BUSN COMM TODAY&PERILS PAULINE&EBIZ PH/G PK

by BOVEE

• ISBN13: 9780130554000
• ISBN10: 0130554006
• Format: Package
• Copyright: 2000-01-01
• Publisher: Pearson (Manual)

Summary

A classic of pure mathematics, this advanced graduate-level text explores the intersection of functional analysis and analytic function theory. Close in spirit to abstract harmonic analysis, it is confined to Banach spaces of analytic functions in the unit disc. The first four chapters examine proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the factorization, and a brief description of the classical approach to the theorems of the first five chapters. The remainder of the book addresses the structure of various Banach spaces and Banach algebras of analytic functions in the unit disc. Enhanced with 100 challenging exercises, a bibliography, and an index, this text belongs in the libraries of students, professional mathematicians, as well as anyone interested in a rigorous, high-level treatment of this topic. Book jacket.

Table of Contents

Preliminaries	p. 1
Measure and Integration	p. 1
Banach Spaces	p. 5
Hilbert Space and Fourier Series	p. 8
Fourier Series	p. 15
Cesaro Means	p. 16
Characterization of Types of Fourier Series	p. 22
Notes	p. 24
Exercises	p. 25
Analytic and Harmonic Functions in the Unit Disc	p. 27
The Cauchy and Poisson Kernels	p. 28
Boundary Values	p. 32
Fatou's Theorem	p. 34
H[superscript p] Spaces	p. 39
Notes	p. 39
Exercises	p. 40
The Space H[superscript 1]	p. 42
The Helson-Lowdenslager Approach	p. 42
Szego's Theorem	p. 48
Completion of the Discussion of H[superscript 1]	p. 50
Dirichlet Algebras	p. 54
Notes	p. 57
Exercises	p. 58
Factorization for H[superscript p] Functions	p. 61
Inner and Outer Functions	p. 61
Blaschke Products and Singular Functions	p. 63
The Factorization Theorem	p. 67
Absolute Convergence of Taylor Series	p. 70
Remarks on the Classical Approach	p. 72
Functions of Bounded Characteristic	p. 73
Notes	p. 74
Exercises	p. 74
Analytic Functions with Continuous Boundary Values	p. 77
Conjugate Harmonic Functions	p. 78
Theorems of Fatou and Rudin	p. 80
The Closed Ideals of A	p. 82
Commutative Banach Algebras	p. 89
Wermer's Maximality Theorem	p. 93
Notes	p. 95
Exercises	p. 95
The Shift Operator	p. 98
The Shift Operator on H[superscript 2]	p. 98
More about Dirichlet Algebras	p. 101
Invariant Subspaces for H[superscript 2] of the Half-plane	p. 103
Isometries	p. 108
The Shift Operator on L[superscript 2]	p. 111
The Vector-valued Case	p. 114
Representations of H[superscript infinity]	p. 116
Notes	p. 119
Exercises	p. 119
H[superscript p] Spaces in a Half-plane	p. 121
H[superscript p] of the Half-plane	p. 121
Boundary Values for H[superscript p] Functions	p. 124
The Paley-Wiener Theorem	p. 131
Factorization for H[superscript p] Functions in a Half-plane	p. 132
Notes	p. 133
Exercises	p. 133
H[superscript p] as a Banach Space	p. 136
Extreme Points	p. 136
Isometries	p. 142
Projections from L[superscript p] to H[superscript p]	p. 149
Notes	p. 155
Exercise	p. 156
H[superscript infinity] as a Banach Algebra	p. 158
Maximal Ideals in H[superscript infinity]	p. 159
Topological Structure of M(H[superscript infinity])	p. 162
Discs in Fibers	p. 166
L[superscript infinity] as a Banach Algebra	p. 169
The Silov Boundary	p. 172
Inner Functions and the Silov Boundary	p. 175
Representing Measures and Annihilating Measures	p. 180
Algebras on the Fibers	p. 187
Maximality	p. 193
Interpolation	p. 194
Notes	p. 206
Exercises	p. 207
Bibliography	p. 209
Index	p. 214
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