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To give an unmathematical catchy answer, let's look at Fraunhofer diffraction in double slit experiment. Interference at the observation plane depends on slit parameter $d$. What is the frequency of slits? E.g. $1\,\text{mm}\frac{1}{d}$: number of slits per length. Concluding frequiency in the setup. The following argumentation links this frequency to the ...

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In general, what you're trying to do is called the "spectral method" for solving PDEs. Wikipedia has a little on it, including some useful references, and a solution of the Poisson equation. http://en.wikipedia.org/wiki/Spectral_method As Peter Kravchuck says, the kernel will always be $k^{-2}$ for the Poisson equation. In the linked PDF, ...

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In Fourier space the Poisson equation is $k^2\phi=\rho$ (up to a convention-dependent constant factor). So in every dimension the kernel is $1/k^2$. As for the real space, it is, up to a constant $|r|,\,\log|r|,\,1/r$ in 1D,2D,3D respectively.

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