  We study conditions on a Banach frame that ensures the validity of a
reconstruction formula. In particular, we show that any Banach frames for (a
subspace of) $L_p$ or $L_{p,q}$ ($1\le p < \infty$) with respect to a solid
sequence space always satisfies an unconditional reconstruction formula. The
existence of reconstruction formulae allows us to prove some James-type results
for atomic decompositions: an unconditional atomic decomposition (or
unconditional Schauder frame) for $X$ is shrinking (respectively, boundedly
complete) if and only if $X$ does not contain an isomorphic copy of $\ell_1$
(respectively, $c_0$).
