A Bridge Between Lie Theory and Frame Theory : Applications of Lie Theory to Harmonic Analysis
(Visualizza in formato marc)
(Visualizza in BIBFRAME)
| Autore: |
Oussa Vignon
|
| Titolo: |
A Bridge Between Lie Theory and Frame Theory : Applications of Lie Theory to Harmonic Analysis
|
| Pubblicazione: | Newark : , : John Wiley & Sons, Incorporated, , 2025 |
| ©2025 | |
| Edizione: | 1st ed. |
| Descrizione fisica: | 1 online resource (599 pages) |
| Disciplina: | 512/.482 |
| Soggetto topico: | Frames (Vector analysis) |
| Lie groups | |
| Geometry, Differential | |
| Harmonic analysis | |
| Nota di contenuto: | 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 -- References -- 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 -- References -- 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 -- References -- Chapter 5 Frame Theory -- 5.1 Series Expansions in Hilbert Spaces -- 5.2 Riesz Bases -- 5.3 Frames -- References -- 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 -- References -- 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 -- References -- 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 -- References -- Chapter 9 Groups with Frames of Translates -- 9.1 Frames and Bases of Translates on the ax+b Lie Group -- References -- 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 -- References -- 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 -- References -- Index -- EULA. | |
| Sommario/riassunto: | "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."-- |
| Titolo autorizzato: | A Bridge Between Lie Theory and Frame Theory |
| ISBN: | 9781119712145 |
| 1119712149 | |
| 9781119712169 | |
| 1119712165 | |
| 9781119712152 | |
| 1119712157 | |
| Formato: | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione: | Inglese |
| Record Nr.: | 9911019626503321 |
| Lo trovi qui: | Univ. Federico II |
| Opac: | Controlla la disponibilità qui |