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======Continuous Fourier transform======
Consider the Fourier transform:
\begin{eqnarray}
D(x) & = & \frac{1}{4\pi^2} \int_{-\infty}^\infty d\kappa e^{-i\kappa x}\tilde{D}(\kappa) \\
 & \approx & \frac{1}{4\pi^2} \int_{-K}^K d\kappa e^{-i\kappa x}\tilde{D}(\kappa) \\
 & \approx & \frac{1}{4\pi^2} \sum_{k=0}^{N-1} \frac{2K}{N} \exp\left[-i\left(\frac{2K}{N}k-K\right) x\right]\tilde{D}_k \\
D_m & \propto & e^{iKx}\sum_{k=0}^{N-1}\exp\left[-i\frac{2K}{N}k\left(\frac{2L}{N}m-L\right)\right]\tilde{D}_k \\
 & \propto & \exp\frac{i2KLm}{N} \sum_{k=0}^{N-1}
  \exp\frac{-i 4KLkm}{N^2}\exp\frac{i2KLk}{N} \tilde{D}_k
\end{eqnarray}
Let $2KL=\pi N$, we get
\begin{equation}
e^{-i\pi m}D_m\propto\sum_{k=0}^{N-1}\exp\left(-2\pi i\frac{km}{N}\right)e^{i\pi k}\tilde{D}_k
\end{equation}
=====Fourier integral theorem and Dirac δ-function=====
{{  :notes:delta_dirichlet.svg?320|}}
[[https://en.wikipedia.org/wiki/Dirichlet_integral|Dirichlet integral]]
\[
\int_{-\infty}^\infty dx \frac{\sin(xL)}{x} = \pi
\]
As $L$ goes to infinity, the integrand becomes more and more like a δ-function.

  - [[http://www1.maths.leeds.ac.uk/~frank/math3383/AppA.pdf|Fourier Integrals & Dirac δ-function]] {{ :notes:appa.pdf|PDF}}

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