Overview

From the Preface: """"The functional-analytic approach to uniform algebras is inextricably interwoven with the theory of analytic functions ...[T]he concepts and techniques introduced to deal with these problems [of uniform algebras], such as """"peak points"""" and """"parts,"""" provide new insights into the classical theory of approximation by analytic functions. In some cases, elegant proofs of old results are obtained by abstract methods. The new concepts also lead to new problems in classical function theory, which serve to enliven and refresh that subject. In short, the relation between functional analysis and the analytic theory is both fascinating and complex, and it serves to enrich and deepen each of the respective disciplines."""" This volume includes a Bibliography, List of Special Symbols, and an Index. Each of the chapters is followed by notes and numerous exercises.

Full Product Details

Author:   Theodore W. Gamelin
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Edition:   New edition
Volume:   No. 311
Dimensions:   Width: 23.70cm , Height: 2.00cm , Length: 16.60cm
Weight:   0.532kg
ISBN:  

9780821840498

ISBN 10:   0821840495
Pages:   269
Publication Date:   01 March 2006
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Temporarily unavailable  
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Table of Contents

Commutative Banach Algebras: 1. Spectrum and resolvent 2. The maximal ideal space 3. Examples 4. The Shilov boundary 5. Two basic theorems 6. Hulls and kernels 7. Commutative algebras 8. Compactifications 9. The algebra 10. Normal operators on Hilbert space Notes Exercises Uniform Algebras: 1. Algebras on planar sets 2. Representing measures 3. Dirichlet algebras 4. Logmodular algebras 5. Maximal subalgebras 6. Hulls 7. Decomposition of orthogonal measures 8. Cauchy transform 9. Mergelyan's theorem 10. Local algebras 11. Peak points 12. Peak sets 13. Antisymmetric algebras Notes Exercises Methods of Several Complex Variables: 1. Polynomial convexity 2. Rational convexity 3. Circled sets 4. Functional calculus 5. Polynomial approximation 6. Implicit function theorem 7. Cohomology of the maximal ideal space 8. Local maximum modulus principle 9. Extensions of uniform algebras Notes Exercises Hardy Spaces: 1. The conjugation operator 2. Representing measures 3. The uniqueness subspace 4. Enveloped measures 5. Core measures 6. The finite dimensional case 7. Logmodular measures 8. Hypodirichlet algebras Notes Exercises Invariant Subspace Theory: 1. Uniform integrability 2. The Hardy algebra 3. Jensen measures 4. Characterization 5. Invertible elements of $H$ 6. Invariant subspaces 7. Embedding of analytic discs 8. Szego's theorem 9. Extremal functions in Notes Exercises Parts: 1. Representing measures for a part 2. Characterization of parts 3. Parts of $R(K)$ 4. Finitely connected case 5. Pointwise bounded approximation 6. Finitely generated ideals 7. Extremal methods Notes Exercises Generalized Analytic Functions: 1. Preliminaries 2. Algebras associated with groups 3. A theorem of Bochner 4. Generalized analytic functions 5. Analytic measures 6. Local product decomposition 7. The Hardy spaces 8. Weak-star maximality 9. Weight functions 10. Invariant subspaces 11. Structure of cocycles 12. Cocycles and invariant subspaces Notes Exercises Analytic Capacity and Rational Approximation: 1. Analytic capacity 2. Elements of analytic capacity 3. Continuous analytic capacity 4. Peaking criteria 5. Criteria for $R(K)=C(K)$ 6. Analytic diameter 7. A scheme for approximation 8. Criteria for 9. Failure of approximation 10. Pointwise bounded approximation 11. Pointwise bounded approximation with same norm 12. Estimates for integrals 13. Analytically negligible sets Notes Exercises Bibliography List of special symbols Index.

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