  In this paper we deal with a scale of reproducing kernel Hilbert spaces
$H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian
Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as
ranges of the Laplace transform in extended versions of the Paley-Wiener
theorem which involve absolutely continuous functions of higher degree. An
explicit integral formula is given for the reproducing kernel $K_{z,n}$ of
$H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert
z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_\varphi
:H_2^{(n)} \to H_2^{(n)}$, $C_\varphi f=f\circ \varphi $, on these spaces are
discussed, giving some necessary and some sufficient conditions for analytic
maps $\varphi: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition
operators.
