convolution product of characteristic functions

Question

Consider the characteristic function $f(x)=1_{[0,r]}(x)$. How to compute $(f*f)(x)$ where $r \in [0,1[$ and by definition $(f*g)(x)=\int_{\mathbb R} f(y)g(x-y) \ dy$ ? thanks.

Answer 1

Note that for any two sets $A,B\subseteq\mathbb R$ we have that $${\bf 1}_A(x)\cdot{\bf 1}_B(x)={\bf 1}_{A\cap B}(x),\quad,\forall x\in \mathbb R.$$ Additionally, note that if $x\in\mathbb R$ is fixed, then $g_x(y)={\bf 1}_{[0,r]}(x-y)= {\bf 1}_{[x-r,x]}(y).$

Combining the two, it should come out that $$(f*f)(x)=\int_{\mathbb R}{\bf 1}_{[0,r]\cap[x-r,x]}(y)dy,$$ which can be computed quite easily, considering possible cases of $x$.

Answer 2

If you can do $f\star g$ then you can certainly do $f\star f$ as a special case. Note that for any set $E$, $$\int_{\mathbb{R}} 1_E(y)h(y)~dy = \int_E h(y)~dy.$$ This should be enough to get you through the problem.