An Approximation Problem in Multiplicatively Invariant Spaces

TL;DR

This paper addresses the problem of best approximation in multiplicatively invariant spaces within Hilbert spaces, introducing solutions for decomposable MI spaces and their relation to shift invariant spaces, with applications to locally compact abelian groups.

Contribution

It proves existence and construction of best-fit MI spaces for data, introduces decomposable MI spaces, and connects these to shift invariant and translation invariant spaces.

Findings

Existence of best approximation MI spaces for given data

Introduction of decomposable MI spaces and their approximation solutions

Establishment of correspondence between translation invariant spaces and totally decomposable MI spaces

Abstract

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.