**Aufgaben und Lehrsätze aus der Analysis (2 Vols.) (1925), by George Pólya and Gabor Szegö.****1.1. Review by: Anon.**

*The Mathematical Gazette***13**(183) (1926), 169-170.These two volumes of examples in analysis are numbered XIX and XX respectively in the 'Grundlehren der mathematischen Wissenschaften' series, which is producing so many interesting works. The volumes under notice together form an organic whole. They are the result of collaboration of writers whose work lies in Zurich and Berlin respectively, while Professor Polya himself is a member of the London Mathematical Society. To judge from the contents of the books they represent an enormous teaching experience over a wide range of advanced subjects, generally comprised under the title Analysis: and one is not surprised to find in the preface a long list of the names of most of the eminent Continental mathematicians who have in one way or another materially helped in the compilation and verification of the subject-matter. To these many helpers the authors acknowledge their indebtedness; but the chief feature of the whole book, which only reveals itself with systematic reading, is the undoubted skill shown in the choice and grouping of the examples. The care and thought with which this is done, render the authors' own contribution something far different and more important than merely marshalling disjointed exercises into one great compendium. ... The book presupposes a certain elementary knowledge of Analysis. Probably its most useful field, at any rate in England, would be among all who are called upon to teach or lecture to Honours students. It is essentially stimulating and suggestive. It has a peculiar atmosphere of making the reader - who, of course, has to read with a pen in his hand - ask himself again and again, "Have I really grasped the underlying thought below these exercises?" Most of us are content to solve a problem by any means, and only the rare spirits make it their habit to criticise their style amid the welter of analytical technicalities. The claim that is modestly made in the preface (which itself is extremely interesting reading) that the book ushers in an entirely new mode of presenting examples for solution, would seem to be true. The book can be warmly recommended.

**Orthogonal Polynomials (1939), by Gabor Szegö.****2.1. Review by: Edward T Copson.**

*The Mathematical Gazette***24**(258) (1940), 66-67.The general theory of orthogonal polynomials had its origin in the theory of continued fractions of Stieltjes' type and the associated problem of moments. Although much progress has recently been made in the latter subject, the historical line of approach has been gradually abandoned and the orthogonal property itself has been taken as fundamental; and this is the point of view which Professor Szegö adopts. He gives a new and detailed development of the main ideas of the theory of orthogonal polynomials and leads up to a very interesting account of recent investigations concerning asymptotic expansions and the distribution of zeros. He applies the general theory to special classes of orthogonal polynomials, in particular those satisfying linear differential equations of the second order, for which more exhaustive results are obtained. ... The whole work is characterised by the clarity of exposition and the sustaining of interest which one expects of one of the authors of "Pólya-Szegö".

**2.2. Review by: James Alexander Shohat.**

*Bull. Amer. Math. Soc.***46**(7) (1940), 583-587.The book under review gives in 400 pages a detailed and systematic treatment of the theory and applications of Orthogonal Polynomials. It offers a very lucid and elegant exposition of the subject, to which Szegö himself made so many contributions. It is by no means a compilation of results already known. It presents much material which is new and important; many old results are presented in a novel setting: more precise or more general statements, new proofs. The author tried to bring the book as much up-to-date as possible and has generally succeeded. ... It goes without saying that the writer of a scientific book is the supreme judge in choosing its content. It is only natural for him to favour his own brain-children and, from amongst the works of others, those closely related to his own. In the Introduction the author disclaims completeness of treatment. This disarming remark is necessary but not sufficient. In a book of this kind one would like to find as complete a treatment as possible, preparing the reader for future progress in various branches of the subject. The omission of the problem of moments is, we believe, very regrettable. ... the beautifully printed book of Szegö is an excellent addition to the Colloquium Publications of the Society. It is a remarkable source of powerful methods and far-reaching results in an important field, which will inspire and stimulate further research.

**2.3. Review by: Dunham Jackson.**

*Science, New Series***91**(2370) (1940), 526.Twenty-five years ago a theory of orthogonal polynomials would have been made up of apparently heterogeneous elements, more or less forcibly dragged together from their natural context of general function theory, differential equations, integral equations, continued fractions, mathematical physics or statistics, and ranging in mathematical development from the highest degree of analytical perfection to the most naive formalism. Now a depth of critical understanding which scarcely went beyond the fundamental cases of Fourier and Legendre series has come to prevail with unifying authority over a wider range of generalization than had been even tentatively surveyed, and the diverse fields into which the applications extend derive clarification from a common body of coordinated knowledge. To this transformation no one man has made more significant contributions than the author of the book under review. With creative mastery in particular domains he combines an unusually extensive and penetrating acquaintance with the whole background of mathematical analysis, and almost unique experience in presenting the essentials of complicated theories with the greatest possible compactness. The result is a treatise which will be of the greatest value both to the general student and to those seeking more specialized information. A large amount of previously existing material is brought together and made readily accessible, much of it for the first time, new results are presented in their proper place, and the way is prepared for further research in various directions.

