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Unitary systems, wavelet sets, and operator-theoretic interpolation of wavelets and frames. (English) Zbl 1134.42340

Goh, Say Song (ed.) et al., Gabor and wavelet frames. Hackensack, NJ: World Scientific (ISBN 978-981-270-907-3/hbk). Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore 10, 167-214 (2007).
Summary: A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or system, of unitary operators. We will describe the operator-interpolation approach to wavelet theory using the local commutant of a system. This is really an abstract application of the theory of operator algebras to wavelet theory. The concrete applications of this method include results obtained using specially constructed families of wavelet sets. A frame is a sequence of vectors in a Hilbert space which is a compression of a basis for a larger space. This is not the usual definition in the frame literature, but it is easily equivalent to the usual definition. Because of this compression relationship between frames and bases, the unitary system approach to wavelets (and more generally: wandering vectors) is perfectly adaptable to frame theory. The use of the local commutant is along the same lines as in the wavelet theory. Finally, we discuss constructions of frames with special properties using targeted decompositions of positive operators, and related problems.
For the entire collection see [Zbl 1125.42002].

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46C99 Inner product spaces and their generalizations, Hilbert spaces
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