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Wow that was pretty mind-blowing.

His Phi operator almost felt like he was doing some sort of "diagonalization" step of continuous derivatives/integrals into discrete differences/sums in the same way you would diagonalize a matrix.

I wonder if there's something there, maybe phi is some sort of analog to eigenvectors of integrals/derivatives.

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Yeah, the Fourier transform diagonalizes differentiation. Or in other words "differentiation is a Fourier multiplier". Integration is like a low-pass filter (you divide by the amplitude of a component by its frequency and so higher frequencies get quieter) and differentiation is "noisy" since high frequency components get louder.

d^2/dt^2(sin kt) = -k^2 sin(kt)

This idea is used in the study of partial differential equations, especially dispersive.

https://www.math.ucla.edu/~tao/preprints/chapter.pdf

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