  A shift-invariant space is a space of functions that is invariant under
integer translations. Such spaces are often used as models for spaces of
signals and images in mathematical and engineering applications. This paper
characterizes those shift-invariant subspaces S that are also invariant under
additional (non-integer) translations. For the case of finitely generated
spaces, these spaces are characterized in terms of the generators of the space.
As a consequence, it is shown that principal shift-invariant spaces with a
compactly supported generator cannot be invariant under any non-integer
translations.
