TITLE: Condition for an integral to be zero
QUESTION [5 upvotes]: For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ ( not necessarily non-negative ), is it true that
$$
\int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \operatorname{F}\left(x\right)\,{\rm d}x = 0\quad
\forall s > 0  \iff \operatorname{F} \equiv 0
$$
where $k \in \mathbb{N}$ is a positive constant $?$ Of course, one implication ($\leftarrow$) is true. What about the other one $?$.

REPLY [2 votes]: Write
$$f_s(x) = \frac{x^ks}{(s^2+x^2)^{(k+3)/2}}$$
Notice that any such function vanishes at infinity.  Let $A$ be the subalgebra of $\textbf{C}_0((0, \infty), \mathbb{R})$ generated by the family $f_s$.  All these functions are positive on $(0, \infty)$ so the family doesn't vanish at any point, and it clearly separates points.
Thus by the locally compact version of the Stone-Weierstrass Theorem, $A$ is dense in $\textbf{C}_0((0, \infty), \mathbb{R})$, and $F$ satisfies your integral equation with $f_s$ replaced by any element of $A$.  Thus as a functional, using the Riesz Representation Theorem (using the uniqueness part) you get that $F \equiv 0$ a.e.