Find the Fourier transform of the function

$$f(x)= \begin{cases}A, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases}$$

Determine the convolution of the function $f(x)$ with itself.

State the convolution theorem for Fourier transforms. Using it, or otherwise, determine the Fourier transform of the function

$$g(x)= \begin{cases}B(2-|x|), & |x| \leqslant 2 \\ 0, & |x|>2\end{cases}$$