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Orthogonale polynome. (Orthogonal polynomials)
251
Short communications
The value of degf(x) is found from the ratio of the degrees of the numerator and denominator of the rational function F(x). In conclusion I thank V. A. Ditkin for suggesting the problem, and also V. M. Kurochkin for a useful discussion of the draft of the manuscript. Translated by J. Berry. REFERENCES 1.
ABRAMOV,
S.
A. On the summationof rationalfunctions.Zh. vjkhisl. Mat. mat. Fiz., 11, 1071-1075,
2.
LANG, S. Algebra
(Algebra),“Mir”,
1971.1.
Moscow, 1968.
BOOK REVIEWS G . FREUD . Orthogonale polynome. (Orthogonal polynomials). 294 p.
Birkhauser Verl., Basel-Stuttgart, 1969. THIS book comes from the pen of one of the outstanding representatives of the Hungarian school of theorgy of functions. It differs greatly from previously published books on the theory of orthogonal polynomials which have become classics (for example, the book of G. &ego, translated into Russian). In these books particular attention is given to the theory of special orthogonal polynomials, one of the important foundations of computational mathematics. Unlike these Freud devotes his book to the general theory of orthogonal polynomials, embracing the properties of both the individual polynomials and also of systems of such polynomials. This general approach enables the author to obtain extremely simply many known facts of the theory of special orthogonal polynomials (for example, the convergence of iterative processes) as particular cases of the general theory. Being one of the leading world specialists in this field, the author also originally explains many of his own results (some of them are published here for the first time), and the results of other authors. Naturally, the reader of such a book requires, besides the mastery of a general course on computational mathematics, also knowledge of some branches of the theory of functions of real and complex variables. It must be especially mentioned that each chapter ends with a set of well chosen problems many of which can serve as a stimulus to independent investigations. Moreover at the end of the book there is a special supplement in which the author formulates a series of unsolved problems on the theory of orthogonal polynomials. Each chapter is provided with hisotircal information where, with great scrupulousness the services of other authors in the solution of the problems considered in the chapter are mentioned. The book is published at a high level and has an attractive appearance. The book contains 5 chapters, three appendices, an extremely extensive bibliography of books and journal articles, and also the necessary indexes. Chapter I is devoted to some fundamental properties of orthogonal polynomials (for example, recursive formulas, the Christoffel-Darboux summation formula, zero properties, the Markov-Stiltjes inequality etc.). At the end of the chapter these questions are explained for important special
252
Book Reviews
polynomials - the Chebyshev, Legendre and Jacobi polynomials. Chapter II contains the existence and uniqueness theorem for the problem of moments and its applications in the theory of orthogonal polynomials (for example, the connection between the zero properties of orthogonal polynomials and the uniqueness of the solution of the moment problem is considered). In Chapters III and IV the results of Chapters I and II are applied to various questions of the approximation of a given function by orthogonal polynomials. The material of these chapters is especially important for computational mathematics as it is the theoretical apparatus of various sections of it (for example, the theory of quadrature formulas, the theory of interpolation etc.). Chapter V, the last, is devoted to the connection between the theory of orthogonal polynomials in the real domain and on the complex unit circle (Szego’s theory). In particular, explanations are given here of some delicate questions of the convergence of series of orthogonal polynomials, of the estimation of the distance between the zeros of such polynomials, of the Hermite- Fejer formulas etc. Important problems relating to the investigation of the zeros of orthogonal polynomials are discussed both in this, and also in almost all the chapters (wherever necessary). Undoubtedly the book will be of great use not only to specialists in the theory of functions, but also to a large circle of mathematicians and persons engaged in the applications of mathematics. The translation of this book into Russian would be extremely desirable. M. K. Kerimov Translated by J. Berry.
Y. AYANT, M. BORG. Fonctions sp&iales h l’usage des &tudiantsen physique. Maitrise de physique. (Km teorii spetsial’nykh funktsii dlya fizikov). X + 266 p. Dunod, Paris, 1971.
THE THEORY of the special functions of mathematical physics has always been and remains an important subject, forming the basis of the mathematical education of physicists. Despite the powerful development of computational mathematics and various numerical methods making it possible to solve a number of problems which were inaccessible before the advent of high-speed computers, the theory of special functions has never ceased to serve physicists and in general all natural scientists as one of the oldest and most detailed developed working instruments for the approximate and exact solution of many of the problems confronting them. Moreover, recently many generalized hypergeometric functions, whose mathematical theory is still far from perfect, have become firmly established in the physical sciences (alongside the long known special functions such as the Bessel and Legendre functions, and the degenerate hypergeometric functions). This book is of an expository nature and was written as a textbook for student physicists. It is small in size and the coverage of the material is fairly modest. However, those sections of the theory of special functions which are explained in it are described in detail and with great clarity. Since the theory of the Laplace and Fourier integral transformations is important for physicists not only in itself, but also as an apparatus for studying the theory of special functions, the authors have included in the book special chapters devoted to these transformations. Of special value for those studying the theory of special functions (perhaps also for those teaching it) are the problems provided in each chapter. Detailed solutions of these problems or hints for their solution are given at the end of the book.
