I have the convolution of two non-periodic, continuous and differentiable functions $f(t)$ and $g(t)$ is a constant $K \in \mathbb{R}$, i.e., $(f∗g)(t)=K \quad \forall t \in \mathbb{R}$.
Can I conclude that $f$ or $g$ must be a constant? I need to mention that we know $f$ has Fourier Transform but we do not know $g$ has Fourier Transform or not!