11R1. Engineering Applications of Noncummutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups. - GS Chirikjian (Dept of Mech Eng, Johns Hopkins Univ, Baltimore MD) and AB Kyatkin. CRC Press LLC, Boca Raton FL. 2001. 674 pp. ISBN 0-8493-0748-1. $89.95.

Reviewed by AC Buckingham (Center for Adv Fluid Dyn Appl, LLNL, Mail Code L-23, PO Box 808, Livermore CA 94551).

In this book, the authors’ stated and quite apparent aim is to provide a comprehensive, yet essentially self-contained and explicitly informative text and reference work on a mathematically sophisticated, inherently essential branch of analysis familiar to theoretical physics and mathematics students. Here, however, two obviously enthusiastic and effective teachers and applied researchers in physical science, engineering science, and applied mathematics have purposely created a substantial book for training engineers and engineering science students at the upper division or early graduate level. The authors’ emphasis is thereby placed on the instruction of students who are probably not familiar with non-commutative harmonic analysis, generally, as well as lacking the background and preparation in foundation topics such as functional analysis on abstract hyperspaces, algebraic and differential topology, group theory, field theory, etc, more commonly encountered by theoretical physics and mathematics students. It follows that deliberate emphasis is placed on exposition of the title subject by approaching it with methods more familiar to engineering students. To this end, the exposition makes considerable use of linear algebraic matrix theory in physical-coordinate oriented steps proceeding within the framework of differential geometry, but applied to non-abstract configurations readily visualized and familiar to students trained in engineering applications.

The presentation originates with a descriptive overview of the material to be presented and some of the crucial properties and issues associated with the title subject. This is immediately followed by a review of some considerations in well established and familiar engineering applications of commutative harmonic analysis and Fourier transform theory as a foundation for the systematic introduction and instructive discussion of the analysis concepts developed and advanced in this book. Physical coordinate-based examples, descriptions, and many well-planned and positioned figures and sketches enhance the presentation for the student reader’s developing understanding and appreciation. Some of the important fundamental topics treated in the ten essentially developmental chapters include the generation and properties of orthogonal functions on finite intervals; functional mapping, convolution, and functional projection operations; discrete and fast discrete transform procedures; recent advances in discrete polynomial transforms; wavelets and their fundamental scale, translation, and modulation properties; topological properties and parameterizations in plane and spherical projections; description and influence of rotations in 2 and 3 spatial dimensions; appropriately strong emphasis on group theory concepts, symmetries, group representations, and representational theory with examples (some historical) from theoretical physics; harmonic analysis and fast fourier transform methods for specialized motion groups. The mathematical notation adapted and adhered to consistently is easily followed, universally recognized symbolic language from set theory and modern analysis. Seven chapters of interesting and sometimes surprising applications complete the basic body of the text. These include applications in analysis and development of robotics for mobile and stationary, manipulative operations (with several very illustrative photographs from the authors’ own laboratory); image analysis, pattern recognition, and forward as well as inverse tomography for medical imaging and radiation therapy planning and scheduling in medical applications; stochastic processes in control systems design and analysis; Brownian motion and diffusion studies; and the statistical mechanics and analysis of polymeric macromolecular structure dynamics in physical chemistry, to name just a few. Six supplemental appendices are positioned conveniently for the reader’s reference at the end of the general text with useful reminders and information, descriptions, labels, and basic mathematical identities and rules that are used throughout the body of the text.

The presentation is deliberately descriptive and effective for self-instruction. Sections often begin with a brief verbal introduction and explanation of the mathematical relations to follow. These relations are carefully developed with comprehensive steps and most of the matrix operations explicitly identified, followed by verbal explanations connecting the material presented with that which is to follow. The authors avoid the terseness often followed in theoretical physics and mathematics monographs, but pay the price in the added space required to develop (and in many cases) redevelop the material at a different level and in a somewhat different context for reinforcement. Formal mathematical treatment such as the establishment of existence theorems, proofs, and lemmas are avoided. Explicitly developed proofs and the consequences of the development appear only when the issue of clarity and confidence in the results in relation to what has been established appear to warrant them. Each chapter and the six supplemental appendices end with a paragraph or two of summary, reinforcing identification of the primary topics described in the chapter and their role in developing and pursuing the instructional themes of the book. Hundreds of classical historical and newer references are conveniently listed at the end of each chapter or section to which they pertain. There is also a usefully generous, two-level cross index of about 1400 items which is particularly helpful for location, study, and reference of the book’s material.

In the opinion of this reviewer, the authors have accomplished their stated aims. Engineering Applications of Noncommutative Harmonic Analysis is a comprehensive and generally self-contained exposition appropriate for guiding the engineering student to familiarity and, with practice, perhaps competence in an elegant and useful branch of analysis. The drawback, if one exists is that the length and thoroughness of the descriptions required to provide what the authors perceive to be adequate coverage and instruction results in an abundance of material, with some of it revisited at repeated intervals for emphasis, clarity, and at times additional information. While this is an effective device for thorough instruction on unfamiliar topics, it penalizes and may deter the more accomplished individual seeking to use the book for review or fresh perspective rather than instruction. The authors, themselves, point out that the abundance of material is substantially greater than that associated with a single semester or quarter course. They suggest combinations of the material would be adequate for about seven independent semester/quarter courses. Its size and substance precludes carrying the book about casually for occasional inspection as a researcher or a student, but it certainly must be considered as a solid reference addition to personal or institutional libraries. This reviewer welcomes it in his.