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Rational approximation of real functions. (English) Zbl 0644.41010
Encyclopedia of Mathematics and its Applications, Vol. 28. Cambridge etc.: Cambridge University Press. XI, 371 p.; £40.00; \$ 69.50 (1987).
In recent years great interest has been centered on the topic of rational approximation. The source of the interest is the famous paper of D. Neuman published in 1964 and proving that the function $$| x|$$ can uniformly be approximated by rational functions much better than by algebraic polynomials. In the last near 25 years many mathematicians and especially the members of the “Bulgarian school” including the authors have achieved a great number of interesting results in this field. There is no doubt that it was great need for writing the first monograph about the real rational approximation; and the authors have solved this task excellently.
The exposition of their work is clear, the enormous quantity of results is lucidly arranged on the near 400 pages, the hints to the literature are almost complete, so the book will certainly serve as a good reference source. I am convinced that the book of Petrushev and Popov will be welcomed by anyone who wants to know the present stage of the rational approximation and it will be a useful guidepost for further research.
In the preface the authors sketch briefly the contents of the book as follows: “Chapters 1 and 3 contain some basic facts concerning linear approximation theory. A basic problem in approximation theory is to find complete direct and converse theorems. In our opinion the most natural way to obtain such theorems in linear and nonlinear approximations is to prove pairs of adjusted inequalities of Jackson and Bernstein type and then to characterize the corresponding approximations by the K-functional of Peetre. This main viewpoint is given and illustrated at the end of Chapter 3 and next applied to the spline approximation in Chapter 7.
Chapter 2 is devoted to the study of the qualitative theory of rational approximation such as the existence, the uniqueness and the characterization problems, the problem of continuity of metric projection and numerical methods. The heart of the book is contained in Chapters 4 to 11. Chapter 4 presents the uniform rational approximation of some important functions such as $$| x|$$, $$\sqrt{x}$$, $$e^ x$$. In Chapter 5 the uniform rational approximation of a number of classes is considered. The exact orders of approximation are established. The basic methods for rational approximation are given. In Chapter 6 some converse theorems for rational uniform approximation are proved. In Chapter 7 complete direct and converse theorems for the spline approximation in $$L_ p$$, C, BMO are proved using Besov spaces. Chapter 8 investigates the relations between the rational and spline approximations. Chapter 9 deals with rational approximation in Hausdorff metric. A characteristic particularity of rational approximation is the appearance of the so- called ‘o small’ effect in the order of rational approximation of each individual function of some function classes. This phenomenon is investigated and characterized for some function classes in Chapter 10. the exactness of the proved estimates is established and discussed in Chapter 11. Chapter 12 considers some special problems, connected with Padé approximants - some of the so-called direct and converse problems for convergence of the rows and diagonal of the Padé-table. Finally some numerical results and graphs are presented in the Appendix.” The authors are to be congratulated in writing such a readable and self- contained monograph.
Reviewer: L.Leindler

##### MSC:
 41A20 Approximation by rational functions 41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions 41A21 Padé approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A25 Rate of convergence, degree of approximation 41A15 Spline approximation