# Convergence of one integral - books vs. me

1. Jun 17, 2008

### hooker27

Let $$f(t)$$ be a function in $$L^2$$. I am interested under which conditions converges the integral
$$\int_0^\infty \frac{|F(\omega)|^2}{\omega} d\omega$$
where $$F(\omega)$$ denotes the Fourier transform of $$f(t)$$.
My book, well, several books actually, say the sufficient conditions are
- $$F(0) = 0$$ (naturally)
and
- $$F$$ is continuously differentiable ($$C^1$$)

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

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

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted