- #1

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[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.