About this item
Highlights
- About the Author: Vignon Oussa, PhD, is Professor of Mathematics at Bridgewater State University in Bridgewater, MA, USA.
- 384 Pages
- Mathematics, Applied
Description
About the Book
"Frame construction is currently a very active area of research, and a book that provides a systematic introduction of the Lie theoretic tools for such an endeavor, together with thorough demonstrations how these tools can be employed, is in my view a very timely project." Duffin and Schaeffer developed frame theory in the fifties as a tool to solve problems in non-harmonic Fourier series. The search for redundant and flexible basis-like reproducing systems for signal analysis led to the rediscovery of frames in the early eighties. The foundational work of Daubechies, Meyer, Grossman, and others highlighted the influential role that frames play in studying signal analysis through wavelet theory and time-frequency analysis. Frame theory is a branch of harmonic analysis that has now blossomed into a dynamic and active field, drawing its strengths from a wide range of areas such as representation theory, and Lie theory. The proposed book is concerned with the discretization problem of representations of Lie groups, which can be formulated as follows. Given a representation of a Lie group, under which conditions is it possible to sample one of its orbits for the construction of frames with prescribed properties? This book aims to give a systematic, coherent, and detailed treatment of the mathematics encountered in searching for a satisfactory solution to the discretization problem."--From the Back Cover
Comprehensive textbook examining meaningful connections between the subjects of Lie theory, differential geometry, and signal analysis
A Bridge Between Lie Theory and Frame Theory serves as a bridge between the areas of Lie theory, differential geometry, and frame theory, illustrating applications in the context of signal analysis with concrete examples and images.
The first part of the book gives an in-depth, comprehensive, and self-contained exposition of differential geometry, Lie theory, representation theory, and frame theory. The second part of the book uses the theories established in the early part of the text to characterize a class of representations of Lie groups, which can be discretized to construct frames and other basis-like systems. For instance, Lie groups with frames of translates, sampling, and interpolation spaces on Lie groups are characterized.
A Bridge Between Lie Theory and Frame Theory includes discussion on:
- Novel constructions of frames possessing additional desired features such as boundedness, compact support, continuity, fast decay, and smoothness, motivated by applications in signal analysis
- Necessary technical tools required to study the discretization problem of representations at a deep level
- Ongoing dynamic research problems in frame theory, wavelet theory, time frequency analysis, and other related branches of harmonic analysis
A Bridge Between Lie Theory and Frame Theory is an essential learning resource for graduate students, applied mathematicians, and scientists who are looking for a rigorous and complete introduction to the covered subjects.
About the Author
Vignon Oussa, PhD, is Professor of Mathematics at Bridgewater State University in Bridgewater, MA, USA. Dr. Oussa received his PhD in Mathematics from Saint Louis University. His areas of specialty are harmonic analysis, Lie theory, and representation theory. He is the winner of the 2014-2015 Bridgewater State University Distinguished Faculty Research Award.