Preface

Summary

Rational functions are a classical tool for approximation. They turn out to be a more convenient tool for approximation in many cases than polynomials which explains the constant increase of interest in them. On the other hand rational functions are a nonlinear approximation tool and they possess some intrinsic peculiarities creating a lot of difficulties in their investigation. After the classcial results of Zolotarjov from the end of the last century substantial progress was achieved in 1964 when D. Newman showed that |x| is uniformly approximated by rational functions much better than by algebraic polynomials.

Newman's result stimulated the appearance of many substantial results in the field of rational real approximations. Our aim in this book is to present the basic achievements in rational real approximations. Nevertheless, for the sake of completeness we have included some results referring to the field of complex rational approximations in Chapters 6 and 12. Also, in order to stress some peculiarities of rational approximations we have included for comparison some classical and more recent results from the linear theory of approximation. On the other hand, since rational approximations are closely connected with spline approximations, we have included as well some results concerning spline approximations.