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Convergence of an integral - book vs. me

  1. Jun 17, 2008 #1
    Let f(t) be a function in [tex]L^2[/tex]. I am interested under which conditions converges the integral

    [tex]\int_0^\infty \frac{|F(\omega)|^2}{\omega} d\omega[/tex]

    where F(omega) denotes the Fourier transform of f.

    My book, well, several books actually, say the sufficient conditions are
    1) [tex]F(0) = 0[/tex] (naturally)
    2) F is continuously differentiable ([tex]C^1[/tex])

    I don't understand why the differentiability is neccessary. My conjecture - if F is continuous (not neccessarily C1), then the integral converges around zero because of the first condition and thus everywhere since F is in L2 because f was in L2 and the Fourier transform maps L2 onto L2, so there is no problem around infinity.

    Where am I wrong? Thanks for any ideas, H.
  2. jcsd
  3. Jun 17, 2008 #2
    Sorry for the duplicate.
  4. Jun 22, 2008 #3
    Hooker27, I'd like to know too. Deacon John
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