Convergence of an integral - book vs. me

  • Thread starter hooker27
  • Start date
  • #1
16
0

Main Question or Discussion Point

Let f(t) 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 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.
 

Answers and Replies

  • #2
16
0
Sorry for the duplicate.
 
  • #3
122
0
Hooker27, I'd like to know too. Deacon John
 

Related Threads for: Convergence of an integral - book vs. me

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
5K
Replies
2
Views
969
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
3
Views
728
Replies
15
Views
931
  • Last Post
Replies
20
Views
5K
  • Last Post
Replies
8
Views
2K
Top