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You are right, there is no relation. And your intuition gives one way to see why. If you take an average function f and convolve it with a nice, smooth, function g, this smooths f more but also spreads it out more: the smoothness becomes better, but the support of f, the domain where it is non-zero, becomes "worse", so to speak. So the convolution is ...

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Expounding on the 1st part of @Einchenlaub's answer (and describing my favourite example), The Cochlea in the ear (in which the hair follicles are present) is a beautiful example of how the fourier transform is carried out physically. The changing diameter of the cochlea's tube causes different frequencies to resonate in different parts of the cochlea, so ...

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I'm not sure if this will make the Fourier transform easier, but nevertheless you should be able to do a change of variables in the vein of $n = j+l$ and $m = j-l$. My advice would be to create a table of $j$,$l$,$j+l$, and $j-l$. What you will find is that there is a weighting function associated with each variable. For instance, the $(j,l)$ pairs ...

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Two points from a very practical point of view based on laser systems. Imagine there is a small dust particle on an optical element in your path. Diffraction from that little particle will create a pattern that expands with the propagation along the path. If you insert a $4f$ system, you "cut out the propagation" on a distance of $4f$: the plane at a ...

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This is more of a comment than an answer, but I can't fit this into the amount of characters; Writing a quick bit of code, it looks to me like there's not much wrong with the method: The numerical and the analytical solution go on top of one another. N = 256 T = 256*128 L = 1. dt = 0.000001 x = linspace(0., N-1, N)*L/N psix = exp(1j*2*pi*x) psik = ...

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Short answer: The Hartley transform is a subset of the results given by Fourier transforms, which is only the real part (assuming your signal is real, which is almost always the case in physics). Long answer: Practically, you need the amplitude and phase of the signal, and the Fourier transform gives you both amplitude and phase by taking the magnitude: ...

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