E Ward Cheney wrote seven books and, in addition, edited two conference proceedings. Below we give extracts from Prefaces and reviews of the seven books.

**1. Introduction to approximation theory (1966), by E Ward Cheney.**

**1.1. From the Preface.**

In writing this book I have sought to bring a large segment of approximation theory within the comfortable grasp of undergraduate and beginning graduate mathematics students. This goal would not be a reasonable one were it not for the fact that traditionally much of the activity in this discipline has been directed toward uniform approximation on the real line, and that many of the most appealing results in the theory are based upon elementary real-variable arguments. Nevertheless, this pedagogical framework dictates a certain bias in the choice of topics. In particular, approximation with integral norms has been deemphasized in favour of uniform (or "Tchebycheff") approximation. This has made it unnecessary to presuppose the Lebesgue integral or measure theory. Further, in order not to require of the reader a knowledge of complex function theory, the discussion has been directed to problems involving only real-valued functions of real variables. We do not presuppose any "modern" algebra except elementary linear space theory. What

*is*prerequisite of the reader is a familiarity with such topics as sequences, vector spaces, series, uniform convergence, continuity, and the mean-value theorem - all of which are normally acquired in a good calculus course. If he is thus equipped, the reader should find the book self-contained. As a practical matter, a course in advanced calculus or real analysis would smooth the way. Since some material in the text (as, for example, the Bolzano-Weierstrass theorem) is included solely for completeness, the reader should be prepared to skip boldly over what is familiar to him. ... A preliminary version of the book was prepared during the 1961 and 1962 summer institutes in numerical analysis which were held at the University of California (Los Angeles) under the National Science Foundation sponsorship.**1.2. Review by: Theodore Rivlin.**

*Math. Comp.*

**23**(106) (1969), 443-444.

This eminently readable book is intended to be used as a text for a first course in approximation theory. Uniform approximation of functions is emphasized and the discussion is not only theoretical, but provides usable algorithms as well. An introductory chapter presents some of the major theoretical tools, and as- signs an important role to convexity considerations. There follow chapters on the Chebyshev solution of inconsistent linear equations and Chebyshev approximation by polynomials and other linear families. The next chapter treats least-squares approximation and related topics. The scene then shifts back to uniform approximation by rational functions. The final chapter offers a miscellany of topics too interesting to be omitted from the book. Throughout the book the author provides interesting proofs and occasionally new approaches of his own.

**1.3. Review by: Paul Leo Butzer.**

*Mathematical Reviews*MR0222517

**(36 #5568)**.

In this book, which is intended to be an introduction to the subject, the author steers a middle course between the various viewpoints. On the one hand, he presents his material within the framework of (elementary) functional analysis ..., and on the other hand he treats various algorithms which prepare the way for the numerical solution of various types of approximation problems. One of the highlights of the present book is Chapter V on rational approximation which is an important case of non-linear approximation. This is a modern field of research in the subject to which the author and his colleagues have contributed substantial results. ... The author has provided a usable and very versatile text which is certainly to be recommended.

**2. Numerical Mathematics and Computing (1980), by David Kincaid and Ward Cheney.**

**2.1. From the Preface.**

Our basic objective is to acquaint students of science and engineering with the potentialities of the modern computer for solving numerical problems that may arise in their professions. A secondary objective is to give students an opportunity to hone their skills in programming and problem solving. A final objective is to help students arrive at an understanding of the important subject of

