Editing Dirac delta function (section)

===Fourier transform===
The delta function is a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]], and therefore it has a well-defined [[Fourier transform]].  Formally, one finds<ref>The numerical factors depend on the [[Fourier transform#Other conventions|conventions]] for the Fourier transform.</ref>

<math display="block">\widehat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi} \,\delta(x)dx = 1.</math>

Properly speaking, the Fourier transform of a distribution is defined by imposing [[self-adjoint]]ness of the Fourier transform under the [[Dual_system|duality pairing]] <math>\langle\cdot,\cdot\rangle</math> of tempered distributions with [[Schwartz functions]].  Thus <math>\widehat{\delta}</math> is defined as the unique tempered distribution satisfying

<math display="block">\langle\widehat{\delta},\varphi\rangle = \langle\delta,\widehat{\varphi}\rangle</math>

for all Schwartz functions {{mvar|φ}}. And indeed it follows from this that <math>\widehat{\delta}=1.</math>

As a result of this identity, the [[convolution]] of the delta function with any other tempered distribution {{mvar|S}} is simply {{mvar|S}}:

<math display="block">S*\delta = S.</math>

That is to say that {{mvar|δ}} is an [[identity element]] for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an [[associative algebra]] with identity the delta function.  This property is fundamental in [[signal processing]], as convolution with a tempered distribution is a [[linear time-invariant system]], and applying the linear time-invariant system measures its [[impulse response]].  The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for {{mvar|δ}}, and once it is known, it characterizes the system completely. See {{section link | LTI system theory |Impulse response and convolution}}.

The inverse Fourier transform of the tempered distribution {{math|1=''f''(''ξ'') = 1}} is the delta function. Formally, this is expressed as
<math display="block">\int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x)</math>
and more rigorously, it follows since
<math display="block">\langle 1, \widehat{f}\rangle = f(0) = \langle\delta,f\rangle</math>
for all Schwartz functions {{mvar|''f''}}.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on {{math|'''R'''}}.  Formally, one has
<math display="block">\int_{-\infty}^\infty e^{i 2\pi \xi_1 t} \left[e^{i 2\pi \xi_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (\xi_2 - \xi_1) t} \,dt = \delta(\xi_2 - \xi_1).</math>

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution
<math display="block">f(t) = e^{i2\pi\xi_1 t}</math>
is
<math display="block">\widehat{f}(\xi_2) = \delta(\xi_1-\xi_2)</math>
which again follows by imposing self-adjointness of the Fourier transform.

By [[analytic continuation]] of the Fourier transform, the [[Laplace transform]] of the delta function is found to be{{sfn|Bracewell|1986}}
<math display="block"> \int_{0}^{\infty}\delta(t-a)\,e^{-st} \, dt=e^{-sa}.</math>

====Fourier kernels====
{{See also|Convergence of Fourier series}}
In the study of [[Fourier series]], a major question consists of determining whether and in what sense the Fourier series associated with a [[periodic function]] converges to the function.  The {{mvar|n}}-th partial sum of the Fourier series of a function {{mvar|f}} of period {{math|2π}} is defined by convolution (on the interval {{closed-closed|−π,π}}) with the [[Dirichlet kernel]]:
<math display="block">D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left(\left(N+\frac12\right)x\right)}{\sin(x/2)}.</math>
Thus,
<math display="block">s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx}</math>
where
<math display="block">a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy.</math>
A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval&nbsp;{{closed-closed|−π,π}} tends to a multiple of the delta function as {{math|''N'' → ∞}}.  This is interpreted in the distribution sense, that
<math display="block">s_N(f)(0) = \int_{-\pi}^{\pi} D_N(x)f(x)\,dx \to 2\pi f(0)</math>
for every compactly supported {{em|smooth}} function {{mvar|f}}.  Thus, formally one has
<math display="block">\delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx}</math>
on the interval {{closed-closed|−π,π}}.

Despite this, the result does not hold for all compactly supported {{em|continuous}} functions: that is {{math|''D<sub>N</sub>''}} does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of [[summability methods]] to produce convergence.  The method of [[Cesàro summation]] leads to the [[Fejér kernel]]{{sfn|Lang|1997|p=312}}

<math display="block">F_N(x) = \frac1N\sum_{n=0}^{N-1} D_n(x) = \frac{1}{N}\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)^2.</math>

The [[Fejér kernel]]s tend to the delta function in a stronger sense that<ref>In the terminology of {{harvtxt|Lang|1997}}, the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not.</ref>

<math display="block">\int_{-\pi}^{\pi} F_N(x)f(x)\,dx \to 2\pi f(0)</math>

for every compactly supported {{em|continuous}} function {{mvar|f}}.  The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.