Cover
Grundlehren der mathematischen Wissenschaften 304
Constructive Approximation: Advanced Problems
Copyright Springer-Verlag Berlin Heidelberg 1996
ISBN 3-540-57028-4
ISBN 0-387-57028-4
SPIN: 10124042
OA221.L63 1996 515'.83--dc20
LCCN 96-298
Preface
Contents
Chapter 1. Problems of Polynomial Approximation
§ 1. Examples of Polynomials of Best Approximation
§ 2. Distribution of Alternation Points of Polynomials of Best Approximation
§ 3. Distribution of Zeros of Polynomials of Best Approximation
§ 4. Error of Approximation
§ 5. Approximation on (-oo, oo) by Linear Combinations of Functions (x - c)-1
§ 6. Weighted Approximationby Polynomials on (-oo, oo)
§ 7. Spaces of Approximation Theory
§ 8. Problems and Notes
Chapter 2. Polynomial Approximation with Constraints
§ 1. Introduction
§ 2. Growth Restrictions for the Coefficients
§ 3. Monotone Approximation
§ 4. Polynomials with Integral Coefficients
§ 5. Determination of the Characteristic Sets
§ 6. Markov-Type Inequalities
§ 7. The Inequality of Remez
§ 8. One-sided Approximation by Polynomials
§ 9. Problems
§10. Notes
Chapter 3. Incomplete Polynomials
§ 1. Incomplete Polynomials
§ 2. Incomplete Chebyshev Polynomials
§ 3. Incomplete Trigonometric Polynomials
§ 4. Sequences of Polynomials with Many Real Zeros
§ 5. Problems
§ 6. Notes
Chapter 4. Weighted Polynomials
§ 1. Essential Sets of Weighted Polynomials
§ 2. Weighted Chebyshev Polynomials
§ 3. The Equilibrium Measure
§ 4. Determination of Minimal Essential Sets
§ 5. Weierstrass Theorems and Oscillations
§ 6. Weierstrass Theorem for Freud Weights
§ 7. Problems
§ 8. Notes
Chapter 5. Wavelets and Orthogonal Expansions
§ 1. Multiresolutions and Wavelets
§ 2. Scaling Functions with a Monotone Majorant
§ 3. Periodization
§ 4. Polynomial Schauder Bases
§ 5. Orthonormal Polynomial Bases
§ 6. Problems and Notes
Chapter 6. Splines
§ 1. General Facts
§ 2. Splines of Best Approximation
§ 3. Periodic Splines
§ 4. Convergence of Some Spline Operators
§ 5. Notes
Chapter 7. Rational Approximation
§ 1. Introduction
§ 2. Best Rational Approximation
§ 3. Rational Approximation of |x|
§ 4. Approximation of ex on [-1, 1]
§ 5. Rational Approximation of e-Ic on [0, oo)
§ 6. Approximation of Classes of Functions
§ 7. Theorems of Popov
§ 8. Properties of the Operator of Best Rational Approximation in C and LP
§ 9. Approximation by Rational Functions with Arbitrary Powers
§ 10. Problems
§ 11. Notes
Chapter 8. Stahl's Theorem
§ 1. Introduction and Main Result
§ 2. A Dirichlet Problem on [1/2, l/p]
§ 3. The Second Approach to the Dirichlet Problem
§ 4. Proof of Theorem 1.1
§ 5. Notes
Chapter 9. Pad Approximation
§ 1. The Pade Table
§ 2. Convergence of the Rows of the Pade Table
§ 3. The Nuttall-Pommerenke Theorem
§ 4. Problems
§ 5. Notes
Chapter 10. Hardy Space Methods in Rational Approximation
§ 1. Bernstein-Type Inequalities for Rational Functions
§ 2. Uniform Rational Approximation in Hardy Spaces
§ 3. Approximation by Simple Functions
§ 4. The Jackson-Rusak Operator; Rational Approximation of Sums of Simple Functions
§ 5. Rational Approximation on T and on [-1, 1]
