Overview

This text deals with classical and contemporary topics in function theory and is designed to be used after a one-year course in real and complex analysis. It can be used as a text for topics courses or courses in function theory, operator theory and applied areas. The first six chapters supplement the authors' book ""Hardy Classes and Operator Theory"". The theory of harmonic majorants for subharmonic functions is used to introduce Hardy-Orlicz classes, which are specialized to standard Hardy classes on the unit disk. The theorem of Szegoe-Solomentsev characterizes boundary behaviour. Half-plane function theory receives equal treatment and features the theorem of Flett and Kuran on existence of harmonic majorants and applications of the Phragmen-Lindeloef principle. The last three chapters contain an introduction to univalent functions, leading to a self-contained account of Loewner's differential equation and de Branges' proof of the Milin conjecture.

Full Product Details

Author:   Marvin Rosenblum ,  James Rovnyak
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1994 ed.
Dimensions:   Width: 15.50cm , Height: 1.50cm , Length: 23.50cm
Weight:   1.210kg
ISBN:  

9783764351113

ISBN 10:   376435111
Pages:   250
Publication Date:   01 September 1994
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Harmonic Functions.- 1.1 Introduction.- 1.2 Uniqueness principle.- 1.3 The Poisson kernel.- 1.4 Normalized Lebesgue measure.- 1.5 Dirichlet problem for the unit disk.- 1.6 Properties of harmonic functions.- 1.7 Mean value property.- 1.8 Harnack’s theorem.- 1.9 Weak compactness principle.- 1.10 Nonnegative harmonic functions.- 1.11 Herglotz and Riesz representation theorem.- 1.12 Stieltjes inversion formula.- 1.13 Integral of the Poisson kernel.- 1.14 Examples.- 1.15 Space h1(D).- 1.16 Characterization of h1(D).- 1.17 Nontangential convergence.- 1.18 Fatou’s theorem.- 1.19 Boundary functions.- Examples and addenda.- 2 Subharmonic Functions.- 2.1 Introduction.- 2.2 Upper semicontinuous functions.- 2.3 Subharmonic functions.- 2.4 Some properties of subharmonic functions.- 2.5 Maximum principle.- 2.6 Convergence of mean values.- 2.7 Convex functions.- 2.8 Structure of convex functions.- 2.9 Jensen’s inequality.- 2.10 Composition of convex and subharmonic functions.- 2.11 Vector- and operator-valued functions.- 2.12 Subharmonic functions from holomorphic functions.- 3 Part I Harmonic Majorants Part II Nevanlinna and Hardy-Orlicz Classes.- 3.1 Introduction.- 3.2 Least harmonic majorant.- 3.3 Existence of least harmonic majorants.- 3.4 Construction of harmonic majorants.- 3.5 Class shl(D).- 3.6 Characterization of sh1(D).- 3.7 Absolutely continuous component of a related measure.- 3.8 Uniformly integrable family.- 3.9 Strongly convex functions.- 3.10 Theorem of de la Vallée Poussin and Nagumo.- 3.11 Singular component of associated measures.- 3.12 Sufficient conditions for absolute continuity.- 3.13 Theorem of Szegö-Solomentsev.- 3.14 Remark.- 3.15 Hardy and Nevanlinna classes.- 3.16 Linearity of the classes.- 3.17 Properties of log+x.- 3.18 Majorants for stronglyconvex functions.- 3.19 Compositions and restrictions.- 3.20 Quotients of bounded functions.- Examples and addenda.- 4 Hardy Spaces on the Disk.- 4.1 Introduction.- 4.2 Inner and outer functions.- 4.3 Rational inner functions.- 4.4 Infinite products.- 4.5 An infinite product.- 4.6 Blaschke products.- 4.7 Inner functions with no zeros.- 4.8 Singular inner functions.- 4.9 Factorization of inner functions.- 4.10 Boundary functions for N(D).- 4.11 Characterization of N(D).- 4.12 Condition on zeros.- 4.13 N(D) as an algebra.