This book is a classic. It has been for many years one of the most important sources for many mathematicians interested in analysis, especially the connections between and combinations of real analysis, complex analysis and functional analysis. The first edition appeared in 1981 and has been ever since a major reference for analysts working in complex analysis and function algebras, as well as a graduate text and a source of material for learning seminars. The present edition appears as number 236 in Springer's Graduate Texts in Mathematics series.
The topics covered in the book are very interesting (Hp Spaces, Conjugate Functions, Bounded Mean Oscillation, Interpolating Sequences, The Corona Construction, Douglas Algebras, etc) but what really sets this text apart is the way it has been written: very elegant, very detailed proofs. It remains one of the best books in the field even 26 years afaater the first edition.
Although the content and the Bibliography from the original publication have not been changed much, the author mentions in the Preface to the present edition that corrections have been made. Also some books that appeared in the field since 1981 are mentioned in the Preface, as possible references.
Professor Garnett mentions in the Preface that he “had planned to prepare a second edition with an updated bibliography and an appendix on results new in the field since 1981.” I think many people would love to see that plan materialize. Until then, we are all happy to see this revised first edition.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
Preliminaries.- H_P Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.- Interpolating Sequences and Maximal Ideals.- Bibliography.- Index.