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Is this some sort of correlation or convolution?
No. For that you'd need to build products from the elements. You only do differences.
It should be meaningful for calculating ifft(X), but I can't see how...
I'm not immediately recognizing these terms, and the length of 5 seems very odd¹; so, this is probably an algorithmic step in a specific FFT algorithm ...
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In the continuous time-function formalism, and accepting the framework or distributions or generalized functions, the answer is direct. Taking $\delta$ for the Dirac delta function, for a sufficiently well-behaved function $f$:
$$\delta' * f = \delta * f' = f'\,.$$
Therefore, the convolution mask is obvious: it would be the derivative of the Dirac delta. The ...
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