# Why are window functions used when calculating spectral estimates, really?

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).

For example,

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.

• One main reason is control of spectral leakage. – Jason R Aug 15 '14 at 17:49
• windows also is using to analyse short segments of signals which is assumed to be stationary,for example STFT use windows for non stationary signals – dato datuashvili Aug 15 '14 at 19:46