the fourier transform of a "double convolution" Suppose I have a function
$$
m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x)
$$
I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the life of me I can't figure out a way to do it. If anyone has some insight, please help?

In general, we have the product-to-convolution formulas: 
\begin{align*}
\mathcal{F}[f\cdot g]=\hat{f}*\hat{g}\\
\mathcal{F}[f* g]=\hat{f}\cdot\hat{g}
\end{align*}  In your case, we have 
$$
\mathcal{F}[f\cdot(h*g)]=\hat{f}*\mathcal{F}[h *g]=\hat{f}*(\hat{h}\cdot\hat{g})
$$ So essentially it swaps the convolution and the product.
