1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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:

    \langle f, \varphi \rangle

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

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

    3. The attempt at a solution

    Fourier transform on tempered distribution is:

    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

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


    Any help would be fantastically appreciated,
    Last edited: Mar 3, 2010
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted