- #1
hooker27
- 15
- 0
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.
My book, well, several books actually, say the sufficient conditions are
1) F(0) = 0 (naturally)
2) F is continuously differentiable (C^1)
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.
\int_0^\infty \frac{|F(\omega)|^2}{\omega} d\omega
where F(omega) denotes the Fourier transform of f.
My book, well, several books actually, say the sufficient conditions are
1) F(0) = 0 (naturally)
2) F is continuously differentiable (C^1)
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.
