Wandering Vectors for Unitary Systems and Orthogonal Wavelets

Describes an operator-theoretic perspective on wavelet theory that is accessible to functional analysts. 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.

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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.It describes an operator-theoretic perspective on wavelet theory that is accessible to functional analysts. It describes some natural generalizations of standard wavelet systems. It contains numerous examples of computationally elementary wavelets. It 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.