A Bridge Between Lie Theory and Frame Theory: Applications of Lie Theory to Harmonic Analysis

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.

Table of Contents:
Preface ix  Acknowledgments xi  1 Introduction 1  1.1 Organization of the Book 12  1.2 Proficiency Expectations 15  1.3 Aims 15  1.4 Scope and Material Selection 16  1.5 Catering to Diverse Learning Approaches and Expertise Levels 16  References 19  2 Differentiable Manifolds 21  2.1 Calculus on Euclidean Space 22  2.1.1 The Inverse Function Theorem and Its Applications 26  2.1.1.1 The Implicit Function and Constant Rank Theorems 26  2.2 Topological Manifolds 27  2.2.1 Differentiable Structures 30  2.2.2 Submanifolds 39  2.2.3 Derivations 42  2.2.4 Tangent Vectors 52  2.2.4.1 Tangent Vector As Equivalent Classes of Smooth Curves 55  2.2.4.2 Tangent Vectors As Derivations at a Point 60  2.2.5 Tangent Bundles 66  2.2.6 1-Forms 70  2.2.7 Pull-Backs 75  2.2.8 Tensor Fields 75  References 81  3 Lie Theory 83  3.1 Lie Derivatives 84  3.2 Lie Groups and Lie Algebras 94  3.2.1 Lie Groups and Examples 94  3.2.2 Left and Right Translations 100  3.2.3 Lie Algebras 102  3.3 Exponential Map 116  3.4 Invariant Measure on Lie Groups 121  3.5 Homogeneous Spaces 132  3.6 Matrix Lie Theory 140  3.6.1 The Adjoint Maps 155  3.6.1.1 Lie’s Theorem 159  3.7 Construction of Spline-Type Partitions of Unity 165  References 170  4 Representation Theory 173  4.1 Representations of Lie Groups and Lie Algebras 173  4.2 A Survey on the Theory of Direct Integrals 179  4.3 Induced Representations 182  4.3.1 Quasi-invariant Measures on Cosets 182  4.3.2 Induced Unitary Characters 188  4.4 Integrability of Induced Characters 191  References 205  5 Frame Theory 207  5.1 Series Expansions in Hilbert Spaces 207  5.2 Riesz Bases 210  5.3 Frames 213  References 220  6 Frames on Euclidean Spaces 221  6.1 Wavelets and the ax+b Group 221  6.1.1 The Wavelet Representation 231  6.2 Gabor Systems and the Heisenberg Group 234  References 238  7 Frames on Lie Groups 241  7.1 Discretization of Induced Characters 242  7.1.1 Connection to Wavelet Theory and Time-Frequency Analysis 242  7.1.2 A Toy Example 247  7.1.3 Proofs of Main Results 252  7.2 Localized Frames on Matrix Lie Groups 269  7.3 A Generalization 272  References 275  8 Frames on Homogeneous Spaces 277  8.1 Localized Frames on Homogeneous Spaces 277  8.2 Frames on Spheres 281  8.3 Frames on the Klein Bottle 299  References 301  9 Groups with Frames of Translates 303  9.1 Frames and Bases of Translates on the ax+b Lie Group 309  References 315  10 Sampling and Interpolation on Unimodular Lie Groups 317  10.1 Admissible Representations 317  10.2 Gröchenig–Führ’s Method of Oscillations 324  10.3 Sampling on Locally Compact Groups 331  10.4 Bandlimitation for Extensions of R n 337  10.4.1 The Mautner Group and Its Relatives 338  10.4.2 Bandlimitation on a Class of Lie Groups 341  10.4.2.1 Spectral Analysis of Induced Representations 347  References 349  11 Finite Frames Maximally Robust to Erasures 351  11.1 Inductive Construction of All Complex n-Frames 356  11.2 Infinite Singly Generated Subgroups of U (n) 363  11.3 Random Sampling 366  References 367  Index 369

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.