Is there any practical interest in either differing convergence rates with the type of discontinuity, the characteristics of the partial series (e.g. if FFT shows different ringing than a conventional Fourier), or the behaviour as you include more and more terms?
The Wikipedia entry on the Gibbs phenomenon is pretty informative, g_w. As for practicalities, the entry points out that there's no overshoot/undershoot if wavelet transforms are used. Something else to study (tentatively scheduled for the spring of 2093)...
Cheers!