CBMS Regional Conference Series in Mathematics
Volume: 128;
2018;
266 pp;
Softcover
MSC: Primary 28; 81; 11; 60; 42; 37;
Print ISBN: 978-1-4704-4880-6
Product Code: CBMS/128
List Price: $52.00
AMS Member Price: $41.60
MAA Member Price: $46.80
Electronic ISBN: 978-1-4704-4978-0
Product Code: CBMS/128.E
List Price: $52.00
AMS Member Price: $41.60
MAA Member Price: $46.80
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Supplemental Materials
Harmonic Analysis: Smooth and Non-smooth
Share this pagePalle E.T. Jorgensen
A co-publication of the AMS and CBMS
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 via Fourier duality, the author
also covers multiresolution wavelet approaches as well as a
third tool, namely, \(L^2\) spaces derived from appropriate
Gaussian processes. The book is based on a series of ten
lectures delivered in June 2018 at a CBMS conference held at Iowa
State University.
Readership
Undergraduate and graduate students and researchers interested in harmonic analysis and fractals.
Reviews & Endorsements
The book covers various aspects of a recently developed theory of non-smooth harmonic analysis as seen by one of the main experts in the field, who has devoted most of his extensive work to the subject. Through different techniques ranging from Fourier analysis to operator theory, stochastic processes and more, the reader will find a very interesting guide to the field.
-- Javier Duoandikoetxea, Mathematical Reviews
Table of Contents
Table of Contents
Harmonic Analysis: Smooth and Non-smooth
- Cover Cover11
- Title page iii4
- Contents vii8
- Preface ix10
- Acknowledgments xi12
- Chapter 1. Introduction. Smooth vs the non-smooth categories 114
- Chapter 2. Spectral pair analysis for IFSs 2336
- 2.1. The scale-4 Cantor measure, and its harmonic analysis 2336
- 2.2. The middle third Cantor measure 2740
- 2.3. Infinite Bernoulli convolutions 2942
- 2.4. The scale-4 Cantor measure, and scaling by 5 in the spectrum 3346
- 2.5. IFS measures and admissible harmonic analyses 3750
- 2.6. Harmonic analysis of IFS systems with overlap 3851
- Chapter 3. Harmonic analyses on fractals, with an emphasis on iterated function systems (IFS) measures 4558
- Chapter 4. Four kinds of harmonic analysis 6982
- Chapter 5. Harmonic analysis via representations of the Cuntz relations 135148
- Chapter 6. Positive definite functions and kernel analysis 175188
- 6.1. Positive definite kernels and harmonic analysis in 𝐿²(𝜇) when 𝜇 is a gap IFS fractal measure 175188
- 6.2. Positive definite kernels and harmonic analysis in 𝐿²(\ensuremath{𝜇}) when 𝜇 is a general singular measure in a finite interval 180193
- 6.3. Positive definite kernels and the associated Gaussian processes 211224
- Chapter 7. Representations of Lie groups. Non-commutative harmonic analysis 219232
- Bibliography 245258
- Index 265278
- Back Cover Back Cover1281