{"product_id":"a-bridge-between-lie-theory-and-frame-theory-applications-of-lie-theory-to-harmonic-analysis-hardback-9781119712138","title":"A Bridge Between Lie Theory and Frame Theory; Applications of Lie Theory to Harmonic Analysis (Hardback) 9781119712138","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eA Bridge Between Lie Theory and Frame Theory\u003c\/font\u003e\u003cbr\u003e\r\n\u003cfont size=\"5\"\u003eApplications of Lie Theory to Harmonic Analysis\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eVignon Oussa (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9781119712138, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 7 February 2025\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e384 pages\u003cbr\u003e22.9 x 15.2 x 2.4 cm, 0.794 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003e\u003cp\u003e\u003cb\u003eComprehensive textbook examining meaningful connections between the subjects of Lie theory, differential geometry, and signal analysis\u003c\/b\u003e \u003c\/p\u003e\n\u003cp\u003e\u003ci\u003eA Bridge Between Lie Theory and Frame Theory\u003c\/i\u003e 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. \u003c\/p\u003e\n\u003cp\u003eThe 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. \u003c\/p\u003e\n\u003cp\u003e\u003ci\u003eA Bridge Between Lie Theory and Frame Theory\u003c\/i\u003e includes discussion on: \u003c\/p\u003e\n\u003cul\u003e\n\u003cli\u003eNovel constructions of frames possessing additional desired features such as boundedness, compact support, continuity, fast decay, and smoothness, motivated by applications in signal analysis\u003c\/li\u003e\n\u003cli\u003eNecessary technical tools required to study the discretization problem of representations at a deep level\u003c\/li\u003e\n\u003cli\u003eOngoing dynamic research problems in frame theory, wavelet theory, time frequency analysis, and other related branches of harmonic analysis\u003c\/li\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003ci\u003eA Bridge Between Lie Theory and Frame Theory\u003c\/i\u003e 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.\u003c\/p\u003e\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e\u003cp\u003ePreface ix \u003c\/p\u003e \u003cp\u003eAcknowledgments xi \u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e1.1 Organization of the Book 12 \u003c\/p\u003e \u003cp\u003e1.2 Proficiency Expectations 15 \u003c\/p\u003e \u003cp\u003e1.3 Aims 15 \u003c\/p\u003e \u003cp\u003e1.4 Scope and Material Selection 16 \u003c\/p\u003e \u003cp\u003e1.5 Catering to Diverse Learning Approaches and Expertise Levels 16 \u003c\/p\u003e \u003cp\u003eReferences 19 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Differentiable Manifolds 21\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e2.1 Calculus on Euclidean Space 22 \u003c\/p\u003e \u003cp\u003e2.1.1 The Inverse Function Theorem and Its Applications 26 \u003c\/p\u003e \u003cp\u003e2.1.1.1 The Implicit Function and Constant Rank Theorems 26 \u003c\/p\u003e \u003cp\u003e2.2 Topological Manifolds 27 \u003c\/p\u003e \u003cp\u003e2.2.1 Differentiable Structures 30 \u003c\/p\u003e \u003cp\u003e2.2.2 Submanifolds 39 \u003c\/p\u003e \u003cp\u003e2.2.3 Derivations 42 \u003c\/p\u003e \u003cp\u003e2.2.4 Tangent Vectors 52 \u003c\/p\u003e \u003cp\u003e2.2.4.1 Tangent Vector As Equivalent Classes of Smooth Curves 55 \u003c\/p\u003e \u003cp\u003e2.2.4.2 Tangent Vectors As Derivations at a Point 60 \u003c\/p\u003e \u003cp\u003e2.2.5 Tangent Bundles 66 \u003c\/p\u003e \u003cp\u003e2.2.6 1-Forms 70 \u003c\/p\u003e \u003cp\u003e2.2.7 Pull-Backs 75 \u003c\/p\u003e \u003cp\u003e2.2.8 Tensor Fields 75 \u003c\/p\u003e \u003cp\u003eReferences 81 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Lie Theory 83\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e3.1 Lie Derivatives 84 \u003c\/p\u003e \u003cp\u003e3.2 Lie Groups and Lie Algebras 94 \u003c\/p\u003e \u003cp\u003e3.2.1 Lie Groups and Examples 94 \u003c\/p\u003e \u003cp\u003e3.2.2 Left and Right Translations 100 \u003c\/p\u003e \u003cp\u003e3.2.3 Lie Algebras 102 \u003c\/p\u003e \u003cp\u003e3.3 Exponential Map 116 \u003c\/p\u003e \u003cp\u003e3.4 Invariant Measure on Lie Groups 121 \u003c\/p\u003e \u003cp\u003e3.5 Homogeneous Spaces 132 \u003c\/p\u003e \u003cp\u003e3.6 Matrix Lie Theory 140 \u003c\/p\u003e \u003cp\u003e3.6.1 The Adjoint Maps 155 \u003c\/p\u003e \u003cp\u003e3.6.1.1 Lie’s