By iterating or nesting a sine curve like in this question
I get curves like these:
that seem to tend to a square wave.
The eight case of these looks like:
which here on purpose was chosen for it's roundness.
The Mathematica code for these plots can be found here: http://pastebin.com/6UK1u1uX
I don't know much about signal processing but I recalled the Gibbs phenomenon in square waves after I saw these curves.
Would they solve the problem with the Gibbs phenomenon in the case of square waves?
In the Fourier transform this kind of function is not of any use though I understand.
Sawtooth wave: http://pastebin.com/JNg7bzzB
Triangular wave (partial sums instead of integrals): http://pastebin.com/wRCBV7NF
Dirac comb http://pastebin.com/QMSMQf26