  We study the approximation of SPDEs on the whole real line near a change of
stability via modulation or amplitude equations, which acts as a replacement
for the lack of random invariant manifolds on extended domains. Due to the
unboundedness of the underlying domain a whole band of infinitely many
eigenfunctions changes stability. Thus we expect not only a slow motion in
time, but also a slow spatial modulation of the dominant modes, which is
described by the modulation equation.
  As a first step towards a full theory of modulation equations for nonlinear
SPDEs on unbounded domains, we focus, in the results presented here, on the
linear theory for one particular example, the Swift-Hohenberg equation. These
linear results are one of the key technical tools to carry over the
deterministic approximation results to the stochastic case with additive
forcing. One technical problem for establishing error estimates rises from the
spatially translation invariant nature of space-time white noise on unbounded
domains, which implies that at any time we can expect the error to be always
very large somewhere in space.
