Generalized low-pass filters and multiresolution analyses

TL;DR

This paper explores generalized low-pass filters linked to multiresolution analyses on abelian groups, characterizing when associated Ruelle operators are pure isometries and extending existing theorems on subspace intersections.

Contribution

It introduces a framework for generalized filters based on multiplicity functions and homomorphisms, and generalizes a theorem on subspace intersections in multiresolution analyses.

Findings

Characterization of when the Ruelle operator is a pure isometry.

Conditions under which non-trivial intersections imply trivial intersections.

Extension of Bownik and Rzeszotnik's theorem to broader settings.

Abstract

We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.