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Analysis and Probability: Wavelets, Signals, Fractals

Palle E. T. Jorgensen
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 234
[Reviewed by
Tom Schulte
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I find that the best texts for self study (and often for coursework) are those drawn from lectures delivered to varied audiences over time. This is such a book: it is based on interdisciplinary courses that the author taught over several years as well as on his work with his current and former students.

Analysis and Probability is part of Springer’s distinctive yellow-covered Graduate Texts in Mathematics series. As such, a firm grounding in at least one subject is required to get the most out the book. In this case, analysis and probability serve as a basis from which to introduce and compare wavelets, signals and fractals. Tools from signal and image processing, harmonic analysis, and operator theory are recurring themes.

While this book is obviously a hands-on approach for the serious student, it also goes further than many to clear a pathway to success for the diligent reader. For instance, while a reference list of cited and relevant works is to be expected, Jorgensen goes a bit further. He 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 and can go from quoting Lewis Carroll to Mandelbrot without becoming breezy.

Consider the outsized glossary of terms. I have long been a fan of glossaries: they provide more help than an index entry with more economy than the pages the index entry points to. Jorgensen’s glossary has a full paragraph on each entry, from Cuntz relations to martingales. In fact, it also serves as a compact overview of the entire work. Of course, the book also has the standard index, more than 50 figures, and a list of references and of symbols used. Each chapter concludes with exercises. While there are occasional hints, no answers are provided.

The focus is on the unity of basis constructions and expansions in terms of a basis, applied to diverse areas, from wavelets to fractals. For example, a methodology drawn from probability, random walks on trees and their path-space measures, is used to tackle harmonic analysis and the infinite products arising in analysis of wavelets and fractals.

This work resides at the crossroads of signal and image processing engineering with the mathematics of harmonic analysis and operator theory. Chapters are consistently laid out, supported by the bookends of prerequisites, terminology, and other preliminaries at the beginning and conclusions, exercises, and references on the tail end. In between are such notable chapters as a case study on duality for Cantor sets, the minimal eigenfunction, and refining localization through the use of pyramid algorithms.

Appendices touch on filters for signals, Hilbert spaces, and more. An Afterward section ties concepts to practical applications such as image file formats (JPEG, GIF), aliasing, Parseval frames, and computational mathematics. The target audience is graduate students, researchers, engineers, applied mathematicians, operator theorists, communications circuit designers and signal/image processing specialists.

Tom Schulte waves and writes from Michigan, where he is a graduate mathematics student at Oakland University.

 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.