I previously asked a similar question on the signal processing community, but I think may be more easily solved from a mathematical perspective.
I start out with an image, $I$. To be able to process this (apply FFTs and such), I taper the edges to zero (making it left-right and top-bottom continuous). In doing so, some power is lost (quite obviously, and explicitly seen from Parceval's theorem).
This is problematic however if I want to study the power spectrum of an image. My question therefore comes down to: how do I get an unbiased prediction of the lost power per $k$-mode, given the apodizing mask $M$?
Since applying the mask comes down to a multiplication of $I$ and $M$ in real space, in Fourier space the Fourier transform of $I$ is convolved with the Fourier transform of $M$. What I'm looking for, supposedly, is a multiplicative function as a function of $k$ which gives me a properly normalized power spectrum.
Is this possible at all, and if so: how do I make the leap from $FT(M)$ to the function I'm looking for?
I would not be surprised if what I'm missing is very naive. I'm relatively new to this kind of signal processing. Any help is appreciated!