*errors*that inevitably accompany scientific computing, and to arm them with methods for detecting, predicting, and controlling these errors. Much of science today involves complex computations built upon mathematical software systems. The users may have little knowledge of the underlying numerical algorithms used in these problem-solving environments. By studying numerical methods one can become a more informed user and be better prepared to evaluate and judge the accuracy of the results. What this implies is that students should study algorithms to learn not only how they work but also how they can fail. Critical thinking and constant scepticism are attitudes we want students to acquire. Any extensive numerical calculation, even when carried out by state-of-the-art software, should be subjected to independent verification, if possible. Since this book is to be accessible to students who are not necessarily advanced in their formal study of mathematics and computer sciences, we have tried to achieve an elementary style of presentation. Toward this end, we have provided numerous examples and figures for illustrative purposes and fragments of pseudocode, which are informal descriptions of computer algorithms. Believing that most students at this level need a*survey*of the subject of numerical mathematics and computing, we have presented a wide diversity of topics, including some rather advanced ones that play an important role in current scientific computing. We recommend that the reader have at least a one-year study of calculus as a prerequisite for our text. Some knowledge of matrices, vectors, and differential equations is helpful.**3. Approximation theory in tensor product spaces (1985), by W A Light and E W Cheney.**

**3.1. From the Preface.**

In the past two decades, a new branch of approximation theory has emerged; it concerns the approximation of multivariate functions by combinations of univariate ones. The setting for these approximation problems is often a Banach space which is the tensor product of two or more simpler spaces. Approximations are usually sought in subspaces which are themselves tensor products. While these are infinite dimensional, they may share some of the characteristics of finite-dimensional subspaces. The usual questions from classical approximation theory can be posed for these approximating subspaces, such as (i) Do best approximations exist? (ii) Are best approximations unique? (iii) How are best approximations characterized? (iv) What algorithms can be devised for computing best approximations? (v) Do there exist simple procedures which provide "good" approximations, in contrast to "best" approximations? (vi) What are the projections of least norm on these subspaces? and (vii) what are the projection constants of these subspaces? This volume surveys only a part of this growing field of research. Its purpose is twofold: first, to provide a coherent account of some recent results; and second, to give an exposition of the subject for those not already familiar with it. We cater for the needs of this latter category of reader by adopting a deliberately slow pace and by including virtually all details in the proofs. We hope that the book will be useful to students of approximation theory in courses and seminars.

**3.2. Review by: Werner Haussmann.**

*Mathematical Reviews*MR0817984

**(87g:41064)**.

This book is concerned with approximation of multivariate functions by certain combinations of univariate ones. The setting for these approximation problems is a Banach space which is the tensor product of two or more simpler spaces. Approximations are sought in subspaces which are themselves built up by tensor products. While these subspaces are infinite-dimensional, they also share some of the characteristics of finite-dimensional ones. ... The purpose of this book is twofold: first, to provide a coherent account of some recent results, and second to give an exposition of the subject for those not already familiar with it. ... There has been a real need for a book in this area, and the reviewer is convinced that this book will be welcomed by both experts and advanced students interested in this field.

**4. Multivariate approximation theory (1986), by E Ward Cheney.**

**4.1. From the Preface.**

These notes provide an account of lectures given at a Regional Conference on Approximation Theory and Numerical Analysis that was sponsored by the Conference Board of the Mathematical Sciences and supported financially by the National Science Foundation. The host institution was the University of Alaska in Fairbanks, which provided not only all facilities but also additional financial support. ... My principal objective in the lectures and in these notes was to describe the current status of several branches of multivariate approximation theory and, if possible, to entice more mathematicians into undertaking research on these matters. I especially had in mind the topics of best approximation, algorithms, and projection operators, since these topics are rife with challenging problems. As part of the survey, I tried to point out the many gaps in the current body of knowledge and to furnish copious references. The central theme is the perennial problem of "best approximation," usually formulated in a normed linear space whose elements are functions of several real variables. First we ask, "What subclasses of functions are suitable for approximating other functions?" Here interest focuses naturally on functions that are simple combinations of univariate functions. The important tensor-product subspaces play the principal role here because of their simple linear structure. One chapter is devoted to introductory material on the tensor product of Banach spaces, as seen from the perspective of approximation theory. I extend my thanks to all the participants in the conference, for they cheerfully suffered the lectures and offered interesting points of view in addition to stimulating questions.

