For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (*not necessarily non-negative*), if
$$
\int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \operatorname{F}\left(x\right)\,{\rm d}x = 0\quad
\forall s > 0
$$
where $k \in \mathbb{N}$ is a positive constant,
is it true that
$$
\int_{0}^{\infty}\left(\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\right)^{h} \, \operatorname{F}\left(x\right)\,{\rm d}x = 0\quad
\forall s > 0
$$
where $h \in \mathbb{N}$?

This question is inspired by a comment to the answer in https://math.stackexchange.com/questions/3996738/condition-for-an-integral-to-be-zero (which required to check the assumptions of the Stone-Weierstrass theorem).

More generally, ignoring the question above, my main concern is this:

- Use Stone-Weierstrass to prove that $$ \int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \operatorname{F}\left(x\right)\,{\rm d}x = 0\quad \forall s > 0 $$ if and only if $F \equiv 0$.