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The Resource Szegő's theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials, Barry Simon

Szegő's theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials, Barry Simon

Label
Szegő's theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials
Title
Szegő's theorem and its descendants
Title remainder
spectral theory for L2 perturbations of orthogonal polynomials
Statement of responsibility
Barry Simon
Creator
Subject
Language
eng
Summary
  • This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gabor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical backgrund that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first booklength treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line
  • In addition to the Szego and Killip--Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC. --Book Jacket
Cataloging source
DLC
http://library.link/vocab/creatorDate
1946-
http://library.link/vocab/creatorName
Simon, Barry
Dewey number
515/.55
Illustrations
illustrations
Index
index present
LC call number
QC20.7.S64
LC item number
S56 2011
Literary form
non fiction
Nature of contents
bibliography
http://library.link/vocab/subjectName
  • Spectral theory (Mathematics)
  • Orthogonal polynomials
  • Théorie spectrale (mathématiques)
  • Polynômes orthogonaux
  • Orthogonal polynomials
  • Spectral theory (Mathematics)
Label
Szegő's theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials, Barry Simon
Instantiates
Publication
Note
"M. B. Porter lectures"--half t.p. verso
Bibliography note
Includes bibliographical references (pages [607]-640) and indexes
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Favard's Theorem, the Spectral Theorem, and the Direct Problem for OPRL
  • 1.4.
  • Gems of Spectral Theory
  • 1.5.
  • Sum Rules and the Plancherel Theorem
  • 1.6.
  • Polya's Conjecture and Szego's Theorem
  • 1.7.
  • OPUC and Szego's Restatement
  • 1.8.
  • Machine generated contents note:
  • Verblunsky's Form of Szego's Theorem
  • 1.9.
  • Back to OPRL: Szego Mapping and the Shohat -- Nevai Theorem
  • 1.10.
  • The Killip -- Simon Theorem
  • 1.11.
  • Perturbations of the Periodic Case
  • 1.12.
  • Other Gems in the Spectral Theory of OPUC
  • ch. 2
  • ch. 1
  • Szego's Theorem
  • 2.1.
  • Statement and Strategy
  • 2.2.
  • The Szego Integral as an Entropy
  • 2.3.
  • Caratheodory, Herglotz, and Schur Functions
  • 2.4.
  • Weyl Solutions
  • 2.5.
  • Gems of Spectral Theory
  • Coefficient Stripping, Geronimus' and Verblunsky's Theorems, and Continued Fractions
  • 2.6.
  • The Relative Szego Function and the Step-by-Step Sum Rule
  • 2.7.
  • The Proof of Szego's Theorem
  • 1.1.
  • What Is Spectral Theory?
  • 1.2.
  • OPRL as a Solution of an Inverse Problem
  • 1.3.
  • 2.12.
  • The Variational Approach to Szego's Theorem
  • 2.13.
  • Another Approach to Szego Asymptotics
  • 2.14.
  • Paraothogonal Polynomials and Their Zeros
  • 2.15.
  • Asymptotics of the CD Kernel: Weak Limits
  • 2.16.
  • Asymptotics of the CD Kernel: Continous Weights
  • 2.8.
  • 2.17.
  • Asymptotics of the CD Kernel: Locally Szego Weights
  • ch. 3
  • The Killip -- Simon Theorem: Szego for OPRL
  • 3.1.
  • Statement and Strategy
  • 3.2.
  • Weyl Solutions and Coefficient Stripping
  • 3.3.
  • Meromorphic Herglotz Functions
  • A Higher-Order Szego Theorem
  • 3.4.
  • Step-by-Step Sum Rules for OPRL
  • 3.5.
  • The P2 Sum Rule and the Killip-Simon Theorem
  • 3.6.
  • An Extended Shohat -- Navai Theorem
  • 3.7.
  • Szego Asymptotics for OPRL
  • 3.8.
  • The Moment Problem: An Aside
  • 2.9.
  • 3.9.
  • The Krein Density Theorem and Indeterminate Moment Problems
  • 3.10.
  • The Nevai Class and Nevai Delta Convergence Theorem
  • 3.11.
  • Asymptotics of the CD Kernel: OPRL on [-2,2]
  • The Szego Function and Szego Asymptotics
  • 2.10.
  • Asymptotics for Weyl Solutions
  • 2.11.
  • Additional Aspects of Szego's Theorem
  • 4.3.
  • Coefficient Stripping
  • 4.4.
