"Not obvious": yes, I was originally writing up the paper just to contain the derivation and results of the Fourier transform calculation — whatever it turned out to be — and was dumbstruck when it turned out to be sinc^3 x. The original title was just "The Fourier Transform of Magic Kernel Sharp."
It all turned out to be more interesting than a boring calculation. Some of it also doesn't quite flow properly because the story kept changing as the mystery unfolded, but I think it's close enough to summarize it all.
I also realized while writing it that the original doubling "magic" kernel (the 1/4, 3/4) was just the linear interpolation kernel for a factor of 2 with "tiled" positions, which I added to the page after writing the paper. I have since seen that others had previously made that comment (which I missed, but they are completely right). It's interesting that the infinitely repeated application of this "second" order kernel, i.e. what I now call m_2(x), originally led me (i.e. in 2011) to the "next" one, i.e. m_3(x). I still haven't fully sorted through how that works, but it's lucky that it did.
I'm always curious when/where/how we see Fourier Transforms of any sort -- because we see them in so many diverse fields, and in so much diverse Mathematics to consider them almost ubiquitous and fundamental for any meaningful understanding in many areas of Physics...
I also realized while writing it that the original doubling "magic" kernel (the 1/4, 3/4) was just the linear interpolation kernel for a factor of 2 with "tiled" positions, which I added to the page after writing the paper. I have since seen that others had previously made that comment (which I missed, but they are completely right). It's interesting that the infinitely repeated application of this "second" order kernel, i.e. what I now call m_2(x), originally led me (i.e. in 2011) to the "next" one, i.e. m_3(x). I still haven't fully sorted through how that works, but it's lucky that it did.