Theorem 159 \u003c\/p\u003e \u003cp\u003e3.7 Construction of Spline-Type Partitions of Unity 165 \u003c\/p\u003e \u003cp\u003eReferences 170 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Representation Theory 173\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e4.1 Representations of Lie Groups and Lie Algebras 173 \u003c\/p\u003e \u003cp\u003e4.2 A Survey on the Theory of Direct Integrals 179 \u003c\/p\u003e \u003cp\u003e4.3 Induced Representations 182 \u003c\/p\u003e \u003cp\u003e4.3.1 Quasi-invariant Measures on Cosets 182 \u003c\/p\u003e \u003cp\u003e4.3.2 Induced Unitary Characters 188 \u003c\/p\u003e \u003cp\u003e4.4 Integrability of Induced Characters 191 \u003c\/p\u003e \u003cp\u003eReferences 205 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Frame Theory 207\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e5.1 Series Expansions in Hilbert Spaces 207 \u003c\/p\u003e \u003cp\u003e5.2 Riesz Bases 210 \u003c\/p\u003e \u003cp\u003e5.3 Frames 213 \u003c\/p\u003e \u003cp\u003eReferences 220 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Frames on Euclidean Spaces 221\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e6.1 Wavelets and the ax+b Group 221 \u003c\/p\u003e \u003cp\u003e6.1.1 The Wavelet Representation 231 \u003c\/p\u003e \u003cp\u003e6.2 Gabor Systems and the Heisenberg Group 234 \u003c\/p\u003e \u003cp\u003eReferences 238 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Frames on Lie Groups 241\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e7.1 Discretization of Induced Characters 242 \u003c\/p\u003e \u003cp\u003e7.1.1 Connection to Wavelet Theory and Time-Frequency Analysis 242 \u003c\/p\u003e \u003cp\u003e7.1.2 A Toy Example 247 \u003c\/p\u003e \u003cp\u003e7.1.3 Proofs of Main Results 252 \u003c\/p\u003e \u003cp\u003e7.2 Localized Frames on Matrix Lie Groups 269 \u003c\/p\u003e \u003cp\u003e7.3 A Generalization 272 \u003c\/p\u003e \u003cp\u003eReferences 275 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Frames on Homogeneous Spaces 277\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e8.1 Localized Frames on Homogeneous Spaces 277 \u003c\/p\u003e \u003cp\u003e8.2 Frames on Spheres 281 \u003c\/p\u003e \u003cp\u003e8.3 Frames on the Klein Bottle 299 \u003c\/p\u003e \u003cp\u003eReferences 301 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Groups with Frames of Translates 303\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e9.1 Frames and Bases of Translates on the ax+b Lie Group 309 \u003c\/p\u003e \u003cp\u003eReferences 315 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Sampling and Interpolation on Unimodular Lie Groups 317\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e10.1 Admissible Representations 317 \u003c\/p\u003e \u003cp\u003e10.2 \u003ci\u003eGröchenig\u003c\/i\u003e–\u003ci\u003eFühr\u003c\/i\u003e’s Method of Oscillations 324 \u003c\/p\u003e \u003cp\u003e10.3 Sampling on Locally Compact Groups 331 \u003c\/p\u003e \u003cp\u003e10.4 Bandlimitation for Extensions of R n 337 \u003c\/p\u003e \u003cp\u003e10.4.1 The Mautner Group and Its Relatives 338 \u003c\/p\u003e \u003cp\u003e10.4.2 Bandlimitation on a Class of Lie Groups 341 \u003c\/p\u003e \u003cp\u003e10.4.2.1 Spectral Analysis of Induced Representations 347 \u003c\/p\u003e \u003cp\u003eReferences 349 \u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Finite Frames Maximally Robust to Erasures 351\u003c\/b\u003e \u003c\/p\u003e \u003cp\u003e11.1 Inductive Construction of All Complex n-Frames 356 \u003c\/p\u003e \u003cp\u003e11.2 Infinite Singly Generated Subgroups of U (n) 363 \u003c\/p\u003e \u003cp\u003e11.3 Random Sampling 366 \u003c\/p\u003e \u003cp\u003eReferences 367 \u003c\/p\u003e \u003cp\u003eIndex 369\u003c\/p\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematics [\u003ca title=\"See our other books on Mathematics\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematics%20%5BPB%5D%22\"\u003ePB\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley","offers":[{"title":"Brand New","offer_id":52430885224728,"sku":"9781119712138","price":89.27,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/files\/9781119712138.jpg?v=1784764915","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/a-bridge-between-lie-theory-and-frame-theory-applications-of-lie-theory-to-harmonic-analysis-hardback-9781119712138","provider":"Freshly Printed Books","version":"1.0","type":"link"}