Bratteli O., Jorgensen P. Wavelets Through a Looking Glass. The World of the Spectrum

Springer, 2002. — 413 p.Wavelet theory stands on the interface between signal processing and harmonic analysis, the mathematical tools involved in digitizing continuous data with a view to storage, and the synthesis process, recreating, for example, a picture or time signal from stored data. The algorithms involved go under the name of filter banks, and their spectacular efficiency derives in part from the use of hidden self-similarity, relative to some scaling operation, in the data being analyzed. Observations or time signals are functions, and classes of functions make up linear spaces. Numerical correlations add structure to the spaces at hand, Hilbert spaces. There are operators in the spaces deriving from the discrete data and others from the spaces of continuous signals. The first type are good for computations, while the second reflect the real world. The operators between the two are the focus of the present monograph. Relations between operations in the discrete and continuous domains are studied as symbols. The mathematics involved in assigning operators to the symbolic relations is developed as a representation theory. The presentation is self-contained, and may serve as an introduction for readers who encounter these ideas for the first time and who would like to learn them from scratch.
A main point is the study of intertwining operators between, on the one side, the discrete world of high-pass/low-pass filters of signal processing, and on the other side, the continuous world of wavelets. There are significant issues in operator algebra and representation theory on both sides of the divide, and the intertwining operators shed light on central issues for wavelets in higher dimensions. Tools from diverse areas of analysis, as well as from dynamical systems and operator algebra, merge into the wavelet analysis. The diversity of techniques also adds to the charm of the subject, which continues to generate new mathematics.
The purpose of this book is twofold: first, to give a general presentation of sorne recent developments in wavelet theory, with an emphasis on techniques that are both fundamental and relatively timeless, and that have a geometric and spectral-theoretic flavor. It is our hope that it can be used equally well as a text for graduate students, as a reference book for specialists and researchers in neighboring fields, and in applications. Secondly, we are presenting some new results for the first time that have not previously appeared in papers, for example on the homotopy of multiresolutions, on approximation theory, and on the spectrum of associated transfer and subdivision operators.Introduction.
Overture: Why wavelets?
Subband filters and sieves.
Matrix functions and multiresolutions
Qubits: The Oracle of Feynman and the algorithm of Shor.
Chaos and cascade approximation.
Spectral bounds for the transfer and subdivision operators.
Connections to group theory.
Wavelet packets.
The Gabor transform
Exercises.
Terminology.
Homotopy theory and cascades.
The dangers of navigating the Iandscape of wavelets.
Homotopy classes of wavelets.
Multiresolution analysis and tight frames.
Generality of multiresolution analysis.
Global homotopy and an index theorem.
Cascades in L² (R).
An open problern.
Exercises.
Can you hear the shape of a wavelet?
The world of the spectrum.
Transfer operators.
Galerkin projections and spectra of transfer operators.
Spectrum and regularity.
The spectra of restrictions of the transfer operator.
The spectral radius.
Exercises.
The transfer operator and Perron - Frobenius theory.
A slanted matrix from dynamics.
The duality between kneading and chopping.
A Green's function.
The Perron - Frobenius cigenfunction.
Approximation theory.
Ergodie means.
Exercises.
The fixed points of the transfer operator.
The fine structure of correlations.
The minimal function g suchthat R(g) = g. 0≤g≤1, and g(1) = 1.
The C*-algebra ker (1 - Rl_C(T)): Estimates and identities.
Inverse spectral theory.
Cycles.
Pure states.
Historical notes and remarks.
Exercises.
Orthogonalization and isospectral approximation.
The other side of wavelets.
Examples.
Spectral properties of biorthogonal wavelets.
Isospectral approximation.
Biorthogonal frames: Multiresolution analysis and index theorem.
Exercises.
References.
Index.