I am a bit new to fourier transforms, I've been reading some literature and have a question about the following derivation.
Consider the function $f(x)$ , whose fourier transform is $F(k)$
The fourier transform of second derivative of this function $f''(x)$ is represented as $F''(k)$
It can be shown that $F''(k) = -|k|^2 F(k)$
now we apply inverse fourier transform for the above equation on both sides, and the product becomes a convolution due to the convolution theorem. We have
$f''(x) = \mathcal{F}^{-1}(-|k|^2)*f(x)$
now we know that in the discrete domain, the second derivative of a function can be obtained by convolving a kernel onto the function ( using finite differences: https://en.wikipedia.org/wiki/Discrete_Laplace_operator ).
My question is: does this kernel in some way relate to the value of $\mathcal{F}^{-1}(-|k|^2)$? I've tried searching for it but haven't found any relationsip, apart from that it has something to do with the dirichlet delta functions.
I would really appreciate anyone enlightening me on this!