Having just read the rather long article on Wikipedia about window functions, I still do not see the mathematical point in using window functions when calculating spectral estimates (using e.g. Welch's method).
The Fourier transform of the function cos ωt is zero, except at frequency ±ω. However, many other functions and waveforms do not have convenient closed form transforms
what is this supposed to mean? I doubt any sampled and digitized signal has a convenient closed form transform anymore.