Assuming I have a function $f(x)$. I can use a numerical Fourier transformation:
NFourierTransform[] to find some points of $F(\omega)$. How can I do the same when I have not $f(x)$ but some points of it, i.e. $f(x_i)$.
For example:
PDF[NormalDistribution[0, 1], x] f[x_] := E^(-(x^2/2))/Sqrt[2 \[Pi]] Testdata = Array[f, 100, {-2, 2}]; Fourier[Testdata] gives a complex dataset back, but the Fourier transform should be again a real Gaussian?
The Gaussian f[x] you are transforming is given by your PDF statement. The corresponding frequency-domain Gaussian is given by
FourierTransform[f[x], x, w] which is the same function with w replacing x, that is, f[w]. The discrete Fourier transform on numerical data, implemented by Fourier, assumes periodicity of the input function. Hence, the Testdata you supply is seen by Fourier as a function of the following form, with an infinite number of peaks ranging from minus infinity to infinity.
Testdata = Array[f, 100, {-2, 2}]; ListLinePlot[Join[Testdata, Testdata, Testdata], Frame -> True, DataRange -> {-150, 150}] 
Note that the function does not reach zero, as an actual Gaussian would (in the limit). This offset means that you are not actually transforming a Gaussian function when you input Testdata to Fourier. A better approximation would be to sample more of the tails of the true Gaussian. In addition, with your Array formulation, the replications assumed by Fourier cause a double-sampling the points at f[-2] and at f[2]. It is better to match endpoints so that, when replicated, the (approximate) Gaussians match seemlessly. The following table of f[x] extends the range to better sample the tails, matches endpoints, and centres on zero (the first sample).
TestdataFull = RotateRight[Table[f[x], {x, -5.0, 5.0 - 1.0/10., 1.0/10.}], 50]; ListLinePlot[TestdataFull, Frame -> True] 
Fourier returns complex data even if the input signal is real. However, by matching endpoints and sampling the Gaussian more fully, the imaginary part is now essentially noise.
GraphicsRow[{ ListLinePlot[Re[Fourier[TestdataFull]][[Range[20]]], PlotRange -> All, Frame -> True, PlotLabel -> "Real Component"], ListLinePlot[Im[Fourier[TestdataFull]][[Range[20]]], PlotRange -> All, Frame -> True, PlotLabel -> "Imaginary Component"]}] 
Take the magnitude of the complex, yet essentially real, data returned by Fourier, and centre the peak. There's your real Gaussian.
ListLinePlot[RotateRight[Abs[Fourier[TestdataFull]], 50], PlotRange -> All, Frame -> True] 
Fourier? $\endgroup$