| Series |
CBMS regional conference series in mathematics ; number 128
Regional conference series in mathematics ; no. 128. ^A58601
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| Contents |
Cover; Title page; Contents; Preface; Acknowledgments; Chapter 1. Introduction. Smooth vs the non-smooth categories; 1.1. Preview; 1.2. Historical context; 1.3. Iterated function systems (IFS); 1.4. Frequency bands, filters, and representations of the Cuntz-algebras; 1.5. Frames; 1.6. Key themes in the book; Chapter 2. Spectral pair analysis for IFSs; 2.1. The scale-4 Cantor measure, and its harmonic analysis; 2.2. The middle third Cantor measure; 2.3. Infinite Bernoulli convolutions; 2.4. The scale-4 Cantor measure, and scaling by 5 in the spectrum. |
| Contents |
2.5. IFS measures and admissible harmonic analyses; 2.6. Harmonic analysis of IFS systems with overlap; Chapter 3. Harmonic analyses on fractals, with an emphasis on iterated function systems (IFS) measures; 3.1. Harmonic analysis in the smooth vs the non-smooth categories; 3.2. Spectral pairs and the Fuglede conjecture; 3.3. Spectral pairs; 3.4. Spectral theory of multiple intervals; Chapter 4. Four kinds of harmonic analysis; 4.1. Orthogonal Fourier expansions; 4.2. Frame and related non-orthogonal Fourier expansions; 4.3. Wavelet expansions. |
| Contents |
4.4. Harmonic analysis via reproducing kernel Hilbert spaces (RKHS)Chapter 5. Harmonic analysis via representations of the Cuntz relations; 5.1. From frequency band filters to signals and to wavelet expansions; 5.2. Stochastic processes via representations of the Cuntz relations; 5.3. Representations in a universal Hilbert space; Chapter 6. Positive definite functions and kernel analysis; 6.1. Positive definite kernels and harmonic analysis in ²() when is a gap IFS fractal measure. |
| Contents |
6.2. Positive definite kernels and harmonic analysis in ²(\ensuremath{ }) when is a general singular measure in a finite interval; 6.3. Positive definite kernels and the associated Gaussian processes; Chapter 7. Representations of Lie groups. Non-commutative harmonic analysis; 7.1. Fundamental domains as non-commutative tiling constructions; 7.2. Symmetry for unitary representations of Lie groups; 7.3. Symmetry in physics and reflection positive constructions via unitary representations of Lie groups; Bibliography; Index; Back Cover. |
| Abstract |
There is a recent and increasing interest in harmonic analysis of non-smooth geometries. Real-world examples where these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. Notions of harmonic analysis focus on transforms and expansions and involve dual variables. In this book on smooth and non-smooth harmonic analysis, the notion of dual variables will be adapted to fractals. In addition to harmonic analysis v. |
| Source of description | Online resource; title from PDF title page (EBSCO, viewed November 20, 2018). |
| Issued in other form | Print version: 9781470448806 1470448807 |
| Genre/form | Electronic books. |
| ISBN | 9781470449780 (electronic bk.) |
| ISBN | 1470449781 (electronic bk.) |
