An Introduction to Finite Tight Frames

This textbook is an introduction to the theory and applications of finite tight frames, an area that has developed rapidly in the last decade. Stimulating much of this growth are the applications of finite frames to diverse fields such as signal processing, quantum information theory, multivariate orthogonal polynomials, and remote sensing.

Key features and topics:

• First book entirely devoted to finite frames

• Extensive exercises and MATLAB examples for classroom use

• Important examples, such as harmonic and Heisenberg frames, are presented in preliminary chapters, encouraging readers to explore and develop an intuitive feeling for tight frames

• Later chapters delve into general theory details and recent research results

• Many illustrations showing the special aspects of the geometry of finite frames

• Provides an overview of the field of finite tight frames

• Discusses future research directions in the field

Featuring exercises and MATLAB examples in each chapter, the book is well suited as a textbook for a graduate course or seminar involving finite frames. The self-contained, user-friendly presentation also makes the work useful as a self-study resource or reference for graduate students, instructors, researchers, and practitioners in pure and applied mathematics, engineering, mathematical physics, and signal processing.

Auteur

Shayne Waldron is a senior lecturer in the Mathematics Department of the University of Auckland. He received his Ph.D. in Mathematics from the University of Wisconsin-Madison and served as a postdoctoral fellow at the Israel Institute of Technology. His areas of research include approximation theory and numerical analysis and the construction of tight frames from abstract groups,

Contenu

Preface.- Tight Frames.- Frames.- Canonical Coordinates for Vector Spaces and Affine Spaces.- Combining and Decomposing Frames.- Variational Characterizations of Tight Frames.- The Algebraic Variet of Tight Frames.- Projective Unitary Equivalence and Fusion Frames.- Symmetries of Tight Frames.- Group Frames.- Harmonic Frames.- Equiangular and Grassmannian Frames.- Tight Frames Generated by Nonabelian Groups.- Weyl-Heisenberg SICs.- Tight Frames of Orthogonal Polynomials on the Simplex.- Continuous Tight Frames for Finite Dimensional Spaces.- Solutions.- References.- Index.-