This volume concerns some general methods for the analysis of those orthonormal bases for a separable complex infinite dimensional Hilbert space which are generated by the action of a system of unitary transformations on a single vector, which is called a complete wandering vector for the system. The main examples are the orthonormal wavelet bases. Topological and structural properties of the set of all orthonormal dyadic wavelets are investigated in this way by viewing them as complete wandering vectors for an affiliated unitary system and then applying techniques of operator algebra and operator theory. Features: describes an operator-theoretic perspective on wavelet theory that is accessible to functional analysts describes some natural generalizations of standard wavelet systems contains numerous examples of computationally elementary wavelets poses many open questions and directions for further research This book is particularly accessible to operator theorists and operator algebraists who are interested in a functional analytic approach to some of the pure mathematics underlying wavelet theory.

Wandering Vectors for Unitary Systems and Orthogonal Wavelets (Memoirs of the American Mathematical Society)

Science & Math

Dai, Xingde

Larson, David R.

Mathematics

Amer Mathematical Society

Over the past decade, wavelets and frames have emerged as increasingly powerful tools of analysis on $n$-dimension Euclidean space. Both wavelets and frames were studied initially by using classical Fourier analysis. However, in recent years more abstract tools have been introduced, for example, from operator theory, abstract harmonic analysis, von Neumann algebras, etc. The editors of this volume organized a Special Session on the functional and harmonic analysis of wavelets at the San Antonio (TX) Joint Mathematics Meetings. The goal of the session was to focus research attention on these newly-introduced tools and to share the organizers' view that this modern application holds the promise of providing some deeper understanding and fascinating new structures in pure functional analysis. This volume presents the fruitful results of the lively discussions that took place at the conference.

The Functional and Harmonic Analysis of Wavelets and Frames: Ams Special Session on the Functional and Harmonic Analysis of Wavelets, January 13-14, 1999, San Antonio, Texas (Contemporary Mathematics)

Baggett, Lawrence W.

AMS Special Session on the Functional and Harmonic Analysis of Wavelets (1999 : San Antonio, Texas)

<p>The authors develop elements of a general dilation theory for operator-valued measures. Hilbert space operator-valued measures are closely related to bounded linear maps on abelian von Neumann algebras, and some of their results include new dilation results for bounded linear maps that are not necessarily completely bounded, and from domain algebras that are not necessarily abelian. In the non-cb case the dilation space often needs to be a Banach space. They give applications to both the discrete and the continuous frame theory. There are natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of continuous linear maps between C ∗ -algebras. In this connection frame theory itself is identified with the special case in which the domain algebra for the maps is an abelian von Neumann algebra and the map is normal (i.e. ultraweakly, or σ weakly, or w*) continuous.</p>

Operator-Valued Measures, Dilations, and the Theory of Frames (Memoirs of the American Mathematical Society)

Han, Deguang

Liu, Bei

Liu, Rui

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