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Generalized filters, the low-pass condition, and connections to multiresolution analyses
Authors:Lawrence W Baggett  Veronika Furst  Kathy D Merrill  Judith A Packer  
Institution:aDepartment of Mathematics, University of Colorado, Boulder, CO 80309, USA;bDepartment of Mathematics, Fort Lewis College, Durango, CO 81301, USA;cDepartment of Mathematics, Colorado College, Colorado Springs, CO 80903, USA
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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