Physical implications of the Gibbs phenomenon for Quantum Mechanics

From Wikipedia:

The Gibbs Phenomenon is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit. This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring devices.

Two questions on this effect:

1. Does it carry over from Fourier Series into Fourier Transforms? I have done a search on Google, but no immediate answer jumps out at me, apologies if I missed something obvious, or skipped over something too math heavy for me to follow.

2. If there is a Fourier Transform version of the Gibbs effect, how would that affect, in physical terms, and taking for simplicity (and my chances of understanding any answer), the transformation of a 1D electron wavefunction from the position space representation into it's momentum state equivalent.

EDIT I completely failed to spot this answer: Why no Gibbs Phenomenon in QM, as provided in Qmechanic's comment below. END EDIT