  Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite
measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $
L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by
functions in a fix subset of $L^{\infty}(\Omega).$ Given a finite set of data
$\mathcal{F}\subseteq L^2(\Omega, \mathcal{H}),$ in this paper we prove the
existence and construct an MI space $M$ that best fits $\mathcal{F}$, in the
least squares sense. MI spaces are related to shift invariant (SI) spaces via a
fiberization map, which allows us to solve an approximation problem for SI
spaces in the context of locally compact abelian groups. On the other hand, we
introduce the notion of decomposable MI spaces (MI spaces that can be
decomposed into an orthogonal sum of MI subspaces) and solve the approximation
problem for the class of these spaces. Since SI spaces having extra invariance
are in one-to-one relation to decomposable MI spaces, we also solve our
approximation problem for this class of SI spaces. Finally we prove that
translation invariant spaces are in correspondence with totally decomposable MI
spaces.
