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An Introduction to Finite Tight Frames

An Introduction to Finite Tight Frames
Author: Shayne F. D. Waldron
Publisher: Springer
Total Pages: 590
Release: 2018-02-03
Genre: Mathematics
ISBN: 0817648151
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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. 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.

Finite Frame Theory: A Complete Introduction to Overcompleteness

Finite Frame Theory: A Complete Introduction to Overcompleteness
Author: Kasso A. Okoudjou
Publisher: American Mathematical Soc.
Total Pages: 266
Release: 2016-07-13
Genre: Mathematics
ISBN: 1470420198
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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.

Finite Frames

Finite Frames
Author: Peter G. Casazza
Publisher: Springer Science & Business Media
Total Pages: 492
Release: 2012-09-13
Genre: Mathematics
ISBN: 0817683720
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Hilbert space frames have long served as a valuable tool for signal and image processing due to their resilience to additive noise, quantization, and erasures, as well as their ability to capture valuable signal characteristics. More recently, finite frame theory has grown into an important research topic in its own right, with a myriad of applications to pure and applied mathematics, engineering, computer science, and other areas. The number of research publications, conferences, and workshops on this topic has increased dramatically over the past few years, but no survey paper or monograph has yet appeared on the subject. Edited by two of the leading experts in the field, Finite Frames aims to fill this void in the literature by providing a comprehensive, systematic study of finite frame theory and applications. With carefully selected contributions written by highly experienced researchers, it covers topics including: * Finite Frame Constructions; * Optimal Erasure Resilient Frames; * Quantization of Finite Frames; * Finite Frames and Compressed Sensing; * Group and Gabor Frames; * Fusion Frames. Despite the variety of its chapters' source and content, the book's notation and terminology are unified throughout and provide a definitive picture of the current state of frame theory. With a broad range of applications and a clear, full presentation, this book is a highly valuable resource for graduate students and researchers across disciplines such as applied harmonic analysis, electrical engineering, quantum computing, medicine, and more. It is designed to be used as a supplemental textbook, self-study guide, or reference book.

An Introduction to Frames

An Introduction to Frames
Author: Jelena Kovacevic
Publisher: Now Publishers Inc
Total Pages: 110
Release: 2008
Genre: Computers
ISBN: 160198068X
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An Introduction to Frames is an introduction to redundant signal representations called frames. These representations have recently emerged as yet another powerful tool in the signal processing toolbox, spurred by a host of recent applications requiring some level of redundancy. It asks the question: Why and where should one use frames? And answers emphatically: Anywhere where redundancy is a must. It then goes on to discuss a host of applications that richly illustrate that answer. An Introduction to Frames is geared primarily toward engineering students and those without extensive mathematical training. It is also intended to help researchers and practitioners decide whether frames are the right tool for their application.

Frames for Undergraduates

Frames for Undergraduates
Author: Deguang Han
Publisher: American Mathematical Soc.
Total Pages: 314
Release: 2007
Genre: Mathematics
ISBN: 0821842129
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"The early chapters contain the topics from linear algebra that students need to know in order to read the rest of the book. The later chapters are devoted to advanced topics, which allow students with more experience to study more intricate types of frames. Toward that end, a Student Presentation section gives detailed proofs of fairly technical results with the intention that a student could work out these proofs independently and prepare a presentation to a class or research group. The authors have also presented some stories in the Anecdotes section about how this material has motivated and influenced their students."--BOOK JACKET.

