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 $m$ and matrix-valued filter functions $H$. Given a natural number valued function $m$ and a system of functions encoded in a matrix $H$ satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function $m$ and filter system $H$. An equivalence relation on GMRAs is defined and described in terms of their associated pairs $(m,H)$. This classification system is applied to classical examples in $L^2 (\mathbb R^d)$ as well as to previously studied abstract examples.Comment: 18 pages including bibliograp

Cohomology of Polynomials Under an Irrational Rotation

A new description of cohomology of functions under an irrational rotation is given in terms of symmetry properties of the functions on subintervals of [0, 1]. This description yields a method for passing information about the cohomology classes for a given irrational to the cohomology classes for an equivalent irrational

On Functions That Are Trivial Cocycles for a Set of Irrationals. II

Two results are obtained about the topological size of the set of irrationals for which a given function is a trivial cocycle. An example of a continuous function which is a coboundary with non-L(1) cobounding function is constructed

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 $m$ which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space \H is $L^2(\mathbb R^n)$, the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function $m$ satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function $m$.Comment: 16 pages including bibliograph

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 m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L2(Rn), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m

Safety of Induced Sputum Collection in Children Hospitalized With Severe or Very Severe Pneumonia.

BACKGROUND.: Induced sputum (IS) may provide diagnostic information about the etiology of pneumonia. The safety of this procedure across a heterogeneous population with severe pneumonia in low- and middle-income countries has not been described. METHODS.: IS specimens were obtained as part a 7-country study of the etiology of severe and very severe pneumonia in hospitalized children <5 years of age. Rigorous clinical monitoring was done before, during, and after the procedure to record oxygen requirement, oxygen saturation, respiratory rate, consciousness level, and other evidence of clinical deterioration. Criteria for IS contraindications were predefined and serious adverse events (SAEs) were reported to ethics committees and a central safety monitor. RESULTS.: A total of 4653 IS procedures were done among 3802 children. Thirteen SAEs were reported in relation to collection of IS, or 0.34% of children with at least 1 IS specimen collected (95% confidence interval, 0.15%-0.53%). A drop in oxygen saturation that required supplemental oxygen was the most common SAE. One child died after feeding was reinitiated 2 hours after undergoing sputum induction; this death was categorized as "possibly related" to the procedure. CONCLUSIONS.: The overall frequency of SAEs was very low, and the nature of most SAEs was manageable, demonstrating a low-risk safety profile for IS collection even among severely ill children in low-income-country settings. Healthcare providers should monitor oxygen saturation and requirements during and after IS collection, and assess patients prior to reinitiating feeding after the IS procedure, to ensure patient safety