This book provides an introduction to frame theory in Banach and Hilbert spaces, with a particular focus on the Banach space aspects of the frame theory and its applications. The authors present main developments from recent decades in the theory of frames, dilations, operator-valued measures, sampling, and related applications. By bridging topics from Banach space geometry, applied harmonic analysis, and quantum information theory, the book highlights a novel perspective: frame theory not only serves as a powerful and central tool in mathematics and engineering, but is also deeply intertwined with the structure and theory of Banach spaces. Rather than providing an exhaustive treatment, the book offers a broadly relevant and accessible overview that enables researchers across various fields to quickly grasp the foundational ideas and explore related research directions.

Blends functional analytic aspects of frame theory with Banach space approximation properties and dilation theory Focuses on frames in Banach spaces and the intersection of Banach space geometry including operator space structures Presents applications of frames in quantum information theory such as phase retrieval and quantum detection

Autorentext

Deguang Han, Ph.D., is a Professor in the Department of Mathematics at the University of Central Florida, Orlando, USA. His research interests include the theory of operator and operator algebras and the functional analysis approach to problems in applied harmonic analysis and applications. He has published over 150 publications including three books. Qianfeng Hu, Ph.D., is a Lecturer at the School of Science at the Hebei University of Technology, Tianjin, China. He obtained a Ph.D. in Pure Mathematics from Nankai University. His research interests include functional analysis and its application in quantum information theory. Bei Liu, Ph.D., is a Professor in the College of Science at Tianjin University of Technology, Tianjin, China. Her research interests include approximation theory, wavelet analysis, and signal processing, and she has published many papers on these topics. Rui Liu, Ph.D., is a Professor in the School of Mathematical Sciences and LPMC at Nankai University, Tianjin, China. He has published over 30 journal articles and one book. His research interests include functional analysis, Banach space theory, operator spaces, and related topics.

Klappentext

This book offers a fresh introduction to frame theory in Banach and Hilbert spaces, with a particular focus on the Banach space aspects of the frame theory and its applications. It presents some main developments from recent decades in the theory of frames, dilations, operator-valued measures, sampling, and related applications. By bridging topics from Banach space geometry, applied harmonic analysis, and quantum information theory, the authors highlight a novel perspective: frame theory not only serves as a powerful and central tool in mathematics and engineering, but is also deeply intertwined with the structure and theory of Banach spaces. Rather than providing an exhaustive treatment, the book aims to offer a broadly relevant and accessible overview that enables researchers across various fields to quickly grasp the foundational ideas and explore related research directions.

In addition, this book:

• Blends functional analytic aspects of frames in Banach and Hilbert spaces with approximation properties, dilation theory, operator-valued measure, etc.
• Focuses on frames for Banach spaces and the intersection of Banach space geometry, even operator space structures
• Presents wide applications of frames in globally popular problems in quantum information theory and applied harmonic analysis, such as phase retrieval and quantum detection

Inhalt

Frames in Hilbert Spaces.- Frames in Banach Spaces.- Approximation Properties.- Banach Dilation.- Nonselfadjoint Dilation and Generalized Choi-Kraus Representation.- Quantum Detection and Phase Retrieval.