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Fourier transforms of smooth, compactly supported functions

  1. May 31, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex] f, \hat{f} \in C_c^\infty(\mathbb{R}^n)[/tex]


    2. Relevant equations


    [tex] \hat{f} = \int_{\mathbb{R}^n} f(x) e^{-2\pi i \xi \cdot x} \,dx [/tex]
    [tex] \check{f} = \int_{\mathbb{R}^n} \hat{f}(x) e^{2\pi i \xi \cdot x} \,d\xi [/tex]


    3. The attempt at a solution

    As [tex] C_c^\infty \subset \mathcal{S} [/tex] (the Schwarz space) we know that the Fourier transformation is invertible, and that
    [tex] f = (\hat{f})\check{} = (\check{f})\hat{} [/tex]
    in other words
    [tex] f =
    \int_{\mathbb{R}^n} \hat{f} e^{2 \pi i \xi \cdot x} \,d\xi
    = \int_{\mathbb{R}^n} \check{f} e^{-2 \pi i \xi \cdot x} \,dx [/tex]

    Somehow these must be zero. I am familiar with the idea that the smoother f is, the faster its transform must decay at infinity; and vice versa. Since [tex]C^\infty, C_c[/tex] are the ultimate of both this must somehow indicate that the situation is "too good to be true" in a nontrivial way. But I do not know how to implement this. Advice?
     
  2. jcsd
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