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.

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)"

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.

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.

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.