Constructive Approximation

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 (-?, ?) by Linear Combinations of Functions (x - c)-1.- § 6. Weighted Approximation by Polynomials on (-?, ?).- § 7. Spaces of Approximation Theory.- § 8. Problems and Notes.- 2. Approximation Problems 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.- 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.- 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.- 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.- 6. Splines.- §1. General Facts.- § 2. Splines of Best Approximation.- § 3. Periodic Splines.- § 4. Convergence of Some Spline Operators.- §5. Notes.- 7. Rational Approximation.- §1. Introduction.- §2. Best Rational Approximation.- § 3. Rational Approximation of |x|.- §4. Approximation of e-xon [-1,1].- § 5. Rational Approximation of e-x on [0, ?).- § 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.- 8. StahPs Theorem.- § 1. Introduction and Main Result.- § 2. A Dirichlet Problem on [1/2, l/pn].- § 3. The Second Approach to the Dirichlet Problem.- § 4. Proof of Theorem 1.1.- §5. Notes.- 9. Padé Approximation.- §1. The Padé Table.- § 2. Convergence of the Rows of the Pade Table.- § 3. The Nuttall-Pommerenke Theorem.- §4. Problems.- §5. Notes.- 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 0 < p < ?.- §7. Problems.- §8. Notes.- 11. Müntz Polynomials.- § 1. Definitions and Simple Properties.- § 2. Müntz-Jackson Theorems.- § 3. An Inverse Müntz-Jackson Theorem.- § 4. The Index of Approximation.- § 5. Markov-Type Inequality for Müntz Polynomials.- §6. Problems.- §7. Notes.- 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.- 13. Widths I.- § 1. Definitions and Basic Properties.- § 2. Relations Between Different Widths.- § 3. Widths of Cubes and Octahedra.- §4. Widths in Hilbert Spaces.- § 5. Applications of Borsuk's Theorem.- § 6. Variational Problems and Spectral Functions.- § 7. Results of Buslaev and Tikhomirov.- § 8. Classes of Differentiate Functions on an Interval.- § 9. Classes of Analytic Functions.- § 10. Problems.- §11. Notes.- 14. Widths II: Weak Asymptotics for Widths of 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.- 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 Functions.- § 6. The Birman-Solomyak Theorem.- § 7. Entropy Numbers of Operators.- §8. Notes.- 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.- 17. Representation of Functions by Superpositions.- § 1. The Theorems 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.- §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.- Author Index.