11.3.4 Convolution of functions

Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes a value, equal to zero, when \( x \notin\left[x_{1} ; x_{4}\right] \), and when \( x \in\left[x_{1} ; x_{4}\right] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A\left(x_{1} ; 0\right), \quad B\left(x_{2} ; a\right), \quad C\left(x_{3} ; b\right), \quad D\left(x_{4} ; b\right) . \\ x_{1}=-2, \quad x_{2}=1, \quad x_{3}=3, \quad x_{4}=4, \quad a=2, \quad b=-1 . \end{array} \] The function \( g(x) \) has the form \[ g(x)=\left\{\begin{array}{ll} 0, & x<0 \\ 1, & 0 \leq x<1 \\ 0, & x \geq 0 \end{array}\right. \]