Let [tex]f(t)[/tex] be a function in [tex]L^2[/tex]. I am interested under which conditions converges the integral(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Convergence of one integral - books vs. me

Can you offer guidance or do you also need help?

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