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Math Help - fourier transform

  1. #1
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    fourier transform

    I need to find the Fourier transform for the following function:

    f: \mathbb{R}^n \rightarrow \mathbb{R}
    f(x)=e^{-\pi a |x|^2}

    I'm a newbie when it comes to Fourier transforms so any help would be great.

    Thank you.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by marianne View Post
    I need to find the Fourier transform for the following function:

    f: \mathbb{R}^n \rightarrow \mathbb{R}
    f(x)=e^{-\pi a |x|^2}.
    First note that, by Fubini theorem, the Fourier transform equals: for every \xi\in\mathbb{R}^n,
    F(\xi)=\int e^{-2i\pi x\cdot \xi}e^{-\pi a |x|^2}dx = \prod_{k=1}^n \int e^{-2i\pi x_k \xi_k} e^{-\pi a x_k^2} dx_k
    (you may have a slightly different definition of the Fourier transform; there are several conventions)

    This shows that it suffices to find the Fourier transform F of the function x\mapsto e^{-\pi a x^2} from \mathbb{R} to \mathbb{R}.
    There are various proofs for this, and I think you should have been given hints to find it.
    A possibility is to show that the Fourier transform satisfies a first order differential equation. Here are the steps:
    Differentiate F (with respect to \xi) and integrate by parts (by integrating x e^{-\pi a x^2} and dividing e^{-2 i\pi \xi x}). You get F'(\xi)=-\frac{2\pi\xi}{a} F(\xi).
    Hence there is C such that F(\xi)=C e^{-\frac{\pi}{a}\xi^2}.
    To find the value of C, remark that C=F(0)=\int e^{-\pi a x^2}dx=\frac{1}{\sqrt{\pi a}}\int e^{-u^2}du. And \int e^{-x^2}dx = \sqrt{\pi} (this can be proved by polar change of variable if you don't know that). As a consequence, C=\frac{1}{\sqrt{a}}. (There may be mistakes here)
    Notice that if a=1, the function is equal to its own Fourier transform.
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