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Convergence of one integral - books vs. me

  1. Jun 17, 2008 #1
    Let [tex]f(t)[/tex] 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 [tex]F(\omega)[/tex] denotes the Fourier transform of [tex]f(t)[/tex].
    My book, well, several books actually, say the sufficient conditions are
    - [tex]F(0) = 0[/tex] (naturally)
    and
    - [tex]F[/tex] is continuously differentiable ([tex]C^1[/tex])

    I don't understand why the differentiability is neccessary. My conjecture - if [tex]F[/tex] is continuous (not neccessarily [tex]C^1[/tex]), then the integral converges around zero because of the first condition and thus everywhere since [tex]F[/tex] is in [tex]L^2[/tex] because [tex]f[/tex] was in [tex]L^2[/tex] and Fourier transform maps [tex]L^2[/tex] onto [tex]L^2[/tex], so there is no problem around infinity.

    Where am I wrong? Thanks for any ideas, H.
     
  2. jcsd
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