Comment by gmueckl

Slightly more technical: every function that is continuous on a finite support can be expanded into an infinite Fourier series. The terms of that series form an orthonormal basis in the Hilbert space over the function's support, so this transformation is exact for the infinite series. The truncated Fourier series converges monotonically towards the original function with increasing number of terms. So truncation produces an approximation.

The beauty of the Fourier series is that the individual basis functions can be interpreted as oscillations with ever increasing frequency. So the truncated Fourier transformation is a band linited approximation to any function it can be appolied to. And the Nyquist frequency happens to be the oscillating frequency of the highest order term in this truncation. The Nyquist-Shannon theorem relates it strictly to the sampling frequency of any periodicaly sampled function. So every sampled signal inherently has a band limited frequency space representation and is subject to frequency domain effects under transformation.