"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 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.