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. LinearSuperpositions.-
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.