  In the setting of horizontal curves in the Heisenberg group, we prove a
$C^{m,\omega}$ finiteness principle, a $C^{m,\omega}$ Lusin approximation
result, a $C^{\infty}$ Whitney extension result, and a $C^{\infty}$ Lusin
approximation result. Combined with previous work, this completes the study of
Whitney extension and Lusin approximation for horizontal curves of class
$C^{m}$, $C^{m,\omega}$, and $C^{\infty}$ in the Heisenberg group.
