نظریهٔ فریم محدود: مقدمه‌ای کامل بر فراوانی (مجموعه مقالات سمپوزیوم در ریاضیات کاربردی)

Frames are overcomplete sets of vectors that can be used to stably and faithfully decompose and reconstruct vectors in the underlying vector space. Frame theory stands at the intersection of many areas in mathematics such as functional and harmonic analysis, numerical analysis, matrix theory, numerical linear algebra, algebraic and differential geometry, probability, statistics, and convex geometry. At the same time its applications in engineering, medicine, computer science, and quantum computing are motivating new research problems in applied and pure mathematics. This volume is based on lectures delivered at the 2015 AMS Short Course "Finite Frame Theory: A Complete Introduction to Overcompleteness", held January 8-9, 2015 in San Antonio, TX. Mostly written in a tutorial style, the seven chapters contained in this volume survey recent advances in the theory and applications of finite frames. In particular, it presents state-of-the-art results on foundational frame problems, and on the analysis and design of various frames, mostly motivated by specific applications. Carefully assembled, the volume quickly introduces the non-expert to the basic tools and techniques of frame theory. It then moves to develop many recent results in the area and presents some important applications. As such, the volume is designed for a diverse audience including researchers in applied and computational harmonic analysis, as well as engineers and graduate students. Cover Title page Contents Preface Introduction Bibliography A Brief Introduction to Hilbert Space Frame Theory and its Applications 1. Reading List 2. The Basics of Hilbert Space Theory 3. The Basics of Operator Theory 4. Hilbert Space Frames 5. Constants Related to Frames 6. Constructing Finite Frames 7. Gramian Operators 8. Fusion Frames 9. Infinite Dimensional Hilbert Spaces 10. Major Open Problems in Frame Theory References Unit norm tight frames in finite-dimensional spaces 1. Introduction 2. Motivating applications 3. Examples 4. The frame potential 5. Eigensteps 6. Equiangular tight frames References Algebro-Geometric Techniques and Geometric Insights for Finite Frames 1. Background 2. Notation 3. Intersecting tori and Stiefel manifolds in the Hilbert-Schmidt sphere 4. Explicit, locally-defined, analytic coordinate functions on \F 5. Connectivity and irreducibility of \F 6. A final challenge References Preconditioning techniques in frame theory and probabilistic frames 1. Introduction 2. Preconditioning techniques in frame theory 3. Probabilistic frames Acknowledgment References Quantization, finite frames, and error diffusion 1. Introduction 2. Finite Frames 3. Quantization 4. Memoryless Scalar Quantization 5. First Order ΣΔ Algorithms 6. Stability and Error Bounds 7. Acknowledgements References Frames and Phaseless Reconstruction 1. Introduction 2. Geometry of H and \SS^{p,q} Spaces 3. The Injectivity Problem 4. Robustness of Reconstruction 5. Reconstruction Algorithms References Compressed sensing and dictionary learning 1. Introduction 2. Background to Compressed signal processing 3. Compressed sensing with tight frames 4. \index{dictionary learning}Dictionary Learning: An introduction 5. Dictionary Learning: Algorithms Acknowledgment References Index Back Cover