Titchmarsh convolution theorem

(E. Titchmarsh, 1926). Let $f, g \in L_{1}(\mathbb {R}), f \equiv g \equiv 0$ for $t < 0$ and $f * g = 0$. Prove that then one of the functions $f$ or $g$ is zero almost everywhere.

Let the functions $f, g \in L_{1}(\mathbb {R})$ have support on the segment $[0,1]$, be continuous on this segment, and $f(1) \neq 0$. Prove that if $\operatorname {supp}(f * g) \subseteq [0,1]$, then $g \equiv 0$.