**4.2. Review by: Frank Deutsch.**

*Mathematical Reviews*MR0862115

**(88k:41003)**.

The author, in the book under review, makes a convincing case for the use of tensor products in studying multivariate approximation. In fact, most of the topics discussed can be easily phrased using the language of tensor products. To assist the novice, he has included a readable account of the fundamentals of tensor products. Although only eight pages, this chapter provides an excellent introduction to tensor products and motivates the topics covered. ... The author has woven a pretty tapestry of some important topics in multivariate approximation. The main thread which seems to hold it all together is the tensor product. While the presentation is mainly expository with few proofs given, there is a copious supply of references listed for the reader eager for more detail. ... There are three ingredients necessary for a successful research monograph. It should be enjoyable to read, it should teach, and it should inspire further research. I believe Cheney's book scores highly on all three points.

**5. Numerical analysis: Mathematics of Scientific Computing (1991), by David Kincaid and Ward Cheney.**

**5.1. From the Preface.**

This book has evolved over many years from lecture notes that accompany certain upper-division and graduate courses in mathematics and computer sciences at The University of Texas al Austin. These courses introduce students to the algorithms and methods that are commonly needed in scientific computing. The mathematical underpinnings of these methods are emphasized as much as their algorithmic aspects. The students have been diverse: mathematics, engineering. science, and computer science undergraduates, as welt as graduate students from various disciplines. Portions of the book also have been used to lay the groundwork in several graduate courses devoted to special topics in numerical analysis. such as the numerical solution of differential equations, numerical linear algebra. and approximation theory. Our approach has always been to treat the subject from a mathematical point of view, with attention given to its rich offering of theorems, proofs, and interesting ideas. From these arise many computational procedures and intriguing questions of computer science, Of course, our motivation comes from the practical world of scientific computing, which dictates the choice of topics and the manner of treating each. For example, with some topics it is more instructive to discuss the theoretical foundations of the subject and not attempt to analyze algorithms in detail. In other cases, the reverse is true, and the students learn much from programming simple algorithms themselves and experimenting with them - although we offer a blanket admonition to use well-tested software. such as from program libraries. on problems that arise from applications.

**5.2. Review by: Carroll O Wilde.**

*Mathematical Reviews*MR1099375

**(92c:65002)**.

The present book is intended for a scholarly and mathematically detailed course on the subject for upper division undergraduate mathematics, engineering, and computer science majors. Chapter topics are mathematical preliminaries, computer arithmetic, solution of nonlinear equations, systems of linear equations, eigenvalues and singular-value decomposition, approximation of functions, differentiation and integration, ordinary differential equations, partial differential equations, and linear programming. Several of the topics are unusual in this context, including homotopy and continuation methods, multivariate interpolation of scattered data, Sard's theory of approximating functionals, Bernoulli polynomials and the Euler-Maclaurin formula, and the multigrid method for partial differential equations.

**5.3. Review by: E I.**

*Mathematics of Computation*

**59**(199) (1992), 297-298.

All that is required to understand the mathematics of scientific computing, its algorithms, and its software libraries can be found in the ten chapters of this scholarly exposition. The authors designed a compact, attractive, uncrowded text that contains a wealth of material, by skilfully using TEX to prepare the manuscript. The subject matter has been tested through many years of re- search, classroom use, and practical computing experience-during which the algorithms have been refined and precisely defined in "pseudocode." ... The mathematical treatment is suitable for upper-level undergraduate and first-year graduate students who can work with the ideas of analysis that are so carefully presented here. Other readers may compensate for a not so thorough mathematical background, provided they have the skill to work with computers and are motivated to learn from the many computer exercises that are listed in the ample problem sets.