  • Step-by-Step Sum Rules of MOPRL
  • 4.5.
  • A Shohat -- Nevai Theorem for MOPRL
  • 4.6.
  • A Killip -- Simon Theorem for MOPRL
  • ch. 5
  • Periodic OPRL
  • 3.12.
  • 5.1.
  • Overview
  • 5.2.
  • m-Functions and Quadratic Irrationalities
  • 5.3.
  • Real Floquet Theory and Direct Integrals
  • 5.4.
  • The Discriminant and Complex Floquet Theory
  • 5.5.
  • Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent
  • Asymptotics of the CD Kernel: Lubinsky's Second Approach
  • 5.6.
  • Approximation by Periodic Spectra, I. Finite Gap Sets
  • 5.7.
  • Chebyshev Polynomials
  • 5.8.
  • Approximation by Periodic Spectra, II. General Sets
  • 5.9.
  • Regularity: An Aside
  • 5.10.
  • The CD Kernel for Periodic Jacobi Matrices
  • ch. 4
  • 5.11.
  • Asymptotics of the CD Kernel: OPRL on General Sets
  • 5.12.
  • Meromorphic Functions on Hyperelliptic Surfaces
  • 5.13.
  • Minimal Herglotz Functions and Isospectral Tori Appendix to Section 5.13: A Child's Garden of Almost Periodic Functions
  • Sum Rules and Consequences for Matrix Orthogonal Polynomials
  • 4.1.
  • Introduction
  • 4.2.
  • Basics of MOPRL
  • 6.3.
  • QR Factorization
  • 6.4.
  • Poisson Brackets of OPs, Eigenvalues, and Weights
  • 6.5.
  • Spectral Solution and Asymptotics of the Toda Flow
  • 6.6.
  • Lax Pairs
  • 6.7.
  • The Symes -- Deift -- Li -- Tomei Integration: Calculation of the Lax Unitaries
  • 5.14.
  • 6.8.
  • Complete Integrability of Periodic Toda Flow and Isospectral Tori
  • 6.9.
  • Independence of Toda Flows and Trace Gradients
  • 6.10.
  • Flows for OPUC
  • ch. 7
  • Right Limits
  • 7.1.
  • Overview
  • Periodic OPUC
  • 7.2.
  • The Essential Spectrum
  • 7.3.
  • The Last -- Simon Theorem on A.C. Spectrum
  • 7.4.
  • Remling's Theorem on A.C. Spectrum
  • 7.5.
  • Purely Reflectionless Jacobi Matrices on Finite Gap Sets
  • 7.6.
  • The Denisov -- Rakhmanov -- Remling Theorem
  • ch. 6
  • ch. 8
  • Szego and Killip -- Simon Theorems for Periodic OPRL
  • 8.1.
  • Overview
  • 8.2.
  • The Magic Formula
  • 8.3.
  • The Determinant of the Matrix Weight
  • 8.4.
  • A Shohat -- Nevai Theorem for Periodic Jacobi Matrices
  • Toda Flows and Symplectic Structures
  • 8.5.
  • Controlling the l2 Approach to the Isospectral Torus
  • 6.1.
  • Overview
  • 6.2.
  • Symplectic Dynamics and Completely Integrable Systems
  • 9.2.
  • Fractional Linear Transformations
  • 9.3.
  • Mobius Transformations
  • 9.4.
  • Fuchsian Groups
  • 9.5.
  • Covering Maps for Multiconnected Regions
  • 9.6.
  • The Fuchsian Group of a Finite Gap Set
  • 8.6.
  • 9.7.
  • Blaschke Products and Green's Functions
  • 9.8.
  • Continuity of the Covering Map
  • 9.9.
  • Step-by-Step Sum Rules for Finite Gap Jacobi Matrices
  • 9.10.
  • The Szego -- Shohat -- Nevai Theorem for Finite Gap Jacobi Matrices
  • 9.11.
  • Theta Functions and Abel's Theorem
  • A Killip -- Simon Theorem for Periodic Jacobi Matrices
  • 9.12.
  • Jost Functions and the Jost Isomorphism
  • 9.13.
  • Szego Asymptotics
  • ch. 10
  • A.C. Spectrum for Bethe -- Cayley Trees
  • 10.1.
  • Overview
  • 10.2.
  • The Free Hamiltonian and Radially Symmetric Potentials
  • 8.7.
  • 10.3.
  • Coefficient Stripping for Trees
  • 10.4.
  • A Step-by-Step Sum Rule for Trees
  • 10.5.
