I've found this paper on the logarithm of the discrete fourier transform which proves that

$$ log F = 1/4 i \pi (I - (1 +i)F + F^2 - (1 - i)F^3) $$

where $F$ is the unitary discrete Fourier transform operator.

Is there a similar analogous version known for the infinite dimensional Fourier transform?

I am asking this from the point of view of trying to better understand cumulants, whose generating function is the logarithm of the characteristic function. The characteristic function is the fourier transform of the PDF of a random variable, thus motivating this question.

EDIT: Please take as many "niceness" assumptions as needed to answer the question regarding convergence.

reallyunfamiliar with the "proper" fourier transform. I've only used it from a "physics" standpoint. Could you please expand your comment into an answer, as I do not really understand what you've said? That would be very helpful for me :) $\endgroup$ – Siddharth Bhat Mar 17 '20 at 23:58operatorin the sense of spectral theory. It's the difference between $\log(F[f])$ and $(\log F)[f]$. It's like the fact that if $A$ is a matrix and $v$ is a vector, $A^2 v$ is not the same as taking $Av$ and squaring each entry. $\endgroup$ – Nate Eldredge Mar 18 '20 at 14:57