**5.4. Review of 2nd edition by: Robert Frederick Almgren.**

*Mathematical Reviews*MR1388777

**(97g:65003)**.

This is a textbook aimed at upper-level undergraduate students who have a reasonably sophisticated mathematical background, and who are interested in the mathematical underpinnings of the subject. By comparison with most undergraduate textbooks on numerical analysis, and with the authors' other more elementary textbook, a large amount of material is covered, and in reasonable depth. The book might also be suitable for a beginning graduate course for physical science students. Good textbooks in this area are scarce, and this one is better than several that I have tried to use. The book covers the standard topics of numerical analysis: interpolation and approximation; differentiation and integration; linear algebra, including both solution of linear systems and eigenvalue problems; and time evolution problems, including both ordinary and partial differential equations. It covers a few additional topics not usually found in beginning textbooks: homotopy and continuation methods, multivariate interpolation, multigrid methods, and linear programming. Especially welcome is the material on the iterative solution of linear systems, and the chapter on "Selected topics in numerical linear algebra,'' which covers eigenvalues, the singular-value decomposition, and QR factorization.

**6. A course in approximation theory (1999), by W A Light and E W Cheney.**

**6.1. From the Preface.**

This book offers a graduate-level exposition of selected topics in modern approximation theory. A large portion of the book focuses on multivariate approximation theory, where much recent research is concentrated. Although our own interests have influenced the choice of topics, the text cuts a wide swath through modern approximation theory, as can be seen from the table of contents. We believe the book will be found suitable as a text for courses, seminars, and even solo study. Although the book is at the graduate level, it does not presuppose that the reader already has taken a course in approximation theory. A central theme of the book is the problem of interpolating data by smooth multivariable functions. Several chapters investigate interesting families of functions that can be employed in this task; among them are the polynomials, the positive definite functions, and the radial basis functions. Whether these same families can be used, in general, for approximating functions to arbitrary precision is a natural question that follows; it is addressed in further chapters. The book then moves on to the consideration of methods for concocting approximations, such as by convolutions, by neural nets, or by interpolation at more and more points. Here there are questions of limiting behaviour of sequences of operators, just as there are questions about interpolating on larger and larger sets of nodes. A major departure from our theme of multivariate approximation is found in the two chapters on univariate wavelets, which comprise a significant fraction of the book. In our opinion wavelet theory is so important a development in recent times - and is so mathematically appealing - that we had to devote some space to expounding its basic principles. In style, we have tried to make the exposition as simple and clear as possible, electing to furnish proofs that are complete and relatively easy to read without the reader needing to resort to pencil and paper. Any reader who finds this style too prolix can proceed quickly over arguments and calculations that are routine. To paraphrase Shaw: We have done our best to avoid conciseness! We have also made considerable efforts to find simple ways to introduce and explain each topic. We hope that in doing so, we encourage readers to delve deeper into some areas. It should be borne in mind that further exploration of some topics may require more mathematical sophistication than is demanded by our treatment.

**6.2. Review by: G E Fasshauer.**

*Amer. Math. Monthly*

**111**(5) (2004), 448-452.

Finally, someone who was up to the challenge. Ward Cheney and Will Light, the authors of A Course in Approximation Theory, have managed to fill three major gaps in the existing literature on multivariate approximation by daring to become pioneers and writing a very special book. They have produced the first book surveying such a large variety of topics in this vast and active area of mathematical research, and the first one with an extensive treatment of meshfree approximation. Moreover, they wrote it as a textbook rather than a research monograph. Approximation theory is a rather specialized area of mathematics that is practiced as a stand-alone subject by only a small number of mathematicians. However, the issues investigated in approximation theory are of fundamental importance for a wide array of applications ranging from numerical analysis and the solution of PDEs in many areas of science and engineering to computer graphics, data mining, and artificial intelligence. Therefore, approximation theory is actually practiced to some extent by a wide variety of mathematicians, scientists, and engineers. ... I admire and thank Ward Cheney and Will Light for writing this book, especially since they produced not just a research monograph about their own work but a textbook covering a wide array of topics in modern approximation theory. Let's hope that this book will inspire others to follow in their footsteps.

