I create a stepfunction and calculate the Discrete Fourier Transformation with "Fourier". The result does not seem injective. Is that right?
F = Table[ Piecewise[{{1, Abs[i] < 5}}], {i, -10, 10, .001}]; ListPlot[F]
A=Fourier[F]; ListPlot[Abs[A]]
Thank you David, but still I am wondering isn't the result odd? Why would this curve alternate between two values? Shouldn't the fourier transform of a step function be a Sync function?
Note that since you've chose a very large number of points it appears that function is multivlaued. However this is just an illusion because the point values are so closely spaced and they are alternating between the two "curves". You can see this by choosing a much smaller number of points as in
F = Table[Piecewise[{{1, Abs[i] < 5}}], {i, -10, 10, 1}]; A = Fourier[F]; ListPlot[Abs[A]]
