Function Theory on Planar Domains A Second Course in Complex Analysis

by Fisher, Stephen D.

• ISBN13: 9780486457680
• ISBN10: 0486457680
• Format: Paperback
• Copyright: 2007-02-27
• Publisher: Dover Publications

Summary

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 simple closed curves. The author is Professor of Mathematics at Northwestern University. 1983 edition.

Table of Contents

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