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

fmatter
	Title Page
	Copyright
	Contents
	Preface
	Acknowledgments
ch1
	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
ch2
	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
ch3
	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
ch4
	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
ch5
	5.1 Series Expansions in Hilbert Spaces
	5.2 Riesz Bases
	5.3 Frames
	References
ch6
	6.1 Wavelets and the ax+b Group
		6.1.1 The Wavelet Representation
	6.2 Gabor Systems and the Heisenberg Group
	References
ch7
	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
ch8
	8.1 Localized Frames on Homogeneous Spaces
	8.2 Frames on Spheres
	8.3 Frames on the Klein Bottle
	References
ch9
	9.1 Frames and Bases of Translates on the ax+b Lie Group
	References
ch10
	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
ch11
	11.1 Inductive Construction of All Complex n‐Frames
	11.2 Infinite Singly Generated Subgroups of Un
	11.3 Random Sampling
	References
index