(i) State and prove the convolution theorem for Laplace transforms of two realvalued functions.

(ii) Let the function $f(t), t \geqslant 0$, be equal to 1 for $0 \leqslant t \leqslant a$ and zero otherwise, where $a$ is a positive parameter. Calculate the Laplace transform of $f$. Hence deduce the Laplace transform of the convolution $g=f * f$. Invert this Laplace transform to obtain an explicit expression for $g(t)$.

[Hint: You may use the notation $\left.(t-a)_{+}=H(t-a) \cdot(t-a) .\right]$