I'm looking for a concrete example to understand the computation procedure for convolution:
Let $f, g \in C_{0}^{\infty}(\mathbb R)$ be defined as follows: $$f(x) := e^{-\frac{1}{1-x^2}}1_{(-1,1)} $$ $$g(x) := e^{-\frac{1}{1-(x-1)^2}}1_{(0,2)} $$ where $$1_{(a,b)} := \begin{cases} 1 & \text{if $x \in (a,b)$} \\ 0 & \text{otherwise} \end{cases}$$
By the above definitions, $f$ is supported on $(-1,1)$, and $g$ on $(0,2)$.
Then how would one go about computing $$(f*g)(x) = \int_{\mathbb R}f(y)g(x-y)dy$$ Also, how would you determine the support of $h := f*g$?
Thanks.