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Notation and Numbering xiii
The Dirichlet Problem and Harmonic Measure 1
Introduction 1
The Poisson Formula and Some Preliminaries 1
Subharmonic Functions 6
Solution of the Dirichlet Problem 11
The Green's Function of a Domain 16
Harmonic Measure 19
Logarithmic Capacity 24
Additional Readings and Notes 35
Exercises 35
Uniformization and Conditional Expectation 38
Introduction 38
The Uniformization Theorem 38
Conditional Expectation and the Space N 43
Harmonic Measure and L[superscript 1]/[Characters not reproducible] 48
Additional Readings and Notes 49
Exercises 49
The Hardy Spaces H[superscript p]([Omega]) 51
Introduction 51
Basic Properties of H[superscript p]([Omega]) 51
H[superscript p] on the Unit Disc 55
H[superscript p] and H[superscript p]([Omega]) 62
Null Sets and Essential Boundary Points for H[superscript Infinity]([Omega]) 64
H[superscript p]([Omega]) Determines [Omega] 66
Weak Peak Pointsfor H[superscript Infinity]([Omega]) 70
Additional Readings and Notes 73
Exercises 74
Domains of Finite Connectivity 77
Introduction 77
The Defect of ReR([Omega]) in C[subscript 1]([Gamma]) 78
Measures Orthogonal to R([Omega]) 81
H[superscript p]([Omega]) 85
N Again 93
Functions with Periods 97
The Factorization of H[superscript p]([Omega]) Functions 103
Additional Readings and Notes 106
Exercises 107
Blaschke Products, Inner Functions, and Extremal Problems 109
The Ahlfors Function 109
Blaschke Products 116
Approximation by Inner Functions 120
Pick-Nevanlinna Interpolation 130
Interpolation Sequences 141
The Maximum Principle for Multiple-Valued Bounded Analytic Functions 151
Additional Readings and Notes 160
Exercises 161
The Maximal Ideal Space of H[superscript Infinity]([Omega]) 165
Introduction 165
Peak Points and Parts 165
The Fibers of M([Omega]) 171
Distinguished Homomorphisms 177
The Shilov Boundary of H[superscript Infinity]([Omega]) 190
The Corona Theorem 195
Additional Readings and Notes 202
Exercises 203
Linear Operators on H[superscript p] Spaces 206
The Isometries of H[superscript p]([Omega]) 206
Self-Mappings of a Domain 217
General Properties of Composition Operators 228
Compact Composition Operators on H[superscript Infinity]([Omega]) 232
Optimal Estimation and Widths of Spaces of Holomorphic Functions: Part 1. The H[superscript Infinity] Case 237
Optimal Estimation and Widths of Spaces of Holomorphic Functions: Part 2. The H[superscript 2] Case 251
Additional Readings and Notes 258
Exercises 258
Bibliography 262
Index 267
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Add Function Theory on Planar Domains: A Second Course in Complex Analysis, Suitable for upper-level undergraduates and graduate students, this treatment of complex analysis focuses on function theory on a finitely connected planar domain. Clear and complete, it emphasizes domains bounded by a finite number of disjoint analytic s, Function Theory on Planar Domains: A Second Course in Complex Analysis to the inventory that you are selling on WonderClubX
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Add Function Theory on Planar Domains: A Second Course in Complex Analysis, Suitable for upper-level undergraduates and graduate students, this treatment of complex analysis focuses on function theory on a finitely connected planar domain. Clear and complete, it emphasizes domains bounded by a finite number of disjoint analytic s, Function Theory on Planar Domains: A Second Course in Complex Analysis to your collection on WonderClub |