Functions of Omega-Bounded Type: Basic Theory

✍ Scribed by Armen M. Jerbashian, Joel E. Restrepo

• Publisher: Birkhäuser
• Year: 2024
• Tongue: English
• Leaves: 366
• Series: Frontiers in Mathematics
• Category: Library

✦ Synopsis

The book gives the basic results of the theory of the spaces A^pω of functions holomorphic in the unit disc, halfplane and in the finite complex plane, which depend on functional weights ω permitting any rate of growth of a function near the boundary of the domain. This continues and essentially improves M.M. Djrbashian's theory of spaces A^pα (1945) of functions holomorphic in the unit disc, the English translation of the detailed and complemented version of which (1948) is given in Addendum to the book. Besides, the book gives the ω-extensions of M. M. Djrbashian's two factorization theories of functions meromorphic in the unit disc of 1945-1948 and 1966-1975 to classes of functions delta-subharmonic in the unit disc and in the half-plane.
The book can be useful for a wide range of readers. It can be a good handbook for Master, PhD students and Postdoctoral Researchers for enlarging their knowledge and analytical methods, as well as a useful resource for scientists who want to extend their investigation fields.

✦ Table of Contents

Preface
Acknowledgements
Introduction
Contents
Part I Omega-Weighted Classes of Area Integrable Regular Functions
1 Preliminary Results
1.1 M.M. Djrbashian Operators Lω and His Omega-Kernels
1.2 Evaluation of M.M. Djrbashian Cω-Kernels
1.2.1 ω Decreases Not More Rapidly than a Power Function
1.2.2 ω Decreases Exponentially
1.3 Volterra Equation and Hausdorff Moment Problem
1.4 Notes
2 Spaces Apω(D) in the Unit Disc
2.1 The Spaces Aωp(D), Representations
2.1.1 Definition and Elementary Properties of Aωp(D) Spaces
2.1.2 Representations of Functions from Aωp(D)
2.2 Projection Theorems and the Conjugate Space of Aωp(D)
2.2.1 Orthogonal Projection and Isometry for Aω2(D)
2.2.2 Projection Theorems from Lωp(D) to Aωp(D) (1≤p<+∞, p≠2)
2.3 Dirichlet Type Spaces Apω(D)Hp(D)
2.3.1 The Omega-Capacity
2.3.2 The Spaces Aωp(D)Hp(D), Boundary Properties
2.4 Notes
3 Spaces Apω(C) of Entire Functions
3.1 Spaces Apω(C) of Entire Functions, Representations
3.2 Orthogonal Projection and Isometry
3.3 Biorthogonal Systems of Functions in A2ω(D) and A2ω(C)
3.4 Unitary Operators in A2ω(C)
3.5 Notes
4 Nevanlinna–Djrbashian Classes of Functions Delta-Subharmonic in the Unit Disc
4.1 Blaschke Type Factors and Green -Type Potentials
4.2 The Main Representation Theorem
4.3 Universal Orthogonal Decomposition
4.4 Notes
5 Spaces Apω,γ(G+) in the Halfplane
5.1 The Spaces Apω,γ(G+)
5.2 Representation by an Integral Over a Strip
5.3 General Representation
5.4 Orthogonal Projection and Isometry
5.5 The Projection Lpω,0(G+) to Apω,0(G+), the Space (Apω,0(G+))*
5.6 Biorthogonal Systems, Bases, and Interpolation in A2˜ω(G+)
5.7 Notes
6 Orthogonal Decomposition of Functions Subharmonic in the Halfplane
6.1 The spaces hp(G+) and hpω(G+)
6.2 Orthogonal Projection L2ω(G+)→h2ω(G+)
6.3 Orthogonal Decomposition
6.4 Notes
7 Nevanlinna–Djrbashian Classes in the Halfplane
7.1 Preliminary Definitions and Statements
7.2 Nevanlinna–Djrbashian Classes in the Halfplane
7.3 Notes
Part II Delta-Subharmonic Extension of M.M. Djrbashian Factorization Theory
8 Extension of the Factorization Theory of M.M. Djrbashian
8.1 Green-Type Potentials
8.2 R. Nevanlinna and M.M. Djrbashian Characteristics
8.3 Classes Nω of Delta-Subharmonic Functions with ωΩ(D)
8.4 Classes Nω of Delta-Subharmonic Functions with ω˜Ω(D)
8.5 Embedding of the Classes Nω
8.6 Boundary Properties of Meromorphic Classes N{ω}N
8.7 Notes
9 Banach Spaces of Functions Delta-Subharmonic in the Unit Disc
9.1 Banach Spaces of Green-Type Potentials
9.1.1 Banach Spaces of Potentials Formed by ˜bω
9.1.2 Smaller Banach Spaces of Potentials Formed by ˜bω
9.1.3 Banach Spaces of Potentials Formed by bω
9.1.4 Smaller Banach Spaces of Potentials Formed by bω
9.2 Banach Spaces of Delta-subharmonic Functions in the Disc
9.3 Notes
10 Functions of Omega-Bounded Type in the Halfplane
10.1 Blaschke-Type Factors
10.2 Green-Type Potentials
10.3 One More Property of Green-Type Potentials
10.4 Representations of Classes of Harmonic Functions
10.5 Riesz-Type Representations with a Minimality Property
10.6 Notes
11 Subclasses of Harmonic Functions with Nonnegative Harmonic Majorants in the Halfplane
11.1 Preliminary Lemmas
11.2 Representations
11.3 Boundary Values
11.4 Notes
12 Subclasses of Delta-subharmonic Functions of Bounded Type in the Halfplane
12.1 Blaschke-Type Factors
12.2 Green-Type Potentials
12.3 Weighted Classes of Delta-Subharmonic Functions
12.4 Boundary Property of Subclasses of Functions of Bounded Type in the Halfplane
12.5 Notes
13 Banach Spaces of Functions Delta-subharmonic in the Halfplane
13.1 Additional Statements on Green-Type Potentials
13.2 Banach Spaces of Potentials ˜Pω
13.3 Smaller Banach Spaces of Potentials ˜Pω
13.4 Banach Spaces of Potentials Pω
13.5 Banach Spaces of Delta-subharmonic Functions
13.6 Notes
A Addendum
A.1 On Representation of Some Classes of Functions Holomorphic in the Unit Disc
B.1 On a Generalization of the Jensen–Nevanlinna Formula. Canonical Representations of Meromorphic Functions of Unbounded Type
C.1 On Representability of Some Classes of Entire Functions
References
References
Index