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Fourier transform on a distribution

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Determine the Fourier transform on the tempered distribution:

    [tex]
    \langle f, \varphi \rangle
    [/tex]

    Where [tex] f[/tex] can be given by they taylor series representation:

    [tex]
    f = i\sum_{n=0}^{\infty} \frac {x^{3n+2}}{(2n)!}
    [/tex]


    3. The attempt at a solution

    Fourier transform on tempered distribution is:

    [tex]
    F\langle f, \varphi \rangle = \langle F f, \varphi \rangle = \langle f, F \varphi \rangle

    = i\int \sum_{n=0}^{\infty} \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx

    = i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx
    [/tex]

    I'm stuck as how to resolve the infinite sum involving the Fourier transform of the test function [tex]\varphi[/tex].

    Perhaps:

    [tex]
    i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx = i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} \hat{\delta}(x-x_0) dx
    = i\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} e^{2{\pi}ikx_0}dx


    = ie^{2{\pi}ik{x_0}}\sum_{n=0}^{\infty}\int \frac {x^{3n+2}}{(2n)!} dx

    = ie^{2{\pi}ik{x_0}}\sum_{n=0}^{\infty}\frac {x^{3n+3}}{(2n)! * (3n+3)!}

    [/tex]


    Any help would be fantastically appreciated,
    Jerry109
     
    Last edited: Mar 3, 2010
  2. jcsd
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