  • The Global l2 Theorem
  • 10.6.
  • The Local l2 Theorem
  • Sum Rules for Periodic OPUC
  • ch. 9
  • Szego's Theorem for Finite Gap OPRL
  • 9.1.
  • Overview
Control code
587249070
Dimensions
25 cm
Extent
x, 650 pages
Isbn
9780691147048
Isbn Type
(cloth : alk. paper)
Lccn
2010023223
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other physical details
illustrations
System control number
(OCoLC)587249070
Label
Szegő's theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials, Barry Simon
Publication
Note
"M. B. Porter lectures"--half t.p. verso
Bibliography note
Includes bibliographical references (pages [607]-640) and indexes
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Favard's Theorem, the Spectral Theorem, and the Direct Problem for OPRL
  • 1.4.
  • Gems of Spectral Theory
  • 1.5.
  • Sum Rules and the Plancherel Theorem
  • 1.6.
  • Polya's Conjecture and Szego's Theorem
  • 1.7.
  • OPUC and Szego's Restatement
  • 1.8.
  • Machine generated contents note:
  • Verblunsky's Form of Szego's Theorem
  • 1.9.
  • Back to OPRL: Szego Mapping and the Shohat -- Nevai Theorem
  • 1.10.
  • The Killip -- Simon Theorem
  • 1.11.
  • Perturbations of the Periodic Case
  • 1.12.
  • Other Gems in the Spectral Theory of OPUC
  • ch. 2
  • ch. 1
  • Szego's Theorem
  • 2.1.
  • Statement and Strategy
  • 2.2.
  • The Szego Integral as an Entropy
  • 2.3.
  • Caratheodory, Herglotz, and Schur Functions
  • 2.4.
  • Weyl Solutions
  • 2.5.
  • Gems of Spectral Theory
  • Coefficient Stripping, Geronimus' and Verblunsky's Theorems, and Continued Fractions
  • 2.6.
  • The Relative Szego Function and the Step-by-Step Sum Rule
  • 2.7.
  • The Proof of Szego's Theorem
  • 1.1.
  • What Is Spectral Theory?
  • 1.2.
  • OPRL as a Solution of an Inverse Problem
  • 1.3.
  • 2.12.
  • The Variational Approach to Szego's Theorem
  • 2.13.
  • Another Approach to Szego Asymptotics
  • 2.14.
  • Paraothogonal Polynomials and Their Zeros
  • 2.15.
  • Asymptotics of the CD Kernel: Weak Limits
  • 2.16.
  • Asymptotics of the CD Kernel: Continous Weights
  • 2.8.
  • 2.17.
  • Asymptotics of the CD Kernel: Locally Szego Weights
  • ch. 3
  • The Killip -- Simon Theorem: Szego for OPRL
  • 3.1.
  • Statement and Strategy
  • 3.2.
  • Weyl Solutions and Coefficient Stripping
  • 3.3.
  • Meromorphic Herglotz Functions
  • A Higher-Order Szego Theorem
  • 3.4.
  • Step-by-Step Sum Rules for OPRL
  • 3.5.
  • The P2 Sum Rule and the Killip-Simon Theorem
  • 3.6.
  • An Extended Shohat -- Navai Theorem
  • 3.7.
  • Szego Asymptotics for OPRL
  • 3.8.
  • The Moment Problem: An Aside
  • 2.9.
  • 3.9.
  • The Krein Density Theorem and Indeterminate Moment Problems
  • 3.10.
  • The Nevai Class and Nevai Delta Convergence Theorem
  • 3.11.
  • Asymptotics of the CD Kernel: OPRL on [-2,2]
  • The Szego Function and Szego Asymptotics
  • 2.10.
  • Asymptotics for Weyl Solutions
  • 2.11.
  • Additional Aspects of Szego's Theorem
  • 4.3.
  • Coefficient Stripping
  • 4.4.
  • Step-by-Step Sum Rules of MOPRL
  • 4.5.
  • A Shohat -- Nevai Theorem for MOPRL
  • 4.6.
  • A Killip -- Simon Theorem for MOPRL
  • ch. 5
  • Periodic OPRL
  • 3.12.
  • 5.1.
  • Overview
  • 5.2.
  • m-Functions and Quadratic Irrationalities
  • 5.3.
  • Real Floquet Theory and Direct Integrals
  • 5.4.
  • The Discriminant and Complex Floquet Theory
  • 5.5.
  • Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent
  • Asymptotics of the CD Kernel: Lubinsky's Second Approach
  • 5.6.
  • Approximation by Periodic Spectra, I. Finite Gap Sets
  • 5.7.
  • Chebyshev Polynomials
  • 5.8.
  • Approximation by Periodic Spectra, II. General Sets
  • 5.9.
  • Regularity: An Aside
  • 5.10.
  • The CD Kernel for Periodic Jacobi Matrices
  • ch. 4
  • 5.11.
  • Asymptotics of the CD Kernel: OPRL on General Sets
  • 5.12.
  • Meromorphic Functions on Hyperelliptic Surfaces
  • 5.13.
  • Minimal Herglotz Functions and Isospectral Tori Appendix to Section 5.13: A Child's Garden of Almost Periodic Functions
  • Sum Rules and Consequences for Matrix Orthogonal Polynomials
  • 4.1.
  • Introduction
  • 4.2.
  • Basics of MOPRL
  • 6.3.
  • QR Factorization
  • 6.4.
  • Poisson Brackets of OPs, Eigenvalues, and Weights
  • 6.5.
  • Spectral Solution and Asymptotics of the Toda Flow
  • 6.6.
  • Lax Pairs
  • 6.7.
  • The Symes -- Deift -- Li -- Tomei Integration: Calculation of the Lax Unitaries
  • 5.14.
  • 6.8.
  • Complete Integrability of Periodic Toda Flow and Isospectral Tori
  • 6.9.
  • Independence of Toda Flows and Trace Gradients
  • 6.10.
  • Flows for OPUC
  • ch. 7
  • Right Limits
  • 7.1.
  • Overview
  • Periodic OPUC
  • 7.2.
  • The Essential Spectrum
  • 7.3.
  • The Last -- Simon Theorem on A.C. Spectrum
  • 7.4.
  • Remling's Theorem on A.C. Spectrum
  • 7.5.
  • Purely Reflectionless Jacobi Matrices on Finite Gap Sets
  • 7.6.
  • The Denisov -- Rakhmanov -- Remling Theorem
  • ch. 6
  • ch. 8
  • Szego and Killip -- Simon Theorems for Periodic OPRL
  • 8.1.
  • Overview
  • 8.2.
  • The Magic Formula
  • 8.3.
  • The Determinant of the Matrix Weight
  • 8.4.
  • A Shohat -- Nevai Theorem for Periodic Jacobi Matrices
  • Toda Flows and Symplectic Structures
  • 8.5.
  • Controlling the l2 Approach to the Isospectral Torus
  • 6.1.
  • Overview
  • 6.2.
  • Symplectic Dynamics and Completely Integrable Systems
  • 9.2.
  • Fractional Linear Transformations
  • 9.3.
  • Mobius Transformations
  • 9.4.
  • Fuchsian Groups
  • 9.5.
  • Covering Maps for Multiconnected Regions
  • 9.6.
  • The Fuchsian Group of a Finite Gap Set
  • 8.6.
  • 9.7.
  • Blaschke Products and Green's Functions
  • 9.8.
  • Continuity of the Covering Map
  • 9.9.
  • Step-by-Step Sum Rules for Finite Gap Jacobi Matrices
  • 9.10.
  • The Szego -- Shohat -- Nevai Theorem for Finite Gap Jacobi Matrices
  • 9.11.
  • Theta Functions and Abel's Theorem
  • A Killip -- Simon Theorem for Periodic Jacobi Matrices
  • 9.12.
  • Jost Functions and the Jost Isomorphism
  • 9.13.
  • Szego Asymptotics
  • ch. 10
  • A.C. Spectrum for Bethe -- Cayley Trees
  • 10.1.
  • Overview
  • 10.2.
  • The Free Hamiltonian and Radially Symmetric Potentials
  • 8.7.
  • 10.3.
  • Coefficient Stripping for Trees
  • 10.4.
  • A Step-by-Step Sum Rule for Trees
  • 10.5.
  • The Global l2 Theorem
  • 10.6.
  • The Local l2 Theorem
  • Sum Rules for Periodic OPUC
  • ch. 9
  • Szego's Theorem for Finite Gap OPRL
  • 9.1.
  • Overview
Control code
587249070
Dimensions
25 cm
Extent
x, 650 pages
Isbn
9780691147048
Isbn Type
(cloth : alk. paper)
Lccn
2010023223
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other physical details
illustrations
System control number
(OCoLC)587249070

Library Locations

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      38.944377 -92.326537
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