76 research outputs found
Classification of Generalized Multiresolution Analyses
We discuss how generalized multiresolution analyses (GMRAs), both classical
and those defined on abstract Hilbert spaces, can be classified by their
multiplicity functions and matrix-valued filter functions . Given a
natural number valued function and a system of functions encoded in a
matrix satisfying certain conditions, a construction procedure is described
that produces an abstract GMRA with multiplicity function and filter
system . An equivalence relation on GMRAs is defined and described in terms
of their associated pairs . This classification system is applied to
classical examples in as well as to previously studied
abstract examples.Comment: 18 pages including bibliograp
Generalized multiresolution analyses with given multiplicity functions
Generalized multiresolution analyses are increasing sequences of subspaces of
a Hilbert space \H that fail to be multiresolution analyses in the sense of
wavelet theory because the core subspace does not have an orthonormal basis
generated by a fixed scaling function. Previous authors have studied a
multiplicity function which, loosely speaking, measures the failure of the
GMRA to be an MRA. When the Hilbert space \H is , the
possible multiplicity functions have been characterized by Baggett and Merrill.
Here we start with a function satisfying a consistency condition which is
known to be necessary, and build a GMRA in an abstract Hilbert space with
multiplicity function .Comment: 16 pages including bibliograph
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