Editing Dirac delta function (section)

====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.