A fundamental inequality in the convolution of $L\sb{2}$ functions on the half line

For any positive integer $q$ and ${F_j} \in {L_2}(0,\infty )$, we note the inequality for the iterated convolution $\prod _{j = 1}^{2q} * {F_j}\;$ of ${F_j}$:

$$\int_0^\infty {{{\left\vert {\prod\limits_{j = 1}^{2q} { * {F_j}(... ...s_{j = 1}^{2q} {\int_0^\infty {{{\left\vert {{F_j}(t)} \right\vert}^2}dt.} } }$$