Given two intervals on XY-plane: $I_1$ lying on OX-axis, $I_2$ lying on OY-axis. Compute convolution of two characteristic functions of these intervals.
$$\chi_{I_1} \ast \chi_{I_2}.$$
MY ATTEMPT:
$(\chi_{I_1},w)=\int_{I_1}w(0,x)dx,$
$(\chi_{I_2},w)=\int_{I_2}w(0,x)dx,$
$(\chi_{I_1}\ast \chi_{I_2},w)=(\chi_{I_1}, (\chi_{I_2},w))=\int_{I_1}\int_{I_2}w dy_1 dx_2 =(\chi_{I_2\times I_1},w).$