An Introduction to Frames and Riesz Bases

An Introduction to Frames and Riesz Bases
Author: Ole Christensen
Publisher: Birkhäuser
Total Pages: 704
Release: 2018-05-26
Genre: Mathematics
ISBN: 9783319798141
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This revised and expanded monograph presents the general theory for frames and Riesz bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet analysis, and generalized shift-invariant systems. Compared with the first edition, more emphasis is put on explicit constructions with attractive properties. Based on the exiting development of frame theory over the last decade, this second edition now includes new sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems, duality theory for as well Gabor frames as wavelet frames, and open problems in the field. Key features include: *Elementary introduction to frame theory in finite-dimensional spaces * Basic results presented in an accessible way for both pure and applied mathematicians * Extensive exercises make the work suitable as a textbook for use in graduate courses * Full proofs includ ed in introductory chapters; only basic knowledge of functional analysis required * Explicit constructions of frames and dual pairs of frames, with applications and connections to time-frequency analysis, wavelets, and generalized shift-invariant systems * Discussion of frames on LCA groups and the concrete realizations in terms of Gabor systems on the elementary groups; connections to sampling theory * Selected research topics presented with recommendations for more advanced topics and further readin g * Open problems to stimulate further research An Introduction to Frames and Riesz Bases will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference. Review of the first edition: "Ole Christensen’s An Introduction to Frames and Riesz Bases is a first-rate introduction to the field ... . The book provides an excellent exposition of these topics. The material is broad enough to pique the interest of many readers, the included exercises supply some interesting challenges, and the coverage provides enough background for those new to the subject to begin conducting original research." — Eric S. Weber, American Mathematical Monthly, Vol. 112, February, 2005

Frames and Bases

Frames and Bases
Author: Ole Christensen
Publisher: Springer Science & Business Media
Total Pages: 326
Release: 2008-06-05
Genre: Mathematics
ISBN: 0817646787
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Based on a streamlined presentation of the author’s successful work, An Introduction to Frames and Riesz Bases, this book develops frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field. The book presents basic results in an accessible way and includes extensive exercises.

Frames, Bases and Group Representations

Frames, Bases and Group Representations
Author: Deguang Han
Publisher: American Mathematical Soc.
Total Pages: 111
Release: 2000
Genre: Mathematics
ISBN: 0821820672
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This work develops an operator-theoretic approach to discrete frame theory on a separable Hilbert space. It is then applied to an investigation of the structural properties of systems of unitary operators on Hilbert space which are related to orthonormal wavelet theory. Also obtained are applications of frame theory to group representations, and of the theory of abstract unitary systems to frames generated by Gabor type systems.

Surgery on Frames

Surgery on Frames
Author: Nga Quynh Nguyen
Publisher:
Total Pages:
Release: 2010
Genre:
ISBN:
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In this dissertation, we investigate methods of modifying a tight frame sequence on a finite subset of the frame so that the result is a tight frame with better properties. We call this a surgery on the frame. There are basically three types of surgeries: transplants, expansions, and contractions. In this dissertation, it will be necessary to consider surgeries on not-necessarily-tight frames because the subsets of frames that are excised and replaced are usually not themselves tight frames on their spans, even if the initial frame and the final frame are tight. This makes the theory necessarily complicated, and richer than one might expect. Chapter I is devoted to an introduction to frame theory. In Chapter II, we investigate conditions under which expansion, contraction, and transplant problems have a solution. In particular, we consider the equiangular replacement problem. We show that we can always replace a set of three unit vectors with a set of three complex unit equiangular vectors which has the same Bessel operator as the Bessel operator of the original set. We show that this can not always be done if we require the replacement vectors to be real, even if the original vectors are real. We also prove that the minimum angle between pairs of vectors in the replacement set becomes largest when the replacement set is equiangular. Iterating this procedure can yield a frame with smaller maximal frame correlation than the original. Frames with optimal maximal frame correlation are called Grassmannian frames and no general method is known at the present time for constructing them. Addressing this, in Chapter III we introduce a spreading algorithm for finite unit tight frames by replacing vectors three-at-a-time to produce a unit tight frame with better maximal frame correlation than the original frame. This algorithm also provides a "good" orientation for the replacement sets. The orientation part ensures stability in the sense that if a selected set of three unit vectors happens to already be equiangular, then the algorithm gives back the same three vectors in the original order. In chapter IV and chapter V, we investigate two special classes of frames called push-out frames and group frames. Chapter VI is devoted to some mathematical problems related to the "cocktail party problem."