fourier series converges pointwise but not uniformly. for many use cases you would like uniform convergence (e.g. approximation in an entire neighborhood of a p

fourier series converges pointwise but not uniformly. for many use cases you would like uniform convergence (e.g. approximation in an entire neighborhood of a point) and a fourier series won't give you that no matter how many terms you add. but like i said above - gibbs only occurs at non-removable discontinuities and so you shouldn't expect it to be well behaved at those points regardless of the representation.