This book is based on a series of lectures on “The Constructive Theory of Functions of a Complex Variable,” given at the Leningrad University by Professor V. I. Smirnov and later by Dr. N. A. Lebedev. Professor Smirnov is a full member of the U.S.S.R. Academy of sciences and author of numerous books and papers on advanced mathematics. Both men are distinguished mathematicians with international reputations for research in this field and need no further introduction to readers of this book. The material of the lectures has been greatly expanded for this volume, and the English translation has been edited for the Western world by one of England's leading mathematicians.
The first chapter deals, in the main, with the use of polynomials and rational fractions obtainable by interpolation in the approximate representation of functions that are regular in closed sets. Also covered are the well-known results of Mergelian and Vitushkin's theorem, which employs rational functions that are continuous on a closed bounded set without internal points. The theory of the fundamental and generalized. Faber polynomials and some typical applications are described in the next chapter, together with an account of the necessary parts of the theory of biorthogonal functions. Subsequent chapters deal with mean square approximations with respect to a domain or its boundary.
The final chapter is divided into two parts. The first gives the solution to a series of concrete problems connected with polynomials deviating least from zero, with Mendeleev's problem, and with the results of Markov and Bernstein. The second contains a series of general theorems relating to the theory of best approximations and is followed by a number of typical applications.
In Functions of a Complex Variable, the authors have confined themselves to the simplest fundamental problem using classical methods. This book, which has already established itself as a standard work in Russia, should in its translation engage a truly international readership.