**6.3. Review by: Mircea Ivan.**

*Mathematical Reviews*MR2474372

**(2010d:41001)**.

The purpose of this textbook, written for both research mathematicians and graduate students in the fields of mathematics, physics, engineering and computer science, is to guide the reader to learning about new developments in approximation theory that have come up over the last 20 years. The book splits into 36 chapters, among which Linear Interpolation Operators, Multivariate Polynomials, Completely Monotone Functions, Hilbert Function Spaces and Reproducing Kernels, Wavelets and Quasi-Interpolation, to mention a few. ... The textbook, a clear and concise work written by world-renowned experts in the field of approximation theory, will prove useful not only as a reference for professional mathematicians but also as a text for graduate students.

**7. Analysis for applied mathematics (2001), by Ward Cheney.**

**7.1. From the Preface.**

This book evolved from a course at our university for beginning graduate students in mathematics - particularly students who intended to specialize in applied mathematics. The content of the course made it attractive to other mathematics students and to graduate students from other disciplines such as engineering, physics, and computer science. Since the course was designed for two semesters duration, many topics could be included and dealt with in detail. Chapters 1 through 6 reflect roughly the actual nature of the course, as it was taught over a number of years. The content of the course was dictated by a syllabus governing our preliminary Ph.D. examinations in the subject of applied mathematics. That syllabus, in turn, expressed a consensus of the faculty members involved in the applied mathematics program within our department. The text in its present manifestation is my interpretation of that syllabus: my colleagues are blameless for whatever flaws are present and for any inadvertent deviations from the syllabus. The book contains two additional chapters having important material not included in the course: Chapter 8, on measure and integration, is for the benefit of readers who want a concise presentation of that subject, and Chapter 7 contains some topics closely allied, but peripheral, to the principal thrust of the course. This arrangement of the material deserves some explanation. The ordering of chapters reflects our expectation of our students: If they are unacquainted with Lebesgue integration (for example), they can nevertheless understand the examples of Chapter 1 on a superficial level, and at the same time, they can begin to remedy any deficiencies in their knowledge by a little private study of Chapter 8. Similar remarks apply to other situations, such as where some point-set topology is involved ... To summarize: We encourage students to wade boldly into the course, starting with Chapter 1, and, where necessary, fill in any gaps in their prior preparation. One advantage of this strategy is that they will see the necessity for topology, measure theory, and other topics - thus becoming better motivated to study them. In keeping with this philosophy, I have not hesitated to make forward references in some proofs to material coming later in the book.

**7.2. Review by: Nick Lord.**

*The Mathematical Gazette*

**87**(509) (2003), 394.

Ward Cheney ... has enjoyed a distinguished career in both academia and industry: indeed his earlier book 'Introduction to approximation theory' (1985) will be familiar to many readers. Analysis for applied mathematics grew out of a beginning graduate course for intending applied mathematicians at the University of Texas. This course has a dual bridging role - from undergraduate mathematics to post-graduate mathematics and from undergraduate mathematics towards applications. Its content, which has stabilised over a number of years, consists of those parts of functional analysis and its adjuncts which impinge on modern applications of mathematics. ... This is a very readable, well-organised text: the applied context gives a unifying narrative thrust with a pragmatic emphasis on what the key ideas are and what they may be used for. But the focus is squarely on modem applied mathematics: to those readers for whom it may strike as rather pure in tone and content, Cheney offers the following advice: 'A look at the past would certainly justify my favourite algorithm for creating an applied mathematician: Start with a pure mathematician, and turn him or her loose on real-world problems.'