- 4.14 Characterization of N+(D).- 4.15 N+(D) as an algebra.- 4.16 Estimates from boundary functions for N+(D).- 4.17 Outer functions in N+(D).- 4.18 Characterization of ??(D).- 4.19 Nevanlinna and Hardy-Orlicz classes on the boundary.- 4.20 Szegö’s problem.- 4.21 Classes HP(D) and HP(?).- 4.22 Characterization of HP(D).- 4.23 Characterization of HP(?).- 4.24 Connection between HP(D) and HP(?).- 4.25 Hp(?) as a subspace of LP(?).- 4.26 Hp(D) and HP(?) as Banach spaces.- 4.27 F and M Riesz theorem.- 4.28 H2(D) and H2(?).- 4.29 Sufficient conditions for outer functions.- 4.30 Beurling’s theorem.- 4.31 Theorem of Szegö, Kolmogorov, and Kre?n.- 4.32 Closure of trigonometric functions in Lp(?).- 5 Function Theory on a Half-Plane.- 5.1 Introduction.- 5.2 Poisson representation.- 5.3 Nevanlinna representation.- 5.4 Stieltjes inversion formula.- 5.5 Fatou’s theorem.- 5.6 Boundary functions for N(?).- 5.7 Limits of nondecreasing functions.- 5.8 Nonnegative harmonic functions.- 5.9 Theorem of Flett and Kuran.- 5.10 Nevanlinna and Hardy-Orlicz classes.- 5.11 Notation and terminology.- 5.12 Szegö’s problem on the line.- 5.13 Inner and outer functions.- 5.14 Examples and miscellaneous properties.- 5.15 Hardy classes.- 5.16 Characterization of?P(I?).- 5.17 Inclusions among classes.- 5.18 Poisson representation for ?P(?).- 5.19 Cauchy representation for Hp(?).- 5.20 Characterization of HP(?).- 5.21 Hp(?) as a subspace of N+(?).- 5.22 Condition for mean convergence.- 5.23 Hp(?)and ?P(?) as subspaces of N+(?).- 5.24 HP(?)and ?p(?) as Banach spaces.- 5.25 Local convergence to a boundary function.- 5.26 Remark on the definition of HP(?).- 5.27 Plancherel theorem.- 5.28 Paley-Wiener representation.- 5.29 Natural isomorphisms.- 5.30 Hilbert transforms.- 5.31 Real and imaginary parts of boundary functions.- 5.32 Cauchy transform on Lp(??, ?).- 5.33 Mapping f? f—i f on Lp(-?, ?) to HP(R).- 5.34 M Riesz theorem.- 5.35 Algebraic properties of Hilbert transforms.- Examples and addenda.- 6 Phragmén-Lindelöf Principle.- 6.1 Introduction.- 6.2 Phragmén-Lindelöf principle.- 6.3 Functions on a sector.- 6.4 Estimate from behavior on the imaginary axis.- 6.5 Blaschke products on the imaginary axis.- 6.6 Equivalence of the unit disk and a half-disk.- 6.7 Function theory on a half-disk.- 6.8 Estimates on a half-disk.- 6.9 Test to belong to N(?).- 6.10 Asymptotic behavior of Poisson integrals.- 6.11 Estimate from behavior on semicircles.- 6.12 Blaschke products on semicircles.- 6.13 Factorization of bounded type functions.- 6.14 Nevanlinna factorization and mean type.- 6.15 Formulas for mean type.- 6.16 Exponential type.- 6.17 Kre?n’s theorem.- 6.18 Inequalities for mean type.- Examples and addenda.- 7 Loewner Families.- 7.1 Definitions and overview of the subject.- 7.2 Preliminary results.- 7.3 Riemann mapping theorem.- 7.4 The Dirichlet space and area theorem.- 7.5 Generalization of the Dirichlet space.- 7.6 Bieberbach’s theorem.- 7.7 Size of the image domain.- 7.8Distortion theorem.- 7.9 Carathéodory convergence theorem.- 7.10 Subordination.- 7.11 Technical lemmas.- 7.12 Parametric representation of Loewner families.- 8 Loewner’s Differential Equation.- 8.1 Loewner families and associated semigroups.- 8.2 Estimates derived from Schwarz’s lemma.- 8.3 Absolute continuity.- 8.4 Herglotz functions.- 8.5 Loewner’s differential equation.- 8.6 Solution of the nonlinear equation.- 8.7 Solution of Loewner’s differential equation.- 9 Coefficient Inequalities.- 9.1 Three famous problems.- 9.2 de Branges’ method.- 9.3 Construction of the weight functions.- 9.4 Askey-Gasper inequality.- Notes.

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