Extended Abstracts Fall 2019: Spaces of Analytic Functions: Approximation, Interpolation, Sampling (Trends in Mathematics)

✍ Scribed by Evgeny Abakumov (editor), Anton Baranov (editor), Alexander Borichev (editor), Konstantin Fedorovskiy (editor), Joaquim Ortega-Cerdà (editor)

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 223
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This book collects the abstracts of the mini-courses and lectures given during the Intensive Research Program “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” which was held at the Centre de Recerca Matemàtica (Barcelona) in October–December, 2019.

The topics covered in this volume are approximation, interpolation and sampling problems in spaces of analytic functions, their applications to spectral theory, Gabor analysis and random analytic functions. In many places in the book, we see how a problem related to one of the topics is tackled with techniques and ideas coming from another.

The book will be of interest for specialists in Complex Analysis, Function and Operator theory, Approximation theory, and their applications, but also for young people starting their research in these areas.

✦ Table of Contents

Preface
Contents
Intensive Research Program `Spaces of Analytic Functions: Approximation, Interpolation, Sampling'
1 Background, Objectives and Perspectives of the Program
2 Conference
3 Workshop
4 Advanced Courses and Research Seminar
Comparison of Clark Measures in Several Complex Variables
1 Introduction
1.1 Clark Measures
1.2 Model Spaces and de Branges–Rovnyak Spaces
1.3 Comparison of Clark Measures
2 Clark Measures and de Branges–Rovnyak Spaces
2.1 Cauchy Integrals and Clark Measures
2.2 Partial Isometries Vb, α: L2(σα[b]) tomathcalH(b)
3 Proof of Theorem 2
3.1 Auxiliary Results and Definitions
3.2 Proof of Theorem 2
References
On Spectrum of a Class of Jacobi Matrices on Graph-Trees and Multiple Orthogonal Polynomials
1 Introduction
1.1 Definition of Jacobi Operators
1.2 Multiple Orthogonal Polynomials and Recurrence Relations
2 Angelesco Systems and Main Results
2.1 Angelesco Systems and Ray's Limits of NNRR Coefficients
2.2 Main Results
2.3 Expressions for the Ray's Limits
References
Geometric Properties of Reproducing Kernels in Hilbert Spaces of Entire Functions
1 Introduction
1.1 System of Vectors in Abstract Hilbert Space
2 Exponential Systems on an Interval, Paley–Wiener Spaces and de Branges Spaces
2.1 Paley–Wiener Spaces
2.2 De Branges Spaces
3 Fock Space
References
A New Life of the Classical Szegő Formula
1 Introduction
2 Krein Strings
3 Scattering Theory for Dirac Operators
4 Triangular Factorization of Wiener–Hopf Operators
References
De Branges Canonical Systems with Finite Logarithmic Integral
1 Main Results
References
Rate of Convergence of Critical Interfaces to SLE Curves
1 Introduction: Deterministic Loewner Curves
2 Schramm–Loewner Evolution and Critical Interfaces
3 Critical Site Percolation
4 Polynomial Rate of Convergence: A General Framework
References
Toeplitz and Hankel Operators on Bergman Spaces
1 Preliminaries
2 Toeplitz Operators
3 Hankel Operators
3.1 Trace Estimates
3.2 Critical Decay
References
Bounds for Zeta and Primes via Fourier Analysis
1 The Smallest Bandlimited Function
2 Prime Gaps and RH
References
On Zeros of Solutions of a Linear Differential Equation
References
On Riesz Bases of Exponentials for Convex Polytopes with Symmetric Faces
1 Orthogonal Bases of Exponentials
2 Riesz Bases of Exponentials
3 Convex Polytopes with Symmetric Faces
4 Open Problems
References
Remez-Type Inequalities and Their Applications
1 Introduction
2 Remez Inequality
3 Turán-Nazarov Inequality
4 Discrete Turán-Nazarov Inequality
5 Logvinenko-Sereda Type Estimates
References
Shift-Invariant Spaces of Entire Functions
1 Shift-Invariant Spaces
2 Sampling
2.1 Sampling with Derivatives
3 Interpolation
4 Phase-Retrieval in Shift-Invariant Spaces
5 Further Problems
References
Describing Blaschke Products by Their Critical Points
1 Introduction
2 Inner Functions of Finite Entropy
3 Conformal Metrics and Liouville's Theorem
