Lectures on entire functions. In collab. with Yu. Lyubarskii, M. Sodin, V. Tkachenko. Transl. by V. Tkachenko from an original Russian manuscr. (English) Zbl 0856.30001

Translations of Mathematical Monographs. 150. Providence, RI: American Mathematical Society (AMS). xv, 248 p. (1996).
The first edition of this book was a set of lecture notes for a course taught by B. Ya. Levin at Moscow University in 1969. The present edition is a thorough re-working of the material, to which the author devoted the last ten years of his life. In 28 ‘lectures’ he gives a brilliant survey of the applications of function theory to mathematical, physical and statistical questions. The reader needs no pre-requisite beyond a standard one-term course in complex variable. B. Ya. Levin (1906-1993) was famous for the clarity of his writing. He would not give an exposition of a theory before he had a complete understanding of the logical connections of all its parts. The text under review is a masterpiece. In the opinion of the reviewer it is likely to become a classic, ranking with Euclid’s elements and Euler’s algebra as a masterpiece of exposition. A partial list of topics includes up-to-date versions of Nevanlinna theory, of quasi-analytic classes, of spectral synthesis, of interpolation of linear operators, of approximation by rational functions. A diligent reader of the 28 ‘lectures’ (and they are a pleasure to read) will know all the contents of P. Koosis’ two volume text on the Logarithmic integral and substantial parts of J. Garnett’s Bounded analytic functions, the two most recent and most complete introductions to the higher theory of functions of a complex variable. Readers with some previous knowledge will experience pleasant surprises at the deft way in which the material is presented. David Drasin’s excellent translation is smooth and idiomatic. It reads like a book written in English by a gifted expositor.
Reviewer: W.H.Fuchs (Ithaca)


30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30E10 Approximation in the complex plane
30C35 General theory of conformal mappings