§ 6. Relations Between Spline and Rational Approximation in the Spaces LP, 0 <P < o0
§ 7. Problems
§ 8. Notes
Chapter 11. Müntz Polynomials
§ 1. Definitions and Simple Properties
§ 2. Muntz-Jackson Theorems
§ 3. An Inverse Miintz-Jackson Theorem
§ 4. The Index of Approximation
§ 5. Markov-Type Inequality for Miintz Polynomials
§ 6. Problems
§7. Notes
Chapter 12. Nonlinear Approximation
§ 1. Definitions and Simple Properties
§ 2. Varisolvent Families
§ 3. Exponential Sums
§ 4. Lower Bounds for Errors of Nonlinear Approximation
§ 5. Continuous Selections from Metric Projections
§ 6. Approximation in Banach Spaces: Suns and Chebyshev Sets
§ 7. Problems
§ 8. Notes
Chapter 13. Widths I
§ 1. Definitions and Basic Properties
§ 2. Relations Between Different Widths
§ 3. Widths of Cubes and Octahedra
§ 4. Widths in Hilb ert Spaces
§ 5. Applications of Borsuk's Theorem
§ 6. Variational Problems and Spectral Functions
§ 7. Results of Buslaev and Tikhomirov
§ 8. Classes of Differentiable Functions on an Interval
§ 9. Classes of Analytic Functions
§ 10. Problems
§ 11. Notes
Chapter 14. Widths II: Weak Asymptotics for Lipschitz Balls, Random Approximants
§ 1. Introduction
§ 2. Discretization
§ 3. Weak Equivalences for Widths. Elementary Methods
§ 4. Distribution of Scalar Products of Unit Vectors
§ 5. Kashin's Theorems
§ 6. Gaussian Measures
§ 7. Linear Widths of Finite Dimensional Balls
§ 8. Linear Widths of the Lipschitz Classes
§ 9. Problems
§ 10. Notes
Chapter 15. Entropy
§ 1. Entropy and Capacity
§ 2. Elementary Estimates
§ 3. Linear Approximation and Entropy
§ 4. Relations Between Entropy and Widths
§ 5. Entropy of Classes of Analytic Panctions
§ 6. The Birman-Solomyak Theorem
§ 7. Entropy Numbers of Operators
§ 8. Notes
Chapter 16. Convergence of Sequences of Operators
§ 1. Introduction
§ 2.. Simple Necessary and Sufficient, Conditions
§ 3. Geometric Properties of Dominating Sets
§ 4. Strict Dominating Systems; Minimal Systems; Examples
§ 5. Shadows of Sets of Continuous Functions
§ 6. Shadows in Banach Function Spaces
§ 7. Positive Contractions
§ 8. Contractions
§ 9. Notes
Chapter 17. Representation of Functions by Superpositions
§ 1. The Theorem of Kolmogorov
§ 2. Proof of the Theorems
§ 3. Functions Not Representable by Superpositions
§ 4. Linear Superpositions
§ 5. Notes
Appendix 1. Theorems of Borsuk and of Brunn-Minkowski
§ 1. Borsuk's Theorem
1.1. Introduction; Different Forms of the Theorem
1.2. Properties of the "Equators" Bk.
1.3. Partition and Triangulation of the Cube Qo. F
1.4. Proof of Theorem 1.2.
§ 2. The Brunn-Minkowski Inequality
Appendix 2. Estimates of Some Elliptic Integrals
Appendix 3. Hardy Spaces and Blaschke Products
§ 1. Hardy Spaces
§ 2. Conjugate Functions and Cauchy Integrals
§ 3. Atomic Decompositions in Hardy Spaces
§ 4. Blaschke Products
Appendix 4. Potential Theory and Logarithmic Capacity
§ 1. Logarithmic Potentials
§ 2. Equilibrium Distribution and Logarithmic Capacity
§3. The Dirichlet Problem and Green's Function
§ 4. Balayage Methods
Bibliography
A. Books on Approximation
B. Other Books
C. Articles
Author Index
Subject Index