An Application of the supremum cosine angle between multiplication invariant spaces in $L^2(X; \mc H)$

TL;DR

This paper investigates the supremum cosine angle between multiplication invariant spaces in $L^2(X; \\mathcal{H})$, explores its relation to space sum closedness, and applies findings to sampling theory using the Zak transform.

Contribution

It introduces a new measure for MI spaces, links it to their sum's closedness, and extends results to translation invariant spaces on locally compact groups with applications in sampling theory.

Findings

Supremum cosine angle characterizes MI space relationships.

Closedness of the sum of MI spaces is characterized via the cosine angle.

Results are extended to translation invariant spaces on groups with sampling applications.

Abstract

In this article, we describe the supremum cosine angle between two multiplication invariant (MI) spaces and its connection with the closedness of the sum of those spaces. The results obtained for MI spaces are preserved by the corresponding fiber spaces almost everywhere. Employing the Zak transform, we obtain the results for translation invariant spaces on locally compact groups by action of its closed abelian subgroup. Additionally, we provide the application of our results to sampling theory.