Similar question: mathoverflow.net/questions/11798/…

@JulesLamers O, I missed that. My bad, very thanks for pointing this out

@JulesLamers Yeah, It can be just rectangular window, but usually one adds additional smoothing. Here you can browse various windows: en.wikipedia.org/wiki/… The notation is consistent with the formula in my answer. Regarding $j$, I saw this notation few times; one interesting fact is that in python imaginary unit is in fact denoted by j. Thus, if you write in python 1.0 + 1.0j it will treat it as $1+i$.

@abx Your explanation of 5 makes it clear. I guess I will try to digest 6 next year :P. Wow, you abstracted the essence of the proof in a elegant and compact manner. The one thing which still bothers me is the hidden use of AC in taking maximal ideal $I$ in 5. I wonder if $I$ can be selected in some more concreate manner, but maybe it deserves a separate thread.