An approximation problem in multiplicatively invariant spaces

Abstract

Let H be Hilbert space and (Ω, m) a σ-finite measure space. Multiplicatively invariant(MI) spaces are closed subspaces of L2(Ω, H) that are invariant under point-wise multiplication byfunctions from a fixed subset of L∞(Ω). Given a finite set of data F ⊆ L2(Ω, H), in this paper weprove the existence and construct an MI space M that best fits F, in the least squares sense. MIspaces are related to shift-invariant (SI) spaces via a fiberization map, which allows us to solve anapproximation problem for SI spaces in the context of locally compact abelian groups. On the otherhand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into anorthogonal 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 alsosolve our approximation problem for this class of SI spaces. Finally we prove that translation-invariantspaces are in correspondence with totally decomposable MI spaces.