Generalized Multiresolution Analyses (Applied and Numerical Harmonic Analysis)

This monograph presents the first unified exposition of generalized multiresolution analyses. Expanding on the author’s pioneering work in the field, these lecture notes provide the tools and framework for using GMRAs to extend results from classical wavelet analysis to a more general setting. Beginning with the basic properties of GMRAs, the book goes on to explore the multiplicity and dimension functions of GMRA, wavelet sets, and generalized filters. The author’s constructions of wavelet sets feature prominently, with figures to illustrate their remarkably simple geometric form. The last three chapters exhibit extensions of wavelet theory and GMRAs to other settings. These include fractal spaces, wavelets with composite dilations, and abstract constructions of GMRAs beyond the usual setting of L 2 (R n ). This account of recent developments in wavelet theory will appeal to researchers and graduate students with an interest in multiscale analysis from a pure or applied perspective. Familiarity with harmonic analysis and operator theory will be helpful to the reader, though the only prerequisite is graduate level experience with real and functional analysis. This monograph presents the first unified exposition of generalized multiresolution analyses. Expanding on the author’s pioneering work in the field, these lecture notes provide the tools and framework for using GMRAs to extend results from classical wavelet analysis to a more general setting. Beginning with the basic properties of GMRAs, the book goes on to explore the multiplicity and dimension functions of GMRA, wavelet sets, and generalized filters. The author’s constructions of wavelet sets feature prominently, with figures to illustrate their remarkably simple geometric form. The last three chapters exhibit extensions of wavelet theory and GMRAs to other settings. These include fractal spaces, wavelets with composite dilations, and abstract constructions of GMRAs beyond the usual setting of __L__2(R__n__).This account of recent developments in wavelet theory will appeal to researchers and graduate students with an interest in multiscale analysis from a pure or applied perspective. Familiarity with harmonic analysis and operator theory will be helpful to the reader, though the only prerequisite is graduate level experience with real and functional analysis. Front Matter ....Pages i-x Introduction (Kathy D. Merrill)....Pages 1-8 The Invariance of the Core Subspace (Kathy D. Merrill)....Pages 9-21 The Multiplicity Function (Kathy D. Merrill)....Pages 23-36 Wavelet Sets (Kathy D. Merrill)....Pages 37-54 Generalized Filters (Kathy D. Merrill)....Pages 55-74 Fractal Spaces (Kathy D. Merrill)....Pages 75-83 Composite Dilations and Crystallographic Groups (Kathy D. Merrill)....Pages 85-96 Abstract Constructions of GMRAs (Kathy D. Merrill)....Pages 97-110 Back Matter ....Pages 111-113