because it does not result in [[overshoot (signal)|overshoot]] or undershoot, as the output is a [[convex combination]] of the input values, and thus falls between the maximum and minimum of the input function.  Taking {{math|''η''<sub>1</sub>}} to be any probability distribution at all, and letting {{math|1=''η<sub>ε</sub>''(''x'') = ''η''<sub>1</sub>(''x''/''ε'')/''ε''}} as above will give rise to an approximation to the identity.  In general this converges more rapidly to a delta function if, in addition, {{mvar|η}} has mean {{math|0}} and has small higher moments. For instance, if {{math|''η''<sub>1</sub>}} is the [[uniform distribution (continuous)|uniform distribution]] on {{nowrap|1=<math display="inline">\left[-\frac{1}{2},\frac{1}{2}\right]</math>,}} also known as the [[rectangular function]], then:{{sfn|Saichev|Woyczyński|1997|loc=§1.1 The "delta function" as viewed by a physicist and an engineer, p. 3}}
<math display="block">
\eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)=
\begin{cases}
\frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}, \\
0,                    &\text{otherwise}.
\end{cases}</math>

Another example is with the [[Wigner semicircle distribution]]
<math display="block">\eta_\varepsilon(x)= \begin{cases}
\frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon, \\
0, & \text{otherwise}.
\end{cases}</math>

This is continuous and compactly supported, but not a mollifier because it is not smooth.

====Semigroups====
Nascent delta functions often arise as convolution [[semigroup]]s.<ref>{{Cite book|last1=Milovanović|first1=Gradimir V.|url={{google books |plainurl=y |id=4U-5BQAAQBAJ}}|title=Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava|last2=Rassias|first2=Michael Th|date=2014-07-08|publisher=Springer|isbn=978-1-4939-0258-3|language=en|page=[{{google books |plainurl=y |id=4U-5BQAAQBAJ|page=748 }} 748]}}</ref> This amounts to the further constraint that the convolution of {{mvar|η<sub>ε</sub>}} with {{mvar|η<sub>δ</sub>}} must satisfy
<math display="block">\eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta}</math>

for all {{math|1=''ε'', ''δ'' > 0}}. Convolution semigroups in {{math|''L''<sup>1</sup>}} that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction.

In practice, semigroups approximating the delta function arise as [[fundamental solution]]s or [[Green's function]]s to physically motivated [[elliptic partial differential equation|elliptic]] or [[parabolic partial differential equation|parabolic]] [[partial differential equations]].  In the context of [[applied mathematics]], semigroups arise as the output of a [[linear time-invariant system]].  Abstractly, if ''A'' is a linear operator acting on functions of ''x'', then a convolution semigroup arises by solving the [[initial value problem]]

<math display="block">\begin{cases}
\dfrac{\partial}{\partial t}\eta(t,x) = A\eta(t,x), \quad t>0 \\[5pt]
\displaystyle\lim_{t\to 0^+} \eta(t,x) = \delta(x)
\end{cases}</math>

in which the limit is as usual understood in the weak sense.  Setting {{math|1=''η<sub>ε</sub>''(''x'') = ''η''(''ε'', ''x'')}} gives the associated nascent delta function.

Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.

=====The heat kernel=====
The [[heat kernel]], defined by

<math display="block">\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}</math>

represents the temperature in an infinite wire at time {{math|1=''t'' > 0}}, if a unit of heat energy is stored at the origin of the wire at time {{math|1=''t'' = 0}}. This semigroup evolves according to the one-dimensional [[heat equation]]:

<math display="block">\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.</math>

In [[probability theory]], {{math|1=''η<sub>ε</sub>''(''x'')}} is a [[normal distribution]] of [[variance]] {{mvar|ε}} and mean {{math|0}}. It represents the [[probability density function|probability density]] at time {{math|1=''t'' = ''ε''}} of the position of a particle starting at the origin following a standard [[Brownian motion]]. In this context, the semigroup condition is then an expression of the [[Markov property]] of Brownian motion.

In higher-dimensional Euclidean space {{math|'''R'''<sup>''n''</sup>}}, the heat kernel is
<math display="block">\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},</math>
and has the same physical interpretation, {{lang|la|[[mutatis mutandis]]}}. It also represents a nascent delta function in the sense that {{math|''η<sub>ε</sub>'' → ''δ''}} in the distribution sense as {{math|''ε'' → 0}}.

=====The Poisson kernel=====
The [[Poisson kernel]]
<math display="block">\eta_\varepsilon(x) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{1}{x-\mathrm{i}\varepsilon}\right\}=\frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi x-|\varepsilon \xi|}\,d\xi</math>

is the fundamental solution of the [[Laplace equation]] in the upper half-plane.{{sfn|Stein|Weiss|1971|loc=§I.1}}  It represents the [[electrostatic potential]] in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the [[Cauchy distribution]] and [[Kernel (statistics)#Kernel functions in common use|Epanechnikov and Gaussian kernel]] functions.<ref>{{Cite book|last=Mader|first=Heidy M.|url={{google books |plainurl=y |id=e5Y_RRPxdyYC}}|title=Statistics in Volcanology|date=2006|publisher=Geological Society of London|isbn=978-1-86239-208-3|language=en|editor-link=Heidy Mader|page=[{{google books |plainurl=y |id=e5Y_RRPxdyYC|page=81}} 81]}}</ref> This semigroup evolves according to the equation
<math display="block">\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)</math>

where the operator is rigorously defined as the [[Fourier multiplier]]
<math display="block">\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).</math>

====Oscillatory integrals====
In areas of physics such as [[wave propagation]] and [[wave|wave mechanics]], the equations involved are [[hyperbolic partial differential equations|hyperbolic]] and so may have more singular solutions.  As a result, the nascent delta functions that arise as fundamental solutions of the associated [[Cauchy problem]]s are generally [[oscillatory integral]]s.  An example, which comes from a solution of the [[Euler–Tricomi equation]] of [[transonic]] [[gas dynamics]],{{sfn|Vallée|Soares|2004|loc=§7.2}} is the rescaled [[Airy function]]
<math display="block">\varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). </math>

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense.  Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the [[Dirichlet kernel]] below), rather than in the sense of measures.

Another example is the Cauchy problem for the [[wave equation]] in {{math|'''R'''<sup>1+1</sup>}}:{{sfn|Hörmander|1983|loc=§7.8}}
<math display="block"> \begin{align}
c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\
u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0.
\end{align} </math>

The solution {{mvar|u}} represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the [[sinc function]] (used widely in electronics and telecommunications)
<math display="block">\eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\,dk </math>

and the [[Bessel function]]
<math display="block">  \eta_\varepsilon(x) =  \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). </math>