Encyclopedia of Mathematics and Its Applications. 59. Cambridge: Cambridge Univ. Press. xiv, 746 p. £65.00; $ 110.00 (1996).

The first edition [Part I (1981;

Zbl 0468.30032); Part II (1981;

Zbl 0468.30033)] of “Encyclopedia of Mathematics: Padé approximants” already received rave reviews, and the second edition is even better. The book is an absolute must for every starting researcher in the area of rational approximation theory. The authors, eminencies in the field, have on one hand revised the first edition of the book which appeared in two separate volumes, and have on the other hand enlarged the contents with a new chapter on multiseries approximants, containing a lot of their research from the last few years. The book is written with a smooth progression from elementary ideas to some of the frontiers of research in approximation theory. It consists in total of 11 chapters and covers the theory of Padé, approximation very thoroughly. Let us now discuss the contents in somewhat more detail. The Padé approximant of a given function

$f$ is a rational function of numerator degree

$L$ and denominator degree

$M$ whose power series agrees with the power series of

$f$ up to and including degree

$L+M$. The reader is taken from motivating examples, the Baker definition, over explicit determinant formulas, recursive algorithms and continued fraction representations to the block structure of the table of Padé approximants, all in the first three chapters. A separate chapter is devoted to continued fractions because they play such an essential role in the theory. The next two chapters deal with the problem of convergence, comparing sequences of Padé approximants with the given function

$f$ and describing the approximation power for several function classes. Then two chapters are devoted to generalizations, namely Padé approximants to Laurent series, Fourier series, Chebyshev series, multivariate Taylor series, multipoint Newton series, etc. Also vector Padé approximants to multiseries data, algebraic approximants which are multivalued, integral approximants and many more are discussed. Applications to statistical mechanics and critical phenomena are extensively covered. With a reference list of over 900 references, the volume contains a wealth of material.