Cover

Title page

Contents

Chapter 1. Preface

Frame Theory

Phase retrieval and norm retrieval

1. Introduction

2. Preliminaries

3. Phase retrieval by vectors and Naimark complements

4. Phase retrieval by projections and norm retrieval

5. The identity operator, norm retrieval and phase retrieval

References

On Lipschitz inversion of nonlinear redundant representations

1. Introduction

2. Notations and Statement of Main Results

3. Proof of Results

References

Tight and random nonorthogonal fusion frames

1. Introduction

2. Classification of self adjoint operators via projections

3. Tight nonorthogonal fusion frames

4. Random nonorthogonal fusion frames

Acknowledgements

References

Decompositions of rational Gabor representations

1. Introduction

2. Preliminaries

3. The regular representation and its decompositions

4. Decomposition of 𝜋

5. Application to time-frequency analysis

References

Functional Analysis and 𝐶*-algebras

Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures

1. Introduction

2. Ingredients of the main results

3. Isomorphic semibranching function systems

4. Isomorphic s.f.s. on stationary Bratteli diagrams and equivalent representations of 𝒪_{𝒜}

5. Monic representations of 𝒪_{𝒜}

References

A survey of amenability theory for direct-limit groups

1. Introduction

2. Means on Topological Groups

3. Amenable Groups

4. Unitarizability

5. Invariant Means and Regular Representations

6. Almost Periodic Functions

7. Direct Limits of Abelian Groups

8. Spherical Functions and Direct Limits

References

Explicit construction of equivalence bimodules between noncommutative solenoids

1. Introduction

2. Review of the construction of noncommutative solenoids

3. Directed systems of equivalence bimodules: a method to form equivalence bimodules between direct limits of 𝐶*-algebras

4. The explicit construction of equivalence bimodules at each stage

5. Forming projective modules over noncommutative solenoids using 𝑝-adic fields

6. The Haar multiresolution analysis for 𝐿²(ℚ_{𝕡}) of Shelkovich and Skopina

7. The Haar MRS for 𝐿²(ℚ_{𝕡}) and projective multiresolution structures

References

Harmonic Analysis on Manifolds

An application of hypergeometric shift operators to the 𝜒-spherical Fourier transform

1. Introduction

2. Notation and Preliminaries

3. The Heckman-Opdam Hypergeometric Functions

4. Hypergeometric Shift Operators

5. An Application to the 𝜒-spherical functions

6. The Rank One Case

References

Harmonic analysis of a class of reproducing kernel Hilbert spaces arising from groups

1. Introduction

1.1. Two Extension Problems

1.2. Earlier Papers

1.3. Preliminaries

2. Extensions of Continuous Positive Definite Functions

2.1. Enlarging the Hilbert Space

2.2. 𝐸𝑥𝑡₁(𝐹) and 𝐸𝑥𝑡₂(𝐹)

2.3. The Case of 𝐺=ℝ

2.4. The Case of Locally Compact Abelian Groups

2.5. The Case of 𝐺=ℝⁿ

2.6. Lie Groups

3. Type I v.s. Type II Extensions

3.1. Polya Extensions

4. Models for Operator Extensions

4.1. Model for Restrictions of Continuous p.d. Functions on ℝ

4.2. A Model of ALL Deficiency Index-(1,1) Operators

4.3. The Case of Indices (𝑑,𝑑) where 𝑑>1

4.4. Spectral Representation of Index (1,1) Hermitian Operators

References

Approximations in 𝐿_{𝑝}-norms and Besov spaces on compact manifolds

1. Introduction

2. Kernels and Littlewood-Paley decomposition on compact Riemannian manifolds

3. Interpolation and Approximation spaces

4. Approximation in spaces 𝐿_{𝑝}(𝐌),1≤𝐩≤∞

References

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