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Bounded analytic functions. Revised 1st ed. (English) Zbl 1106.30001
Graduate Texts in Mathematics 236. New York, NY: Springer (ISBN 0-387-33621-4/hbk). xiv, 460 p. EUR 58.80 (2006).

After having been out of print for many years, the long-awaited revised first edition of this wonderful book has finally reappeared, this time with Springer Verlag. In 2003, Garnett was awarded the AMS Steel prize for this book; one of the highest distinctions for mathematical exposition. The following, cited from that prize, appears on the back cover of the new edition: “The book, which covers a wide range of beautiful topics in analysis, is extremely well organized and well written, with elegant, detailed proofs. The book has educated a whole generation of mathematicians with backgrounds in complex analysis and function algebras. It has had a great impact on the early careers of many leading analysts and has been widely adopted as a textbook for graduate courses and learning seminars in both the US and abroad.”

This new edition is, with very few exceptions, the same as the first one [Bounded analytic functions. Pure and Applied Mathematics, 96. (New York) etc.: Academic Press, A subsidiary of Harcourt Brace Javanovich, Publishers. (1981; Zbl 0469.30024)]. Very few corrections of several mathematical and typographical errors were necessary (though not all of them are caught in this new version; an updated list of corrections will be posted soon on the author’s webpage).

The chapters are: (I) Preliminaries (Poisson integrals, Hardy-Littlewood maximal function); (II) H p spaces (inner functions, Blaschke products; Nevanlinna class); (III) conjugate functions; (IV) extremal functions (Helson-Szegö theorem; Nevanlinna’s theory on interpolation by functions in the unit ball of H ); (V) introduction to uniform algebras; (VI) the space BMO of functions of bounded mean oscillation on the unit circle; (VII) interpolating sequences on the disk; (VIII) the Corona construction; (IX) Douglas algebras; and (X) K. Hoffman’s theory on the maximal ideal space of H .

In the preface of this edition, the author lists some of the problems that were posed in the first edition and that have been solved meanwhile. Unfortunately, the exact references were not given (only not very useful MR code numbers, as e.g. MR 1394402 for a paper of A. Nicolau and the author [Pac. J. Math. 173, No. 2, 501–510 (1996; Zbl 0871.30031)] on the closed linear span of the set of all interpolating Blaschke products). An updated bibliography would have been most appreciated, as well as an appendix on new results in the field since 1981. Perhaps, in view of the tremendous success this book has enjoyed, this would have filled several other monographs...

To sum up, Garnett’s book is as useful now as it was in 1981 and is highly recommended to everyone interested in function theory.

30-02Research monographs (functions of one complex variable)
30D50Blaschke products, etc. (MSC2000)
30H05Bounded analytic functions
46J15Banach algebras of differentiable or analytic functions, H p -spaces