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

Xingde Dai (Author), David R. Larson (Author)

Book Description

ISBN-10: 0821808001 | ISBN-13: 978-0821808009 | Publication Date: July 1998

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.

Product Details

• Paperback: 68 pages
• Publisher: Amer Mathematical Society (July 1998)
• Language: English
• ISBN-10: 0821808001
• ISBN-13: 978-0821808009
• Product Dimensions: 9.6 x 6.8 x 0.3 inches
• Shipping Weight: 4.8 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #4,948,404 in Books (See Top 100 in Books)

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2 of 2 people found the following review helpful:

5.0 out of 5 stars A powerful idea!, July 7, 2002

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
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This review is from: Wandering Vectors for Unitary Systems and Orthogonal Wavelets (Memoirs of the American Mathematical Society) (Paperback)

The term "wandering vector" is from the theory of operators in Hilbert space;-- it means that the vector in question is transformed into a family of orthogonal vectors under a prescribed set of unitary operators. Since wavelets represent
orthogonal functions in the Hilbert space of square integrable functions on R, or on R^n for some n, it would only seem natural to merge the two concepts. But it takes originality to carry through the program: The book represents a success story in the application of operator theory to a current problem (in this case, wavelets) in classical analysis. What came out, for the theory of wavelets, was at first quite unexpected: You could have wavelets, it turned out, that are generated by fewer functions on R, or on R^n, than
had been predicted by the more conventional approach to wavelet theory. One generator is enough! The new wavelets turned out to be localized in frequency domain, and they gave rise to a variety of new trends in wavelet theory, the wavelet sets being just one of them, multiplicity, and the dimension function are others.