bandwidth limited pulse,what window function is it? I want to know what kind of shape a multiple cycle sine wave pulse ( or maybe the correct term is burst ) should have to have the narrowest bandwidth possible at given pulse width. In the laser science field, modelocked lasers producing ultrashort pulses are said to make transform limited pulse. Transform limited and bandwidth limited are the same thing: it's a pulse with minimum spectral width possible at the specific length of that pulse. When I was looking at the amplitude envelope shape of these pulses, it reminded me of the window function like it's used for example in fft, and it looked kind of like gaussian (the sort of gaussian that starts and ends with zero):



My question is what amplitude envelope shape should that sinewave multiple cycle pulse have to have minimum spectral bandwidth for its duration? At first I thought, that's easy, it must be gaussian (the "confined gaussian" in wikipedia, zero at edges), but when I thought about it more I am not so sure anymore. Wouldn't for example a pulse with Nuttall shape be narrower spectraly? What about Hanning and Blackman? I know all these look almost the same but hey, the true transform/bandwidth limited pulse can be only one shape, so which is it? https://en.m.wikipedia.org/wiki/Bandwidth-limited_pulse
From the great many types of window functions described on Wikipedia, the familiar rectangular window stand out has having a pretty small bandwidth. It's unfortunately also one with about the worst spectral leakage. 

Now for a given maximum allowable spectral leakage level, you might want to have a look at the Dolph–Chebyshev window, which minimize the norm of the side-lobes for a given main lobe width (or equivalently minimizes the main lobe width for a given maximum side-lobe level). Note however that this window has impulse like discontinuities at the edges in the time-domain due to the constant side-lobe level in the frequency-domain. These discontinuities are more significant for larger side-lobe level as can be seen in the following graph: