## Question Fourier Transform Smoothness/Compactness

Hello,
my question arises from reading the section on Smoothness/Compactness from Bracewell's "The Fourier Transform and Its Applications" page 162.

I don't quite understand the following reasoning:

$$F(\omega) = \ldots = \frac{1}{i\omega}\int_{-\infty}^{+\infty}f'(x)e^{-i\omega x}dx$$

and at this point the author says that when $\omega\to\infty$ then $\omega F(\omega) \to 0$.

But why $\omega F(\omega)$ is supposed to tend to zero, and not just $F(\omega) \to 0$ ?

Thanks.
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 Recognitions: Science Advisor ωF(ω) -> 0 is a stronger statement than F(ω) -> 0 for ω -> ∞.
 Ok, but I thought that if you multiply F by $\omega$ that quantity does not tend to zero anymore. In a similar way 1/x -> 0 but x(1/x) clearly tends to 1. *EDIT: On the other hand, I was thinking that if F(w) itself didnt tend to zero, its inverse FT, that is f(x), would not exist.

Recognitions:
 Quote by mnb96 Hello, my question arises from reading the section on Smoothness/Compactness from Bracewell's "The Fourier Transform and Its Applications" page 162. I don't quite understand the following reasoning: $$F(\omega) = \ldots = \frac{1}{i\omega}\int_{-\infty}^{+\infty}f'(x)e^{-i\omega x}dx$$ and at this point the author says that when $\omega\to\infty$ then $\omega F(\omega) \to 0$. But why $\omega F(\omega)$ is supposed to tend to zero, and not just $F(\omega) \to 0$ ? Thanks.
The author is simply saying that $|\int_{-\infty}^{+\infty}f'(x)e^{-i\omega x}dx|$ tends to 0. This makes sense unless f'(x) is completely pathological.