# Fourier transform on a distribution

1. Mar 3, 2010

### jerry109

1. The problem statement, all variables and given/known data

Determine the Fourier transform on the tempered distribution:

$$\langle f, \varphi \rangle$$

Where $$f$$ 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 $$\varphi$$.

Perhaps:

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

Any help would be fantastically appreciated,
Jerry109

Last edited: Mar 3, 2010