Short communications
The value of degf(x) is found from the ratio of the degrees of the numerator and denominator of the rational function F(x). In conclusion I thank V. A. Ditkin for suggesting the problem, and also V. M. Kurochkin for a useful discussion of the draft of the manuscript. Translated by J. Berry. REFERENCES 1.
ABRAMOV,
S.
A. On the summationof rationalfunctions.Zh. vjkhisl. Mat. mat. Fiz., 11, 1071-1075,
2.
LANG, S. Algebra
(Algebra),“Mir”,
1971.1.
Moscow, 1968.
BOOK REVIEWS G . FREUD . Orthogonale polynome. (Orthogonal polynomials). 294 p.
Birkhauser Verl., Basel-Stuttgart, 1969. THIS book comes from the pen of one of the outstanding representatives of the Hungarian school of theorgy of functions. It differs greatly from previously published books on the theory of orthogonal polynomials which have become classics (for example, the book of G. &ego, translated into Russian). In these books particular attention is given to the theory of special orthogonal polynomials, one of the important foundations of computational mathematics. Unlike these Freud devotes his book to the general theory of orthogonal polynomials, embracing the properties of both the individual polynomials and also of systems of such polynomials. This general approach enables the author to obtain extremely simply many known facts of the theory of special orthogonal polynomials (for example, the convergence of iterative processes) as particular cases of the general theory. Being one of the leading world specialists in this field, the author also originally explains many of his own results (some of them are published here for the first time), and the results of other authors. Naturally, the reader of such a book requires, besides the mastery of a general course on computational mathematics, also knowledge of some branches of the theory of functions of real and complex variables. It must be especially mentioned that each chapter ends with a set of well chosen problems many of which can serve as a stimulus to independent investigations. Moreover at the end of the book there is a special supplement in which the author formulates a series of unsolved problems on the theory of orthogonal polynomials. Each chapter is provided with hisotircal information where, with great scrupulousness the services of other authors in the solution of the problems considered in the chapter are mentioned. The book is published at a high level and has an attractive appearance. The book contains 5 chapters, three appendices, an extremely extensive bibliography of books and journal articles, and also the necessary indexes. Chapter I is devoted to some fundamental properties of orthogonal polynomials (for example, recursive formulas, the Christoffel-Darboux summation formula, zero properties, the Markov-Stiltjes inequality etc.). At the end of the chapter these questions are explained for important special
252
Book Reviews
polynomials - the Chebyshev, Legendre and Jacobi polynomials. Chapter II contains the existence and uniqueness theorem for the problem of moments and its applications in the theory of orthogonal polynomials (for example, the connection between the zero properties of orthogonal polynomials and the uniqueness of the solution of the moment problem is considered). In Chapters III and IV the results of Chapters I and II are applied to various questions of the approximation of a given function by orthogonal polynomials. The material of these chapters is especially important for computational mathematics as it is the theoretical apparatus of various sections of it (for example, the theory of quadrature formulas, the theory of interpolation etc.). Chapter V, the last, is devoted to the connection between the theory of orthogonal polynomials in the real domain and on the complex unit circle (Szego’s theory). In particular, explanations are given here of some delicate questions of the convergence of series of orthogonal polynomials, of the estimation of the distance between the zeros of such polynomials, of the Hermite- Fejer formulas etc. Important problems relating to the investigation of the zeros of orthogonal polynomials are discussed both in this, and also in almost all the chapters (wherever necessary). Undoubtedly the book will be of great use not only to specialists in the theory of functions, but also to a large circle of mathematicians and persons engaged in the applications of mathematics. The translation of this book into Russian would be extremely desirable. M. K. Kerimov Translated by J. Berry.
Y. AYANT, M. BORG. Fonctions sp&iales h l’usage des &tudiantsen physique. Maitrise de physique. (Km teorii spetsial’nykh funktsii dlya fizikov). X + 266 p. Dunod, Paris, 1971.
THE THEORY of the special functions of mathematical physics has always been and remains an important subject, forming the basis of the mathematical education of physicists. Despite the powerful development of computational mathematics and various numerical methods making it possible to solve a number of problems which were inaccessible before the advent of high-speed computers, the theory of special functions has never ceased to serve physicists and in general all natural scientists as one of the oldest and most detailed developed working instruments for the approximate and exact solution of many of the problems confronting them. Moreover, recently many generalized hypergeometric functions, whose mathematical theory is still far from perfect, have become firmly established in the physical sciences (alongside the long known special functions such as the Bessel and Legendre functions, and the degenerate hypergeometric functions). This book is of an expository nature and was written as a textbook for student physicists. It is small in size and the coverage of the material is fairly modest. However, those sections of the theory of special functions which are explained in it are described in detail and with great clarity. Since the theory of the Laplace and Fourier integral transformations is important for physicists not only in itself, but also as an apparatus for studying the theory of special functions, the authors have included in the book special chapters devoted to these transformations. Of special value for those studying the theory of special functions (perhaps also for those teaching it) are the problems provided in each chapter. Detailed solutions of these problems or hints for their solution are given at the end of the book.