We know that the square window gives the lowest main lobe width possible, and that other windows after that trade main lobe width for side lobe height. I also understand that the main lobe width is inversely proportional to the length of the DFT aperature.
Given this, is there then a physical limit to how narrow a main-lobe can be, for a fixed aperture length? For example, can we ever hope to have main lobes that are narrower than that given by a square window, (for a fixed aperture length again), via windowing or any other method?
A second related question I have here has to do with energy conservation - is the interpretation that windows trade side lobe height for main lobe width, because energy must be conserved? Thus any reduction in side lobe height means the energy "has to go somewhere" and ends up in the main lobe? Would this be a correct interpretation?