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

Format:

Book

Author/Creator:

Oussa, Vignon.

Language:

English

Subjects (All):

Frames (Vector analysis).

Lie groups.

Geometry, Differential.

Harmonic analysis.

Physical Description:

1 online resource (599 pages)

Edition:

1st ed.

Place of Publication:

Newark : John Wiley & Sons, Incorporated, 2025.

Summary:

"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."-- Provided by publisher.

Contents:

Cover

Title Page

Copyright

Contents

Preface

Acknowledgments

Chapter 1 Introduction

1.1 Organization of the Book

1.2 Proficiency Expectations

1.3 Aims

1.4 Scope and Material Selection

1.5 Catering to Diverse Learning Approaches and Expertise Levels

References

Chapter 2 Differentiable Manifolds

2.1 Calculus on Euclidean Space

2.1.1 The Inverse Function Theorem and Its Applications

2.1.1.1 The Implicit Function and Constant Rank Theorems

2.2 Topological Manifolds

2.2.1 Differentiable Structures

2.2.2 Submanifolds

2.2.3 Derivations

2.2.4 Tangent Vectors

2.2.4.1 Tangent Vector As Equivalent Classes of Smooth Curves

2.2.4.2 Tangent Vectors As Derivations at a Point

2.2.5 Tangent Bundles

2.2.6 1‐Forms

2.2.7 Pull‐Backs

2.2.8 Tensor Fields

Chapter 3 Lie Theory

3.1 Lie Derivatives

3.2 Lie Groups and Lie Algebras

3.2.1 Lie Groups and Examples

3.2.2 Left and Right Translations

3.2.3 Lie Algebras

3.3 Exponential Map

3.4 Invariant Measure on Lie Groups

3.5 Homogeneous Spaces

3.6 Matrix Lie Theory

3.6.1 The Adjoint Maps

3.6.1.1 Lie's Theorem

3.7 Construction of Spline‐Type Partitions of Unity

Chapter 4 Representation Theory

4.1 Representations of Lie Groups and Lie Algebras

4.2 A Survey on the Theory of Direct Integrals

4.3 Induced Representations

4.3.1 Quasi‐invariant Measures on Cosets

4.3.2 Induced Unitary Characters

4.4 Integrability of Induced Characters

Chapter 5 Frame Theory

5.1 Series Expansions in Hilbert Spaces

5.2 Riesz Bases

5.3 Frames

Chapter 6 Frames on Euclidean Spaces

6.1 Wavelets and the ax+b Group

6.1.1 The Wavelet Representation

6.2 Gabor Systems and the Heisenberg Group

Chapter 7 Frames on Lie Groups.

7.1 Discretization of Induced Characters

7.1.1 Connection to Wavelet Theory and Time‐Frequency Analysis

7.1.2 A Toy Example

7.1.3 Proofs of Main Results

7.2 Localized Frames on Matrix Lie Groups

7.3 A Generalization

Chapter 8 Frames on Homogeneous Spaces

8.1 Localized Frames on Homogeneous Spaces

8.2 Frames on Spheres

8.3 Frames on the Klein Bottle

Chapter 9 Groups with Frames of Translates

9.1 Frames and Bases of Translates on the ax+b Lie Group

Chapter 10 Sampling and Interpolation on Unimodular Lie Groups

10.1 Admissible Representations

10.2 Gröchenig-Führ's Method of Oscillations

10.3 Sampling on Locally Compact Groups

10.4 Bandlimitation for Extensions of Rn

10.4.1 The Mautner Group and Its Relatives

10.4.2 Bandlimitation on a Class of Lie Groups

10.4.2.1 Spectral Analysis of Induced Representations

Chapter 11 Finite Frames Maximally Robust to Erasures

11.1 Inductive Construction of All Complex n‐Frames

11.2 Infinite Singly Generated Subgroups of Un

11.3 Random Sampling

Index

EULA.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781119712145

1119712149

9781119712169

1119712165

9781119712152

1119712157

OCLC:

1505910850