Synopses & Reviews
This book, combining analysis and tools from mathematical probability, focuses on a systematic and novel presentation of recent trends in pure and applied mathematics: the emergence of three fields, wavelets, signals and fractals. The unity of basis constructions and their expansions is emphasized as the starting point for the development of bases that are computationally efficient for use in several areas from wavelets to fractals. The book brings together tools from engineering and math, especially from signal- and image processing, and from harmonic analysis and operator theory. The presentation is aimed at graduate students, as well as users from a diverse spectrum of applications. Key features: • A hands-on approach for students, including tutorials and numerous exercises; • Excellent motivation throughout; • New pedagogical features: glossary of terms, their use in mathematics and in engineering, help for cross-audiences, image processing, visual presentation of key algorithms, structure and geometry of big matrix computations, explanation of uses of the theory in applications outside of mathematics; • Includes more than 50 figures with captions, illustrating the main ideas, plus engineering diagrams, graphic renditions of algorithms, and separate illustrations; • Separate sections in the book explain engineering terms to mathematicians, and operator theory to engineers; • Each chapter concludes with a helpful guide to the literature allowing students to follow up on the topics in the book. Palle E.T. Jorgensen is a Professor of Mathematics at the University of Iowa. This book is based in part on interdisciplinary courses that he has taught over the last several years, and on his work with his current and former students. His most recent book was written jointly with Ola Bratteli and is entitled Wavelets through a Looking Glass, ©2002 Birkhäuser Boston.
Review
From the reviews:
I find that the best texts ? are those drawn from lectures delivered to varied audiences over time. This is such a book ? . it also goes further than many to clear a pathway to success for the diligent reader. ? discusses the content of the recommended works and their specific topical relation to each chapter in concluding sections entitled 'References and remarks'. The additional remarks make all the difference. That form of presentation is why this text has excellent motivation throughout ? ." (Tom Schulte, MathDL, February, 2007)"
Review
From the reviews: "I find that the best texts ... are those drawn from lectures delivered to varied audiences over time. This is such a book ... . it also goes further than many to clear a pathway to success for the diligent reader. ... discusses the content of the recommended works and their specific topical relation to each chapter in concluding sections entitled 'References and remarks'. The additional remarks make all the difference. That form of presentation is why this text has excellent motivation throughout ... ." (Tom Schulte, MathDL, February, 2007) "In this book, an entry in Springer's Graduate Texts in Mathematics series, the author has made the effort to put enough material on probability theory, wavelets and frames, operators, Cuntz algebras, and the Perron-Frobenius theorem ... The illustrations, including graphs of wavelets and signal processing diagrams, are plentiful and illuminating. The multitude of topics in mathematics covered and the links between them that are described in this book mean that there is something for everyone." (Judith A. Packer, Mathematical Reviews, Issue 2008 a)
Synopsis
This text, combining analysis and tools from mathematical probability, focuses on a systematic and novel exposition of a recent trend in pure and applied mathematics. The emphasis is on the unity of basis constructions and their expansions (bases which are computationally efficient), and on their use in several areas: from wavelets to fractals. The aim of this book is to show how to use processes from probability, random walks on branches, and their path-space measures in the study of convergence questions from harmonic analysis, with particular emphasis on the infinite products that arise in the analysis of wavelets. The book brings together tools from engineering (especially signal/image processing) and mathematics (harmonic analysis and operator theory). Key features: * Interdisciplinary focus addressed to a diverse audience of students and workers in a variety of fields, meeting at the crossroads where they merge * Hands-on approach with generous motivation * New pedagogical features to enhance teaching techniques and experience * Includes more than 34 figures with detailed captions, illustrating the main ideas and visualizing the deeper connections in the subject * Separate sections explain engineering terms to mathematicians and operator theory to engineers * Interdisciplinary presentation and approach, combining central ideas from mathematical analysis (with a twist in the direction of operator theory and harmonic analysis), probability, computation, physics, and engineering The presentation includes numerous exercises that are essential to reinforce fundamental concepts by helping both students and applied users practice sketching functions or iterative schemes, as well as tohone computational skills. Graduate students, researchers, applied mathematicians, engineers and physicists alike will benefit from this unique work in book form that fills a gap in the literature.
Synopsis
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. John von Neumann While this is a course in analysis, our approach departs from the beaten path in some ways. Firstly, we emphasize a variety of connections to themes from neighboring fields, such as wavelets, fractals and signals; topics typically not included in a gradu ate analysis course. This in turn entails excursions into domains with a probabilistic flavor. Yet the diverse parts of the book follow a common underlying thread, and to gether they constitute a good blend; each part in the mix naturally complements the other. In fact, there are now good reasons for taking a wider view of analysis, for ex ample the fact that several applied trends have come to interact in new and exciting ways with traditional mathematical analysis as it was taught in graduate classes for generations. One consequence of these impulses from "outside" is that conventional boundaries between core disciplines in mathematics have become more blurred. Fortunately this branching out does not mean that students will need to start out with any different or additional prerequisites. In fact, the ideas involved in this book are intuitive, natural, many of them visual, and geometric. The required background is quite minimal and it does not go beyond what is typically required in most graduate programs."
Synopsis
This book, combining analysis and tools from mathematical probability, focuses on a systematic and novel presentation of recent trends in pure and applied mathematics: the emergence of three fields, wavelets, signals and fractals. The unity of basis constructions and their expansions is emphasized as the starting point for the development of bases that are computationally efficient for use in several areas from wavelets to fractals.
The book brings together tools from engineering and math, especially from signal- and image processing, and from harmonic analysis and operator theory. The presentation is aimed at graduate students, as well as users from a diverse spectrum of applications.
Key features:
A hands-on approach for students, including tutorials and numerous exercises;
Excellent motivation throughout;
New pedagogical features: glossary of terms, their use in mathematics and in engineering, help for cross-audiences, image processing, visual presentation of key algorithms, structure and geometry of big matrix computations, explanation of uses of the theory in applications outside of mathematics;
Includes more than 50 figures with captions, illustrating the main ideas, plus engineering diagrams, graphic renditions of algorithms, and separate illustrations;
Separate sections in the book explain engineering terms to mathematicians, and operator theory to engineers;
Each chapter concludes with a helpful guide to the literature allowing students to follow up on the topics in the book.
Palle E.T. Jorgensen is a Professor of Mathematics at the University of Iowa. This book is based in part oninterdisciplinary courses that he has taught over the last several years, and on his work with his current and former students. His most recent book was written jointly with Ola Bratteli and is entitled Wavelets through a Looking Glass, 2002 Birkhuser Boston.
Synopsis
Combines analysis and tools from probability, harmonic analysis, operator theory, and engineering (signal/image processing) Interdisciplinary focus with hands-on approach, generous motivation and new pedagogical techniques Numerous exercises reinforce fundamental concepts and hone computational skills Separate sections explain engineering terms to mathematicians and operator theory to engineers Fills a gap in the literature
Table of Contents
Preface.- Contents.- Acknowledgments.- Introduction: Measures on Path Space.- List of Figures.- Index of Symbols.- Transition Probabilities: Random Walk.- N0 vs. Z.- A Case Study: Duality for the Cantor Sets.- Infinite Products.- The Minimal Eigenfunction.- Generalizations and Applications.- Pyramids and Operators.- Pairs of Representations of the Cuntz Algebras On and their Application to Multiresolutions.- Appendices: Polyphase Matrices and the Operator Algebra ON.- References.- Comments on Signal/Image Processing Terminology.- Afterword: Computational Math.- List of Names and Discoveries.- General Index.- About the Cover Figure.