**7.3. Review by: David G Schaeffer.**

*Amer. Math. Monthly*

**110**(6) (2003), 550-553.

Resolved: The training of an applied mathematics graduate student should include a thorough grounding in rigorous analysis, including functional analysis. If this proposition were to be debated at a professional meeting, would you support the pro or the con position? For those who support such training, either mandatory or optional, this original book is intended to supply the means. It is intended for use by first-year graduate students. The book's eight chapters cover the following topics: 1. Normed linear spaces 2. Hilbert spaces 3. Calculus in Banach spaces 4. Basic approximate methods of analysis 5. Distributions 6. The Fourier transform 7. Additional topics 8. Measure and integration. The choice of topics and the order of their presentation clearly received careful attention. Organizing this amount of material into a logical form that students can read is a substantial task. Despite the frustrations expressed in what follows, I consider this book a worthwhile addition to the instructional literature. I suspect the book would be a stretch for most first-year students. The exposition of concepts is often rather terse. ... Of course, if the book is used as a textbook for a course, the instructor could ease the students past these difficulties. The book is also suitable for independent study by a more advanced student - say, one preparing for a Ph.D. preliminary exam. ... My own position on the opening proposition of this review is a qualified con. I believe that a grounding in rigorous analysis enhances the power of an applied mathematician, but other training - especially acquiring an insider's knowledge of a secondary scientific field - is at least as valuable. Moreover, some applied mathematics students may be repelled by an intense focus on proofs. I personally would not require rigorous analysis beyond multiple-variable calculus, but I do strongly encourage training beyond this level. Despite my disagreements with Cheney's book, I do not know of a text for such a course better suited to my tastes. Moreover, even in the unlikely event that I summoned the level of energy needed to write such a book myself, I sadly recognize that, six months after it was finished, I would probably find many faults with it.

**7.4. Review by: Karl K Knaub.**

*SIAM Review*

**44**(3) (2002), 494-496.

Applied mathematicians and graduate programs in applied mathematics come in many different varieties, some more mathematically rigorous than others. My experience as a graduate student was in a program that emphasized problem solving rather than mathematical proof; however, a growing number of students in the pro- gram, including myself at the time, sought courses in analysis or formed reading groups to study the subject on our own. Many of us quickly came up against theoretical issues in our research, or weren't sufficiently "impure" to avoid worrying about the foundations of our work. In light of the connections between analysis (in its real, complex, and functional flavours), several authors have written texts attempting to provide the material in analysis that new (or newly interested) applied mathematicians are likely to need in their research. A new entry in this field is 'Analysis for Applied Mathematics' by Ward Cheney. This book grew out of a course designed for beginning graduate students specializing in applied mathematics at Cheney's institution, the University of Texas at Austin. The book succeeds well in providing a sound basis in analysis beyond that which most students will have received as undergraduates ... Overall, this text serves as a wonderful introduction to "useful" analysis for new applied mathematicians.

**7.5. Review by: Robert Gardner Bartle.**

*Mathematical Reviews*MR2474372

**(2010d:41001)**.

he author describes this marvellous book as designed for beginning graduate students in mathematics - in particular for those who intend to specialize in applied mathematics, and for graduate students in other disciplines such as engineering, physics and computer science. The first six chapters contain enough material for a year course, and the final two chapters contain related material (such as topology, measure theory and integration theory) which are mentioned in the earlier portions. The author emphasizes that this book "is

*not*about applied mathematics, but rather about topics in analysis that impinge on applied mathematics''. Those who are familiar with the author's earlier books will not be surprised by its excellence. It is business-like and will be found to be demanding, but it is user-friendly. It is the reviewer's opinion that it will be extremely useful and popular as a text; institutions that do not already require their students to take such a course no longer have an excuse, and should immediately organize one based on this book. ... The author gives a rigorous discussion of a very large number of topics. Careful definitions and clearly stated theorems are given. Most results are proved in detail, but when a result is not proved in detail, a reference is given. Almost every subsection contains exercises; some contain a large number. The writing is clear and readable throughout, with a pleasant and friendly style. While there is room for debate about the subjects that are included or omitted, it is the reviewer's view that the author has presented a rich harvest of useful and interesting analysis in a splendid manner.