Convolution of two functions is a constant

Question

I have the convolution of two functions $f(t)$ and $g(t)$ is a constant $K \in \mathbb{R}$, i.e., $(f*g)(t) = K$ for all $t \in \mathbb{R}$. Taking Fourier transform of the convolution yields $$ F(s)G(s) = K \delta(s) $$ with $F(s)$ and $G(s)$ being the Fourier transform of $f$ and $g$ respectively and $\delta(\cdot)$ being the Dirac Delta function. Can I conclude that $f$ or $g$ must be a constant ?

Answer 1

No, you can't. For example, consider the situation

$$ F(s) = \delta(s) + \delta(s-1) + \delta(s+1)\\ G(s) = e^{-s^2}\cos\left(\frac{\pi}{2} s\right) $$

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Counterexample that I'm not sure of:

$$ F(s) = \delta(s) + \delta(s-1) + \delta(s+1)\\ G(s) = \delta(s) + \delta(s-2) + \delta(s+2) $$