  In this paper, we derive the early-time asymptotics for fixed-frequency
solutions $\phi_\ell$ to the wave equation $\Box_g \phi_\ell=0$ on a fixed
Schwarzschild background ($M>0$) arising from the no incoming radiation
condition on $\mathcal I^-$ and polynomially decaying data, $r\phi_\ell\sim
t^{-1}$ as $t\to-\infty$, on either a timelike boundary of constant area radius
(I) or an ingoing null hypersurface (II). In case (I), we show that the
asymptotic expansion of $\partial_v(r\phi_\ell)$ along outgoing null
hypersurfaces near spacelike infinity $i^0$ contains logarithmic terms at order
$r^{-3-\ell}\log r$. In contrast, in case (II), we obtain that the asymptotic
expansion of $\partial_v(r\phi_\ell)$ near spacelike infinity $i^0$ contains
logarithmic terms already at order $r^{-3}\log r$ (unless $\ell=1$).
  These results suggest an alternative approach to the study of late-time
asymptotics near future timelike infinity $i^+$ that does not assume
conformally smooth or compactly supported Cauchy data: In case (I), our results
indicate logarithmic modifications to Price's law for each $\ell$-mode. On the
other hand, the data of case (II) lead to much stronger deviations from Price's
law. In particular, we conjecture that compactly supported scattering data on
$\mathcal H^-$ and $\mathcal I^-$ lead to solutions that exhibit the same
late-time asymptotics on $\mathcal I^+$ for each $\ell$: $r\phi_\ell|_{\mathcal
I^+}\sim u^{-2}$ as $u\to\infty$.
