This book presents the theory of harmonic analysis for noncommutative compact groups. If G is a commutative locally compact group, there is a well-understood theory of harmonic analysis as discussed in Aspects of Harmonic Analysis on Locally Compact Abelian Groups. If G is not commutative, things are a lot tougher. In the special case of a compact group, there is a deep interplay between analysis and representation theory which was first discovered by Hermann Weyl and refined by Andre Weil. This book presents these seminal results of Weyl and Weil.
Starting with the basics of representations theory, it presents the famous Peter–Weyl theorems and discusses Fourier analysis on compact groups. This book also introduces the reader to induced representations of locally compact groups, induced representations of G-bundles, and the theory of Gelfand pairs. A special feature is the chapter on equivariant convolutional neural networks (CNNs), a chapter which shows how many of the abstract concepts of representations, analysis on compact groups, Peter–Weyl theorems, Fourier transform, induced representations are used to tackle very practical, modern-day problems.
Contents:
- Preface
- Acknowledgments
- Introduction
- Representations of Algebras and Hilbert Algebras
- Unitary Representations of Locally Compact Groups
- Analysis on Compact Groups and Representations
- Matrix Representations of SL(2,ℂ), SU(2) and SO(3)
- Induced Representations
- Constructing Induced Representations a la Mackey
- Equivariant Convolutional Neural Networks
- Harmonic Analysis on Gelfand Pairs
- Bibliography
- Index
- Symbol Index
Readership: Second-year graduate masters/PhD students in mathematics, engineering, or computer vision who are interested in learning about harmonic analysis. Appropriate for mathematical courses on classical and functional analysis, medical imaging and measurement, advanced applied mathematics, mathematical analysis, representation theory of continuous groups, and mathematics for engineering. Supplementary reading for courses in signal/image processing, deep learning/equivariant convolutional neural networks, heat conduction, automatic control, acoustics, optics, and structural analysis.
Professor Jean Gallier has been with the University of Pennsylvania's Computer and Information Science Department for over 40 years. He holds a PhD in computer science from UCLA and is a world-renowned expert in computational logic. He also holds a joint appointment from the University of Pennsylvania's Department of Mathematics. Besides his seminal work on Horn satisfiability, he is known for his mathematical textbooks on geometric modeling, linear algebra, and differential geometry. He is the author of 12 books, a listing of which can be found at .Jocelyn Quaintance is a lecturer at the University of Pennsylvania's Computer and Information Science Department who specializes in linear algebra instruction for the online Masters of Computer and Information Technology degree. She holds a PhD in mathematics from the University of Pittsburgh. She is known for her research in combinatorial identities and power product expansions and has over 30 peer-reviewed publications. She is also known for her textbooks, which combine mathematical theory with computer science applications. Along with her co-author Jean Gallier, she has published books on linear algebra, differential geometry, cohomology, and harmonic analysis.
