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seen May 24 '12 at 19:28

comment Fourier integral/ Fourier transformation of an oscillatory function with FFT
I am still wondering that, since these Fourier transforms are still "well defined" in some sense, shouldn't a dense sampling of N>>1 cycles of f(x) give a good approximation of g(k) for k<klim, with klim determined by the density of the sampling? Naively I expect klim ~ 1/delta_x, but in reality klim is much smaller than that... You are right that the actual klim should have to do with the converging property of the Fourier integral. Is there perhaps a way to quantify this?