Inverse Problems. by D. W. McLaughlinReview by: Victor BarcilonSIAM Review, Vol. 28, No. 1 (Mar., 1986), pp. 97-98Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030616 .

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BOOK REVIEWS 97

Each chapter in this book starts with an introduction summarizing its contents and ends with a section on notes and references, where information is given about the references used in the chapter, about additional references, and about extensions of the topics covered together with historical notes. This is an excellent way to organize a chapter. It facilitates reading and makes it easy to identify relevant references for further exploration. One suggestion here is that the author should have included more references in the text than he already has, especially in connection with the main results. In this way the reader does not have to read the whole section on the notes and references to find the source of interest; however, this is a mild inconvenience.

In summary, this monograph is a valuable and timely contribution to the systems and control literature and an excellent reference book to have. It contains a wealth of information and constructive methods, as well as being a good source for references on state observers and observer-based controllers. It covers topics ranging from the well- known, mature topics of full and reduced-order full-state observers, to the more recent research topics of adaptive observers. The main criticism of this work is that the coverage of many topics and especially of the basic ones is too brief. And the required background of the reader in order to fully appreciate the results presented and the difficulties involved must be, as a consequence, quite strong. Clarity of exposition and ease of study could be greatly enhanced by more numerous and more complete ex- planations and justifications and by deeper coverage of the basic concepts. Simple illustrative examples, completely absent in this work, could help enhance and clarify the material covered.

PANOS J. ANTSAKLIS University of Notre Dame

Inverse Problems. Edited by D. W. McLAUGHLIN. American Mathematical Society, Providence, RI, 1984. viii + 189 pp. No price given. ISBN 0-8218-1334-X. SIAM- AMS Proceedings, Vol. 14. Inverse problems often arise in situations in which one wishes to infer information

about an inaccessible region. Geophysics provides many instances of such situations because portions of the earth, oceans, and atmosphere cannot be probed from within. Similarly, medicine is another source of inverse problems because of the desirability of diagnosing abnormalities in a noninvading manner. In both of these fields, waves of one sort or another are often used to extend one's reach and to see inside these inaccessible regions. In these cases, the mathematical aspects of the related inverse problems consist of determining the speed of the waves everywhere inside the region of interest from measurements at the surface.

Since closely related inverse problems occur in widely different contexts, the need for workers on various frontiers to get together is even greater than in other disciplines. One such meeting was organized recently by D. W. McLaughlin under the AMS-SIAM auspices. The present book contains the proceedings of this conference.

As a rule, conference proceedings do not make the best-sellers list. They are usually produced under less than ideal conditions and they often contain research which has appeared, or will appear, elsewhere. The present volume is not exempt from these shortcomings. Nevertheless, within these limitations, it provides an excellent window onto the world of inverse problems.

There are fourteen articles arranged under four headings: Geophysical Inverse Problems, Computed Tomography and Inverse Problems in Medicine, Developments in Mathematical Inverse Problems, and Methods of Maximum Information Entropy. The

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98 BOOK REVIEWS

articles are by R. L. Parker; D. C. Stickler; A. M. Cormack; K. T. Smith; J. F. Greenleaf; F. A. Griinbaum; R. G. Newton; G. Eskin, J. Ralston and E. Trubowitz; W. W. Symes; R. V. Kohn and M. Vogelius; C. R. Smith, R. Inguva and R. L. Morgan; J. E. Shore; E. T. Jaynes; J. Skilling and S. F. Gull. All of the articles, which are written by leading experts, are uniformly excellent. Some of them provide an up-to-date summary of a given problem, or an introduction to a specific technique, while others contain the latest results on a particular topic. Because of their rather short lengths, one is not frightened to embark onto the reading of even the most abstruse of the articles.

VICTOR BARCILON University of Chicago

Operational Calculus-A Theory of Hyperfunctions. By K. YOSIDA. Springer-Verlag, New York, 1984. x + 170 pp. $22.00, paper. ISBN 0-387-96047-3. Applied Mathematical Sciences, Vol. 55. It is, of course, a programme of some antiquity to seek to solve differential

equations by reducing them to algebraic equations. One method for this is to use Laplace transformations, and a well-known and powerful technique for implementing this method is Mikusinski's operational calculus. The aim of the present work is "to give a simplified exposition as well as an extension of" this calculus. This is surely a worthwhile task. Before discussing the book, let us recall the key features of the calculus: we do this in our, and not the book's, setting and notation.

Let L',c be the set of complex-valued, Lebesgue measurable functions on [0, x) which are locally integrable. Then Llos is a linear space for the pointwise operations, and it is a commutative algebra for the convolution multiplication

(f *g)(t )= f (t-u)g(u)du (t 0).

The key fact is that Llos is an integral domain. This is Titchmarsh's convolution theorem. (More generally, if a(f ) is the infimum of the support of f, then a(f *g) = a(f ) + a(g).) Let h(t)=1 (t _ 0). Then (h*f )(t) = fot f(u) du, and so h corresponds to integration: clearly, h-1 should correspond to differentiation. The operational calculus provides a setting for this. Simply, take F to be the quotient field of the integral domain Ll'p . Then h- 1 E F.

The Laplace transform of f in Llos is

rOO (Yf )(z) =J f (t) e-zdt.

With luck, this integral converges on a right-hand half-plane in C. But, if f (t) = e', for example, the integral does not converge for any z in C. The operational calculus allows one to circumvent this difficulty-and this is genuinely useful.

The book expounds this material but avoids much of the above, seeking an elementary approach. Basically, the author works with C, the set of continuous func- tions on [0, so), rather than Lo, and his integral is the Riemann integral: this is very reasonable. (Eventually, in ?22, he must consider K the set of piecewise continuous functions on [0, x), however.) The class of "hyperfunctions" is C/C= { fg-1:f,gE C, g 0), a subfield of our field F. But earlier, to avoid using Titchmarsh's theorem, the author uses the smaller subfield CH= { f/hn: fe C, n e N); this seems unnecessarily fastidious.

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