**2.4. Review by: James Alexander Shohat.**

*Mathematical Reviews*, MR0000077**(1,14b)**.This is the first detailed systematic treatment of orthogonal polynomials .... As stated in the Preface, no claim is made for completeness. Thus, very little space is devoted to the problem of moments and to the relation of orthogonal polynomials to continued fractions. However, these omissions are compensated by several interesting features: (a) an elaborate treatment of the asymptotic behaviour of orthogonal polynomials, by various methods, with applications, in particular, to the "classical" polynomials of Legendre, Jacobi, Laguerre and Hermite; (b) a detailed study of expansions in series of orthogonal polynomials, regarding convergence and summability; (c) a detailed study of orthogonal polynomials in the complex domain; (d) a study of the zeros of orthogonal polynomials, particularly of the classical ones, based upon an extension of Sturm's theorem for differential equations. The book presents many new results; many results already known are presented in generalized or more precise form, with new simplified proofs.

**Isoperimetric Inequalities in Mathematical Physics (1951), by George Pólya and Gabor Szegö.****3.1. Review by: Arthur Rosenthal.**

*Science New Series***115**(2980) (1952), 155.The classical "isoperimetric inequality" states that for any closed plane curve with given perimeter the area is not greater than the area of the circle with the same perimeter. There are many interesting inequalities of a similar type concerning geometrical or physical quantities that depend on shape and size of curves or solids. Such inequalities are the subject of this book. The general aim is the possibility of estimating certain physical quantities, in which physicists or engineers are interested, by means of geometrical, or other easily accessible quantities. Much work has been done in this direction during the past few decades and, in particular, the authors themselves had already made valuable contributions. In the present book a careful and systematic discussion and a well-organised presentation of these questions are given.

**3.2. Review by: Marcel Brelot.**

*Mathematical Reviews*, MR0043486**(13,270d)**.The book brings together the known results, many of which are due to the authors, by reworking, perfecting, developing the questions of Geometric Extrema, sometimes old and more or less known or previously treated: old isoperimetric problems, maximum rigidity of the torsion of a fixed section area of a cylinder, minimum electrostatic capacity of a body of given volume, minimum fundamental frequency of membranes or plates of a given area, .... These questions considered in recent decades by various authors lead to many inequalities, called here isoperimetric as a generalisation of the famous problem; there are given in addition, because of the difficulty of exact calculations, very practical solutions, compared with special cases for which one can give useful explicit formulas or numerical results and which the authors detail in tables.

**Toeplitz Forms and Their Applications (1958), by Ulf Grenander and Gabor Szegö.****4.1. Review by: Mark Kac.**

*Science, New Series***128**(3316) (1958), 137-138.The present volume is an excellent and virtually complete summary of the work done on [Toeplitz Forms

**]**and related problems up to 1955. Toeplitz matrices and translation kernels occur in a wide variety of branches of pure and applied mathematics, ranging from the theory of analytic functions to crystal statistics and the theory of random noise. To present such wealth of material in the limited space of 240 small pages is a feat in itself. To do it with such skill and elegance should earn the authors the gratitude of all those who might wish to gain acquaintance with this fascinating field. ... Of course, this is a technical book on a technical subject. But discounting this and even my own strong prejudices in favour of the subject, I recommend the book as an outstanding example of the power and beauty of analysis.**4.2. Review by: Frank Ludvig Spitzer.**

*Bull. Amer. Math. Soc.***65**(2) (1959), 97-101.This book owes its timeliness, and much of its importance and unique charm to one particular quality which sets it apart from other research monographs. Its two authors have accomplished a successful synthesis of two important mathematical developments. One of these is the theory of Toeplitz forms, the other, more recent one, the theory of (wide sense) stationary stochastic processes. The theory of Toeplitz forms has its roots in the work of Toeplitz, Féjèr, Carathéodory, F Riesz on trigonometric series and harmonic functions. In two important papers Gábor Szegö unified and extended much of their work by creating a theory the central ideas and results of which also form the core of the present book. In other words, concepts and methods created forty years ago have now gained new interest as the analytical techniques in a branch of mathematics (prediction and estimation theory for stationary stochastic processes) which did not then exist. This remark could not have been made in 1939, when a brief account of the theory of Toeplitz forms first appeared in Szegö's book

*Orthogonal polynomials,*but in any event the present treatment goes further and deeper. ... The mathematical presentation is of the same high calibre as in Szegö's*Orthogonal polynomials,*but even more elegant because the subject matter here is so much more unified. Most of the background theory required in the book is relegated to an introductory chapter. Therefore there are no digressions from the natural development of the theory, and this has enabled the authors to write in a terse but at the same time pleasingly informal style. Not only good students but also serious research workers may find this book difficult if they want to fully bridge the conceptual gap between the two fields which are unified here. But as the book offers so much more than would two separate monographs in the corresponding subjects of analysis and probability, this is precisely the challenge it offers to the reader. The content of the book is evidence enough that this challenge will contribute to the growth of mathematics.**4.3. Review by: Leonard J Savage.**

*J. Amer. Statist. Assoc.***53**(283) (1958), 763.This is a by no means popular monograph in a topic in pure mathematics. It is mentioned here because of its relevance for serious students of (weakly) stationary stochastic processes. ... Beginning with Toeplitz in 1910, there has been a ramified and subtle analytical theory of Toeplitz forms in which Szegö has been one of the principal workers. Interest and progress in the theory have re-blossomed in the last decade or so mainly in consequence of the pioneering work of A N Kolmogoroff and N Wiener on linear prediction and filtering. The book is carefully but dryly written. As often happens with mathematics books, the background required to read it is nominally quite small but actually very considerable. The book will be of direct use for research and advanced study in an important area and promises thus to benefit the whole statistical community.

