Elsevier

Journal of Functional Analysis

Volume 257, Issue 9, 1 November 2009, Pages 2760-2779
Journal of Functional Analysis

Generalized filters, the low-pass condition, and connections to multiresolution analyses

https://doi.org/10.1016/j.jfa.2009.05.004Get rights and content
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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.

Keywords

Wavelet
Generalized multiresolution analysis
Filter
Ruelle operator
Pure isometry

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