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Fourier of Fourier - n level Fourier transforms and scaled waves on waves.

At some point, a few years after college, and after learning about Fourier transforms, I eventually endeavored to take a Fourier transform of a Fourier transform. Conceptually, I can imagine not only taking a "2nd level Fourier transform", but "n-level Fourier transforms", where the transform of the transform of the transform, and so on, is taken a number of times. Mathematically and symbolically this could rapidly grow unwieldy, though it would be interesting but difficult to study the overarching symbolic forms of such a system, and as far as I've read, I've not yet seen it developed. It would also be interesting to see if functions could be found that could withstand multiple iterations without becoming unwieldy.

The following panels show some calculations of 2nd level Fourier transforms. As it was a few years ago, I don't remember the specifics of the first 4 panels other than I'm fairly sure they were just back end calculations setting up the first transform that I could feed into the second, in a way the program would understand. At Panel 5 you can start to see the familiar form of the Fourier technique being applied to the first transform.

Panel 7 through 12 shows the resulting graphs, with labels of the shape of the wave that was used. Among the examples are square waves on squares, and periodic parabolas on periodic parabolas. Though the graphs are not great, if you look closely, you can see the scaled mini wave pattern imposed on the larger transform in a recursive or fractal effect. This technique describes or is related to the "resolution" or "density" of a wave at some level, and could possibly be developed to counter Gibbs Phenomenon, be used as a micro carrier wave a top an already existing signal, or be tailored for noise reduction. If anyone has access to create better or high quality graphs of n-level Fourier transforms, feel free to send the links.

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