4 Invariant Subspaces of Bergman Space
5 Canonical Solutions
5.1 Why Is I' in[H]?
5.2 The Case When H=I'
5.3 Why Is I An Inner Function (For General H)?
5.4 Inner Functions Embed into Invariant Subspaces
5.5 I'0 Generates [H]
5.6 Does I' Generate [H]?
References
Two Problems on Homogenization in Geometry
1 Homogenization in Probability
2 Random Quasiconformal Mappings
3 Circle Packing
4 A Lemma on Percolation
References
Toeplitz Operators Between Distinct Abstract Hardy Spaces
1 Classical Hardy Spaces
2 The Riesz Projection
3 The Brown–Halmos Theorem
4 Banach Function Spaces
5 Abstract Hardy Spaces
6 Pointwise Multipliers
7 Main Result
8 Density of Analytic Polynomials in Abstract Hardy Spaces
9 Formulae for the Norm in a Banach Function Space
References
Polynomial Hermite Padé m-System and Reconstruction of the Values of Algebraic Functions
1 Polynomial Hermite–Padé m-System
2 Hermite–Padé m-System for Germs of Functions That Are Meromorphic on an m-Sheeted Compact Riemann Surface
3 Reconstruction of the Values of an Algebraic Function
4 Ideas of the Proof of Theorems 2 and 3
References
Quantitative Szegő Minimum Problem for Some non-Szegő Measures
1 Introduction
2 Absolutely Continuous Measures
3 Singular Measures
4 Nevai's Conjecture
References
Hausdorff Dimension Exceptional Set Estimates for Projections, Sections and Intersections
1 Introduction
2 Hausdorff Dimension and Exceptional Projections
3 Plane Sections and Radial Projections
4 General Intersections
References
Generic Boundary Behaviour of Taylor Series in Banach Spaces of Holomorphic Functions
1 Universality of Taylor Sections
2 Simultaneous Approximation by Polynomials
References
Szegö-Type ASD for Multiplicative Toeplitz'' Operators 1 Toeplitz Operators 2 Operator Følner Sequences (W. Arveson, A. Connes, E. Bédos) 3 Toeplitz-Like Matrices over Discrete Groups 4 Comments on the Følner Condition (F) 5Multiplicative Toeplitz'' Matrices
6 Toeplitz-Like Operators on Hardy Spaces
7 Examples
8 Asymptotic Spectral Densities as Spectral Approximations (Cum Grano Salis)
References
Around Uncertainty Principle
1 Introduction
2 Spectral Gaps
3 Uniqueness Sets for Paley–Wiener Spaces
References
Inner Functions, Completeness and Spectra
1 Inner Functions and Clark Theory
2 Normalized Cauchy Transform
3 Toeplitz Operators
4 Toeplitz Version of BM Theory
5 Toeplitz Order
6 The General Beurling–Malliavin Problem
7 The Gap Problem
8 The Type Problem
9 Toeplitz Order in Comparison with Similar Relations Among Inner Functions
References
Schmidt Subspaces of Hankel Operators
1 Introduction
1.1 Motivation
1.2 Summary
1.3 Schmidt Subspaces
1.4 Hankel and Toeplitz Matrices
1.5 Hardy Space
1.6 Toeplitz Operators in Hardy Space
1.7 Hankel Operators in Hardy Space
2 Inner Functions, Model Spaces and Isometric Multipliers
2.1 Inner Functions
2.2 Model Spaces
2.3 Isometric Multipliers on Model Spaces
2.4 Frostman Shifts
2.5 Nearly Invariant Subspaces
2.6 Toeplitz Kernels
3 Schmidt Subspaces of Hankel Operators
3.1 Preliminaries
3.2 Main Result
3.3 The Action of Hu on EHu(s)
3.4 Decompositions of Model Spaces
3.5 The Adamyan-Arov-Krein Theorem
3.6 Inverse Spectral Problems
References
Maximum Principle and Comparison of Singular Numbers for Composition Operators
1 Introduction
1.1 General Setting
1.2 Starting Point
2 V. Katsnelson's Result, One New Application
2.1 Maximum Principle
2.2 The Result
2.3 Comments on the Assumptions
3 Singular Numbers, Improvement on Katsnelson
3.1 Singular Numbers
3.2 Subordination and Log-Subordination
3.3 New Theorem
4 Application to Composition Operators
4.1 Our Theorem in This Context
5 Strong'' Points 6Weak'' Points
7 Back to Lens and Cusps
References
Canonical Systems in Classes of Compact Operators
Reference
S-Contours and Convergent Interpolation
1 Multipoint Padé Approximants
2 Stahl–Gonchar–Rakhmanov Theory
3 Szegő-Type Convergence
References
Special Conformal Mappings and Extremal Problems
1 Kharkov's Edition of the Classical Chebyshev Theorem
2 Asymptotics of Chebyshev Polynomials on Cantor Sets
3 Remez Problem for Trigonometric Polynomials
References