[mikegi]

You are confusing two expressions of the waveform here. If you limit yourself to sine waves (as in Fourier) analysis, you need an infinite set of such waves. On the other hand, you still need an infinite number of digits to describ the value at a point. These are only two different descriptions of the same object.

What a bunch of horsesh*t. If you can't see the monumental differences, both conceptually and physically, between the two approaches then you've gulped down far too much of the purple QM Koolaid. Ok, fine. You have your religion and I have mine. See ya!

[Doctor Stochastic]

You are confusing representation with reality. Were Fourier representations not applicable, then extrapolations with such representations would not be possible. However, one can make predictions based on Fourier (or Walsh or Legendre, etc. depending on the basis chosen) which is useful. Bessel's inequality gives sufficient reason for using a Fourier representation.