Counter example let \(\displaystyle h(t)\) be a Gaussian kernel with unit spread parameter (normalise it as you wish). Let \(\displaystyle f(t) \) and \(\displaystyle g(t)\) both be \(\displaystyle 1\) on disjoint intervals and zero elsewhere.
Then \(\displaystyle f(t) \times g(t) \equiv 0\), and so: