  In our recent work, the sampling and reconstruction of non-decaying signals,
modeled as members of weighted-$L_p$ spaces, were shown to be stable with an
appropriate choice of the generating kernel for the shift-invariant
reconstruction space. In this paper, we extend the Strang-Fix theory to show
that, for $d$-dimensional signals whose derivatives up to order $L$ are all in
some weighted-$L_p$ space, the weighted norm of the approximation error can be
made to go down as $O(h^L)$ when the sampling step $h$ tends to $0$. The
sufficient condition for this decay rate is that the generating kernel belongs
to a particular hybrid-norm space and satisfies the Strang-Fix conditions of
order $L$. We show that the $O(h^L)$ behavior of the error is attainable for
both approximation schemes using projection (when the signal is prefiltered
with the dual kernel) and interpolation (when a prefilter is unavailable). The
requirement on the signal for the interpolation method, however, is slightly
more stringent than that of the projection because we need to increase the
smoothness of the signal by a margin of $d/p+\varepsilon$, for arbitrary
$\varepsilon >0$. This extra amount of derivatives is used to make sure that
the direct sampling is stable.
