Fourier transform of product of twho functions that includes characteristic function

Question

I need to find the fourier transform of f(x)=(1-abs(x))(Chi_[-1,1] (x)). In words, I need the fourier transform of one minus the absolute value of x multiplied by the characteristic function on interval -1,1.

I tried using the convolution theorem and finding the fourier transform for both functions separately, but then I realized I didn't understand how to compute the convolution of the separate transforms. Can someone please explain how I would go about finding this transform?

I got an answer of -1/ξ^2 (2isin(ξ) but I am not sure if this correct. I am using the fourier definition of:

(1/2pi) integral of f(x)e^(-iξx) dx

Answer 1

You may compute the Fourier transform directly. Since $f \in L^1(\mathbb{R})$, for $\xi \neq 0$, we have $$ \hat{f}(\xi) = \int_{-1}^1\, (1-|x|)\cdot e^{-2\pi ix\xi}\, dx = \left( \frac{\sin \pi \xi}{\pi \xi}\right)^2. $$ For $\xi=0$, $e^{-2\pi ix\xi}=1$, so $\hat{f}(0)=\int_{-1}^1\, (1-|x|)\, dx= 1$.