The relevant theorem is that variance(position-space function f) * variance(Fourier transform of f) >= some constant.
A wavefunction is a wavefunction, it doesn't have more frequency components at the classical scale. It's just that what looks like a large position spread at the quantum scale is pretty tiny compared to classical length scales, and what looks like a large momentum spread (variance of the Fourier transform) is pretty tiny compared to classical-scale momenta.
You shouldn't think of a wavefunction as being perfectly localized, with infinitely spread out frequency components, because that would mean the particle's momentum is infinitely uncertain. Instead, think of a Gaussian function, whose Fourier transform is a Gaussian. The widths of those two Gaussians are inversely proportional to each other.
Also, you should think about continuous Fourier transforms, not discrete Fourier series. Periodic wavefunctions are only the norm in situations like crystals where the environment itself is periodic.