**4.4. Review by: Michel Loève.**

*Mathematical Reviews*, MR0094840**(20 3 1349)**.Various remarkable problems of the modern theory of functions and theory of probability and statistics may be centred about the study of certain Hermitian forms .... This book unifies the whole subject and frequently completes the known answers. ... The bringing together of the many problems centred about the same basic instrument is invaluable.

**Aufgaben und Lehrsätze aus der Analysis. Dritte Berichtigte Auflage (1964), by George Pólya and Gabor Szegö.****5.1. Review by: Jacob Lionel Bakst Cooper.**

*The Mathematical Gazette***49**(370) (1965), 465.Since their first publication in 1925 these books have served more than one generation of mathematicians as an introduction to important parts of Classical Analysis and number theory: the theory of infinite series; integral calculus, in particular inequalities and asymptotic expansions; functions of a complex variable, for instance geometric function theory and maximal principles; zeroes; polynomials; Determinants and Quadratic forms; number theory; and a little geometry. The virtue of the book is that each section, after a brief introduction to the subject, consists of a series of problems so arranged as to make them of reasonable difficulty, as a whole, and an excellent means of thinking one's way into the subject, together with a very good crib for the occasions when the problems are too hard. The present edition differs at most trivially from the first: the only differences I have detected concern some minor typographical errors. The production is excellent.

**5.2. Review by: Ralph Philip Boas Jr.**

*Mathematical Reviews*, MR0170985**(30 3 1219a)**and**(30 3 1219b)**.Older readers will not need to be reminded of the arrangement of this work: successive chains of related problems, with solutions at the back of each volume. With this technique the authors are able to carry the reader surprisingly far, for instance, from money-changing to summability, from complex numbers to the Phragmén-Lindelöf principle. ... The introduction is full of wise and witty remarks that should be read by all mathematicians.

**Orthogonal Polynomials (3rd edition) (1967), by Gabor Szegö.****6.1. Review by: Jacob Burlak.**

*Mathematical Reviews*, MR0310533**(46 3 9631)**.The second edition of this book differed non-trivially from the first - apart from changes in the text there were three new sections (Further Problems and Exercises, Appendix, and Further References), resulting in about 20 pages of new material. The changes in the third edition are less extensive but they can be found, scattered through the text and reflected in new references. Despite the author's disclaimer ("the aim has purposely been to make the material suggestive rather than exhaustive") this book immediately became the standard treatise in the field. There has been a great deal of progress in recent years, which still awaits comparable systematization; there has also been great activity by an army of subscript hunters and formula manipulators, which has little or no bearing on the subject of this book; it is not surprising then that the book is still indispensable to anyone working in the field.

**Problems and Theorems in Analysis, Vol. 1: Series, Integral Calculus, Theory of Functions (1972), by George Pólya and Gabor Szegö.****7.1. Review by: Stuart Jay Sidney.**

*American Scientist***61**(3) (1973), 376.One is not often afforded the opportunity to review a book which was a classic before one's birth. The original German version of the work under review appeared in 1924 and for nearly half a century has been a major force in the education of countless mathematicians. It may well be the granddaddy of all the learn-by-doing advanced mathematics books, a species which the authors are careful to distinguish from the species of problem collection books. The present English edition is, with relatively minor changes and additions, the same as its German precursor. The first half consists of 776 problems, the second half of indications of their solutions. ... There is an enormous wealth of knowledge here, none of which has lost its interest or value in fifty years. The problems are generally challenging, but the reader, whether graduate student or sophisticated professional, who attacks the book properly - studying together with a problem that which surrounds it and making a serious effort before resorting to the printed solutions - will reap rich dividends. It is hoped that the publishers will soon grace us with an English edition of the second, more advanced, volume.

**Problems and Theorems in Analysis, Vol. 2: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry (1976), by George Pólya and Gabor Szegö.****8.1. Review by: Francesco Brambilla.**

*Giornale degli Economisti e Annali di Economia, Nuova Serie, Anno***35**(7/8) (1976), 516-517.The present volume is the fourth edition of the 2nd volume of a work that constitutes a useful collection of results not only for pure mathematicians but also for all those who, in the course of their research, encountering a difficulty or forgetting a theorem may find a new strategy for action. ... It is a book that should not be missing in any library, not only public but also personal.