Fourier Transform: Decompose Vector Function

[LocationX]

For some reason I can't post everything at once... gives me a "Database error" so I will post in parts...

A vector function can be decomposed to form a curl free and divergence
free parts:

$$\vec{f}(\vec{r})=\vec{f_{\parallel}}(\vec{r'})+\vec{f_{\perp}}(\vec{r'})$$

where

$$\vec{f_{\parallel}}(\vec{r'}) = - \vec{\nabla} \left( \frac{1}{4 \pi} \int d^3 r' \frac{\vec{\nabla'} \cdot \vec{f}(\vec{r'})}{|\vec{r}-\vec{r'}|} \right)$$

and

$$\vec{f_{\perp}}(\vec{r'}) = \vec{\nabla} \times \left( \frac{1}{4 \pi} \int d^3 r' \frac{\vec{\nabla'} \times \vec{f}(\vev{r'})}{|\vec{r}-\vec{r'}|} \right)$$

 

[LocationX]

I am trying to take the Fourier transform of $\vec{f_{\parallel}}(\vec{r'})$ and $\vec{f_{\perp}}(\vec{r})$I am starting at $\vec{f_{\parallel}}(\vec{r'})$. We know that the Fourier transform is given by:

$$\vec{f}(\vec{k}) = \int_{-\infty}^{\infty} d^3r e^{- i \vec{k} \cdot \vec{r}} \vec{f}(\vec{r})$$

$$\vec{f}(\vec{r}) = \frac{1}{(2 \pi)^3} \int_{-\infty}^{\infty} d^3k e^{- i \vec{k} \cdot \vec{r}} \vec{f}(\vec{k})$$

I'm not exactly sure where to begin. If I just plug and chug , we'd have:

$$\vec{f}(\vec{k}) = \int_{-\infty}^{\infty} d^3r e^{- i \vec{k} \cdot \vec{r}} \vec{f}(\vec{r})$$

$$\vec{f}(\vec{k}) = \int_{-\infty}^{\infty} e^{- i \vec{k} \cdot \vec{r}} - \vec{\nabla} \left( \frac{1}{4 \pi} \int \frac{\vec{\nabla'} \cdot \vec{f}(\vec{r'})}{|\vec{r}-\vec{r'}|} d^3 r' \right) d^3r$$I just do not see a simple way of tacking this problem. Any thoughts would be appreciated.