Looking For Periodicity

1. Mar 16, 2010

jerry109

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

Define the periodicity of the function with the following Taylor Series expansion:

$$f=i\sum_{n=0}^{\infty} (-1)^{n} \frac {x^{3n+2}}{(2n)!}$$

2. Relevant equations

To find the periodicity, we take the Fourier transform of f, treated as a tempered distribution:

$$F\langle f, \varphi \rangle = \langle f, F\varphi \rangle = \langle f, \hat{\varphi}\rangle$$

3. The attempt at a solution

Following through with the integral:

$$= i\int_{-\infty}^{\infty} \sum_{n=0}^{\infty}(-1)^{n} \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx$$

$$=i\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}(-1)^{n} \frac {x^{3n+2}}{(2n)!} \hat{\varphi} dx$$

I'm not sure how to finish the problem beyond this point. I was thinking of maybe taking the inverse transform and changing it into a convolution, but that didn't appear to work. I'm also unsure of how to simplify this further, or if you can insert some test function of compact support to resolve the integral.

Any help as how to finish the problem would be greatly appreciated.

Jerry109