General Orthogonal Polynomials

✍ Scribed by Herbert Stahl, Vilmos Totik

• Publisher: Cambridge
• Year: 1992
• Tongue: English
• Leaves: 260
• Category: Library

✦ Table of Contents

Title page
Preface
Acknowledgments
Symbols
1 Upper and Lower Bounds
1.1 Statement of the Main Results
1.2 Some Potential-theoretic Preliminaries
1.3 Proof of the Upper and Lower Bounds
1.4 Proof of the Sharpness of the Upper and Lower Bounds
1.5 Examples
2 Zero Distribution of Orthogonal Polynomials
2.1 Zeros of Orthogonal Polynomials
2.2 Norm Asymptotics and Zero Distribution
2.3 Asymptotic Behavior of Zeros when c_μ>0
3 Regular nth-root Asymptotic Behavior of Orthonormal Polynomials
3.1 Regular Asymptotic Behavior
3.2 Characterization of Regular Asymptotic Behavior
3.3 Regular Behavior in the Case of Varying Weights
3.4 Characterization of Regular Asymptotic Behavior in L^p(μ)
3.5 Examples
3.6 Regular Behavior and Monic Polynomials
4 Regularity Criteria
4.1 Existing Regularity Criteria and Their Generalizations
4.2 New Criteria and Their Sharpness
4.3 Proof of the Regularity Criteria
4.4 Preliminaries for Proving the Sharpness of the Criteria
4.5 Proof of the Sharpness of the Regularity Criteria
4.6 Summary of Regularity Criteria and Their Relations
5 Localization
5.1 Global versus Local Behavior
5.2 Localization at a Single Point
5.3 Localization Theorems
6 Applications
6.1 Rational Interpolants to Markov Functions
6.2 Best Rational Approximants to Markov Functions
6.3 Nondiagonal Padé Approximants to Markov Functions
6.4 Weighted Polynomials in L^p(μ)
6.5 Regularity and Weighted Chebyshev Constants
6.6 Regularity and Best L²(μ) Polynomial Approximation
6.7 Determining Sets
Appendix
A.I Energy and Capacity
A.II Potentials, Fine Topology
A.III Principles
A.IV Equilibrium Measures
A.V Green Functions
A.VI Dirichlet's Problem
A.VII Balayage
A.VIII Green Potential and Condenser Capacity
A.IX The Energy Problem in the Presence of an External Field
Notes and Bibliographical References
Bibliography
Index