Jacobi Matrices and the Moment Problem

• Autore: Berezansky Yurij M
• Pubblicazione: Cham : , : Springer International Publishing AG, , 2024; ©2023
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (489 pages)
• Soggetto topico: Matrius (Matemàtica)
• Soggetto genere / forma: Llibres electrònics
• Altri autori: DudkinMykola E
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Introduction -- 2 Some Aspects of the Spectral Theory of Unbounded Operators -- 2.1 Preliminary Information About Unbounded Operators -- 2.2 Extensions of Hermitian Operators to Self-AdJoint Operators -- 2.3 The Spectral Decomposition of Bounded Operators -- 2.4 The Spectral Decomposition of Unbounded Operators -- 2.5 Representations of One-Parameter Unitary Groups -- 2.6 Self-Adjointness Criteria -- 2.7 Rigged Spaces -- 2.8 Tensor Products -- 2.9 Representations of Continuous Multi-Linear Forms -- 2.10 Semi-Bounded Bilinear Forms -- 2.11 The Generalized Eigenvectors Expansion -- 2.12 A General Case with a Quasi-Scalar Product -- Bibliographical Notes -- 3 Jacobi Matrices and the Classical Moment Problem -- 3.1 Difference Operators, Jacobi Matrices, and Self-Adjointness Conditions -- 3.2 The Generalized Eigenvectors Expansion and the Fourier Transform Corresponding to Jacobi Matrices -- 3.3 The Inverse Problem -- 3.4 Further Spectral Analysis of the Difference Operator -- 3.5 The Classical Moment Problem -- 3.6 Some Other Generalizations of the Moment Problem -- 3.7 Connections with the Theory of Jacobi Matrices -- Bibliographical Notes -- 4 The Strong Moment Problem -- 4.1 Preliminaries to the Strong Moment Problem -- 4.2 The Solution of the Strong Moment Problem -- 4.3 The Orthogonalization Procedure and the Construction of a Tri-Diagonal Block Matrix -- 4.4 Direct and Inverse Spectral Problems Corresponding to Tri-Diagonal Block Jacobi-Laurent Matrices Generating Self-Adjoint Operators -- 4.5 Considerations of Hermitian Block Jacobi-Laurent Type Matrices -- 4.6 The Connection Between the Strong Moment Problem and the Spectral Theory of Jacobi-Laurent Matrices -- 4.7 Two Addition Facts -- 4.8 The Inner Structure of the Jacobi-Laurent Matrix -- Bibliographical Notes. ; 5 Block Jacobi Type Matrices in the Complex Moment Problem -- 5.1 Construction of the Tri-Diagonal Block Jacobi Type Matrix of a Bounded Normal Operator -- 5.2 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices of Bounded Normal Operators -- 5.3 Normality Conditions of Block Jacobi Type Matrices -- 5.4 The Solution of the Complex Moment Problem -- 5.5 The Weyl Function and Polynomials of the Second Kind in the Complex Moment Problem -- Bibliographical Notes -- 6 Unitary Block Jacobi Type Matrices and the Trigonometric Moment Problem -- 6.1 Introduction to the Unitary Block Jacobi Type Matrices -- 6.2 Construction of the Tri-Diagonal Block Jacobi Type Matrices of a Unitary Operator -- 6.3 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices of Unitary Operators -- 6.4 The Detailed Internal Structure of the Block Matrix of the Unitary Operator -- 6.5 The Solution of the Trigonometric Moment Problem -- Bibliographical Notes -- 7 Block Jacobi Type Matrices and the Complex Moment Problem in the Exponential Form -- 7.1 Construction of the Tri-Diagonal Block Matrix Corresponding to the Complex Moment Problem in the Exponential Form -- 7.2 Direct and Inverse Spectral Problems for Tri-Diagonal Block Jacobi Type Matrices Corresponding to the Complex Moment Problem in the Exponential Form -- 7.3 Conditions of Unitarity and Commutativity of Matrices Corresponding to the Complex Moment Problem in the Exponential Form -- 7.4 The Solution of the Complex Moment Problem in the Exponential Form -- Bibliographical Notes -- 8 Block Jacobi Type Matrices and the Two Dimensional Real Moment Problem -- 8.1 Construction of the Tri-Diagonal Block Jacobi Type Matrices Corresponding to the Two Dimensional Real Moment Problem. ; 8.2 Direct and Inverse Spectral Problems for the Tri-Diagonal Block Jacobi Type Matrices Corresponding to the Two-Dimensional Real Moment Problem -- 8.3 An Analogue of the Weyl Function Corresponding to the Two-Dimensional Real Moment Problem -- 8.4 The Solution of the Two-Dimensional Real Moment Problem -- 8.5 Examples of Matrices Corresponding to the Two-Dimensional Real Moment Problem -- Bibliographical Notes -- 9 Applications of the Spectral Theory of Jacobi Matrices and Their Generalizations to the Integration of Nonlinear Equations -- 9.1 The Integration of the Toda Chain on the Semi-axis Using the Spectral Theory of Jacobi Matrices -- 9.2 The Doubly Infinite Toda Chain and Its Equivalent Lax Equation for Block Jacobi Type Matrices -- 9.3 The Spectral Theory of Corresponding to Toda Chains Block Jacobi Type Matrices -- 9.4 Equations for the Weyl Function and Spectral Matrices -- 9.5 The Basic Statement and Applications to Hamiltonian Systems -- Bibliographical Notes -- References -- Subject Index -- Notation Index.
• Titolo autorizzato: Jacobi Matrices and the Moment Problem
• ISBN: 3-031-46387-0
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910767547303321
• Lo trovi qui: Univ. Federico II
• Serie: Operator Theory: Advances and Applications Series