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

Hoboken: Wiley, 2025. — 371 p.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.Preface
AcknowledgmentsOrganization of the Book
Proficiency Expectations
Aims
Scope and Material Selection
Catering to Diverse Learning Approaches and Expertise Levels
ReferencesCalculus on Euclidean Space
The Inverse Function Theorem and Its Applications
The Implicit Function and Constant Rank Theorems
Topological Manifolds
Differentiable Structures
Submanifolds
Derivations
Tangent Vectors
Tangent Vector As Equivalent Classes of Smooth Curves
Tangent Vectors As Derivations at a Point
Tangent Bundles
1‐Forms
Pull‐Backs
Tensor Fields
ReferencesLie Derivatives
Lie Groups and Lie Algebras
Lie Groups and Examples
Left and Right Translations
Lie Algebras
Exponential Map
Invariant Measure on Lie Groups
Homogeneous Spaces
Matrix Lie Theory
The Adjoint Maps
Lie's Theorem
Construction of Spline‐Type Partitions of Unity
ReferencesRepresentations of Lie Groups and Lie Algebras
A Survey on the Theory of Direct Integrals
Induced Representations
Quasi‐invariant Measures on Cosets
Induced Unitary Characters
Integrability of Induced Characters
ReferencesSeries Expansions in Hilbert Spaces
Riesz Bases
Frames
ReferencesWavelets and the ax+b Group
The Wavelet Representation
Gabor Systems and the Heisenberg Group
ReferencesDiscretization of Induced Characters
Connection to Wavelet Theory and Time‐Frequency Analysis
A Toy Example
Proofs of Main Results
Localized Frames on Matrix Lie Groups
A Generalization
ReferencesLocalized Frames on Homogeneous Spaces
Frames on Spheres
Frames on the Klein Bottle
ReferencesFrames and Bases of Translates on the ax+b Lie Group
ReferencesAdmissible Representations
Gröchenig–Führ's Method of Oscillations
Sampling on Locally Compact Groups
Bandlimitation for Extensions of Rn
The Mautner Group and Its Relatives
Bandlimitation on a Class of Lie Groups
Spectral Analysis of Induced Representations
ReferencesInductive Construction of All Complex n‐Frames
Infinite Singly Generated Subgroups of Un
Random Sampling
References
index