Editing Dirac delta function (section)

====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>