Let us consider $$p_t^{(n+2)}(\tilde x) , $$ where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\Delta)^s$ in $n$ dimensions.

It is well known that we can write $p_t^{(n+2)}(\tilde x)= t^{-(n+2)/2s}k(|\tilde x|/t^{1/(2s)}) $. Is it true that the subalgebra generated by $$\Big(f_t(r)\Big)_{t>0} \text{ is dense in $C_0(0,\infty)$},$$ where $f_t(r) =r^{n}k(r/t^{1/(2s)})$?

I wanted to use the locally compact version of Stone-Weierstrass. But, since the expression of $f_t$ is not explicit, I don't know how to prove that the subalgebra separates points and vanishes nowhere. Also, I'm not sure if the fact that $\tilde x=(x_1,…,x_n,0,0)$ (i.e. the last two coordinates are fixed) is a problem or not.