I need to use the discrete fourier transform for function that represented as list of values. I started from the easy task to check am I use it right - try to reach the amplitude values fot 2*Sin[x] function. Looks like it should be somwhere about 2. But using the discrete 'Fourier' operation it is different. Can someone tell me - where is my mistakes?

Please see my observations:

data = Table[2*Sin[x], {x, 0, 100}]; ListPlot[data, Joined -> True]

ListPlot[Abs[Fourier[data]], Joined -> True, PlotRange -> All]

ListPlot[Table[ Abs[FourierTransform[2*Sin[x], x, w]] /. DiracDelta -> DiscreteDelta, {w, 0.1, 10, 0.1}], Joined -> True, PlotRange -> All]

How can I reach fo the 'Fourier' operation the same results as I reach for the 'FourierTransform'?

Thank you for help.