Bergman Spaces (Mathematical Surveys and Monographs)

Over the last ten years, the theory of Bergman spaces has undergone a remarkable metamorphosis. In a series of major advances, central problems once considered intractable were solved, and a rich theory emerged. Although progress continues, the time seems ripe for a full and unified account of the subject, weaving the old and new results together. This thorough exposition provides just that. The subject of Bergman spaces is a masterful blend of complex function theory with functional analysis and operator theory. It has much in common with Hardy spaces, but involves new elements such as hyperbolic geometry, reproducing kernels, and biharmonic Green functions. In this book, the authors develop background material and provide a self-contained introduction to a broad range of topics, including recent advances on interpolation and sampling, contractive zero-divisors, and invariant subspaces. The book is accessible to researchers and advanced graduate students who have studied basic complex function theory, measure theory, and functional analysis In This Book, The Authors Develop Background Material And Provide A Self-contained Introduction To A Broad Range Of Topics, Including Recent Advances On Interpolation And Sampling, Contractive Zero-divisors, And Invariant Subspaces. The Book Is Accessible To Researchers And Advanced Graduate Students Who Have Studied Basic Complex Function Theory, Measure Theory, And Functional Analysis.--book Jacket. The Bergman Kernel Function -- Point-evaluation Functionals -- Orthonormal Bases -- Conformal Invariance -- An Extremal Problem -- Connection With Green's Function -- The Biharmonic Green Function -- Linear Space Properties -- Hardy Spaces -- Strict And Uniform Convexity -- The Bergman Projection -- Dual Spaces -- The Pseudohyperbolic Metric -- The Bloch Space -- Harmonic Conjugates -- Linear Isometries -- Function Multipliers -- Carleson Measures -- Uniformly Discrete Sequences -- Analytic Properties -- More On Hardy Spaces -- Growth Of Functions In Bergman Spaces -- Coefficients Of Functions In Bergman Spaces -- Coefficient Multipliers -- Korenblum's Maximum Principle -- Zero-sets -- Density Of Zero-sets -- Dependence On P -- Unions And Subsets Of Zero-sets -- Blaschke Products As Generators -- Universal Divisors -- Perturbations Of Zero-sets -- Zeros On A Radial Line -- Contractive Zero-divisors -- An Extremal Problem -- Expansive Multipliers -- Proof Of The Integral Formula -- Representation By Kernel Functions -- Analytic Continuation -- Contractive Divisors -- Invariant Subspaces -- Sampling And Interpolation -- Definitions And Motivations -- Interpolation In Hardy Spaces -- A Family Of Sampling And Interpolation Sequences -- Some Explicit Examples -- The Density Theorems -- Direct Calculation Of Densities -- Sharpened Forms Of Horowitz' Theorems -- Sufficient Conditions With The Pseudohyperbolic Metric -- Duality Relations -- Proofs Of Sampling And Interpolation Theorems -- Perturbation Of Sampling Sequences. Peter Duren, Alexander Schuster. Includes Bibliographical References (p. 297-312) And Index. The modern subject of Bergman spaces is a masterful blend of complex function theory with functional analysis and operator theory. It has much in common with Hardy spaces but involves new elements such as hyperbolic geometry, reproducing kernels, and biharmonic Green functions. This book develops background material and provides a self-contained introduction to a broad range of old and new topics in Bergman spaces, including recent advances on interpolation and sampling, contractive zero-divisors, and invariant subspaces. It is accessible to anyone who has studied basic real and complex analysis at the graduate level. Over the years, the theory of Bergman spaces has undergone a remarkable metamorphosis. The subject of Bergman spaces is a masterful blend of complex function theory with functional analysis and operator theory. This book is suitable for those who have studied basic complex function theory, measure theory, and functional analysis.