x PREFACE base our treatment of cyclic singular inner functions (Section 8.2) on his unpublished notes. Christopher Hammond read large portions of the man- uscript with an eagle eye and spotted a number of misprints, minor errors, and obscurities. Anders Bjorn also made helpful remarks. We want to ex- press our sincere appreciation to all of these people, and others whose names we may have overlooked, for helping to improve this book. Any defects that remain, however, are the authors' responsibility. We also had the benefit of the earlier book Theory of Bergman Spaces by Hakan Hedenmalm, Boris Korenblum, and Kehe Zhu (Springer-Verlag, 2000), which served as a useful reference in our approach to several topics. As may be expected, the two books have considerable overlap, but ours de- velops more of the prerequisite material. It also treats topics not discussed in the earlier book, and treats some of the same topics in different ways. A few results appear here for the first time. On the other hand, the book of Hedenmalm, Korenblum, and Zhu contains extensive discussions of sev- eral topics barely touched upon in our book, such as invertible noncyclic functions and logarithmically subharmonic weights. Our book is essentially self-contained. It should be accessible to ad- vanced graduate students who have studied basic complex function theory, measure theory, and functional analysis. Prior knowledge of Hardy spaces is helpful, since that theory often serves as a model for Bergman spaces, but the main facts about Hardy spaces are reviewed in two "crash courses" early in the book and later as motivation for corresponding topics in Bergman spaces. A few Hardy space results are actually needed for the theoretical development of Bergman spaces, and proofs are given. Most of the writing was carried out during summers together in Ann Arbor, where the University of Michigan provided excellent facilities for our work. Thanks also go to the Ann Arbor Diamondbacks, who were an extra incentive for the second-named author to return to Michigan every summer. Over the last decade, the American Mathematical Society held several Special Sessions on Bergman spaces at national and regional meetings, and sponsored a week-long research conference at Mt. Holyoke College in the summer of 1994. That summer conference, in particular, did much to stim- ulate further research in the field. We were therefore especially pleased when the AMS agreed to publish our book. We are grateful to Sergei Gelfand of the AMS for his initial vision that encouraged us not to settle for a revised set of lecture notes, but to do the extra work needed to produce a full ex- pository account of the subject. He showed remarkable patience with the slow pace of the resulting project, but pushed us to finish when the end was in sight and helped with the technical aspects of production. We hope our book may be judged a worthy successor to the classic book by Stefan Bergman, which appeared in the same AMS series many years ago. Peter Duren and Alexander Schuster September 2003
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