Constructive Approximation

1. Problems of Polynomial Approximation.- x 1. Examples of Polynomials of Best Approximation.- x 2. Distribution of Alternation Points of Polynomials of Best Approximation.- x 3. Distribution of Zeros of Polynomials of Best Approximation.- x 4. Error of Approximation.- x 5. Approximation on (-?, ?) by Linear Combinations of Functions (x - c)-1.- x 6. Weighted Approximation by Polynomials on (-?, ?).- x 7. Spaces of Approximation Theory.- x 8. Problems and Notes.- 2. Approximation Problems with Constraints.- x1. Introduction.- x 2. Growth Restrictions for the Coefficients.- x 3. Monotone Approximation.- x 4. Polynomials with Integral Coefficients.- x 5. Determination of the Characteristic Sets.- x 6. Markov-Type Inequalities.- x 7. The Inequality of Remez.- x 8. One-sided Approximation by Polynomials.- x9. Problems.- x 10. Notes.- 3. Incomplete Polynomials.- x 1. Incomplete Polynomials.- x 2. Incomplete Chebyshev Polynomials.- x 3. Incomplete Trigonometric Polynomials.- x 4. Sequences of Polynomials with Many Real Zeros.- x5. Problems.- x6. Notes.- 4. Weighted Polynomials.- x 1. Essential Sets of Weighted Polynomials.- x 2. Weighted Chebyshev Polynomials.- x3. The Equilibrium Measure.- x 4. Determination of Minimal Essential Sets.- x 5. Weierstrass Theorems and Oscillations.- x 6. Weierstrass Theorem for Freud Weights.- x7. Problems.- x8. Notes.- 5. Wavelets and Orthogonal Expansions.- x 1. Multiresolutions and Wavelets.- x 2. Scaling Functions with a Monotone Majorant.- x 3. Periodization.- x 4. Polynomial Schauder Bases.- x 5. Orthonormal Polynomial Bases.- x 6. Problems and Notes.- 6. Splines.- x1. General Facts.- x 2. Splines of Best Approximation.- x 3. Periodic Splines.- x 4. Convergence of Some Spline Operators.- x5. Notes.- 7. Rational Approximation.- x1. Introduction.- x2. Best Rational Approximation.- x 3. Rational Approximation of |x|.- x4. Approximation of e-xon [-1,1].- x 5. Rational Approximation of e-x on [0, ?).- x 6. Approximation of Classes of Functions.- x 7. Theorems of Popov.- x 8. Properties of the Operator of Best Rational Approximation in C and Lp.- x 9. Approximation by Rational Functions with Arbitrary Powers.- x 10. Problems.- x11. Notes.- 8. StahPs Theorem.- x 1. Introduction and Main Result.- x 2. A Dirichlet Problem on [1/2, l/pn].- x 3. The Second Approach to the Dirichlet Problem.- x 4. Proof of Theorem 1.1.- x5. Notes.- 9. Pade Approximation.- x1. The Pade Table.- x 2. Convergence of the Rows of the Pade Table.- x 3. The Nuttall-Pommerenke Theorem.- x4. Problems.- x5. Notes.- 10. Hardy Space Methods in Rational Approximation.- x 1. Bernstein-Type Inequalities for Rational Functions.- x 2. Uniform Rational Approximation in Hardy Spaces.- x 3. Approximation by Simple Functions.- x 4. The Jackson-Rusak Operator; Rational Approximation of Sums of Simple Functions.- x 5. Rational Approximation on T and on [-1,1].- x 6. Relations Between Spline and Rational Approximation in the Spaces 0 ?.- x7. Problems.- x8. Notes.- 11. Muntz Polynomials.- x 1. Definitions and Simple Properties.- x 2. Muntz-Jackson Theorems.- x 3. An Inverse Muntz-Jackson Theorem.- x 4. The Index of Approximation.- x 5. Markov-Type Inequality for Muntz Polynomials.- x6. Problems.- x7. Notes.- 12. Nonlinear Approximation.- x 1. Definitions and Simple Properties.- x 2. Varisolvent Families.- x 3. Exponential Sums.- x 4. Lower Bounds for Errors of Nonlinear Approximation.- x 5. Continuous Selections from Metric Projections.- x 6. Approximation in Banach Spaces: Suns and Chebyshev Sets.- x7. Problems.- x8. Notes.- 13. Widths I.- x 1. Definitions and Basic Properties.- x 2. Relations Between Different Widths.- x 3. Widths of Cubes and Octahedra.- x4. Widths in Hilbert Spaces.- x 5. Applications of Borsuk's Theorem.- x 6. Variational Problems and Spectral Functions.- x 7. Results of Buslaev and Tikhomirov.- x 8. Classes of Differentiate Functions on an Interval.- x 9. Classes of Analytic Functions.- x 10. Problems.- x11. Notes.- 14. Widths II: Weak Asymptotics for Widths of Lipschitz Balls, Random Approximants.- x1. Introduction.- x 2. Discretization.- x 3. Weak Equivalences for Widths. Elementary Methods.- x 4. Distribution of Scalar Products of Unit Vectors.- x 5. Kashin's Theorems.- x 6. Gaussian Measures.- x 7. Linear Widths of Finite Dimensional Balls.- x 8. Linear Widths of the Lipschitz Classes.- x9. Problems.- x10. Notes.- 15. Entropy.- x 1. Entropy and Capacity.- x 2. Elementary Estimates.- x 3. Linear Approximation and Entropy.- x 4. Relations Between Entropy and Widths.- x 5. Entropy of Classes of Analytic Functions.- x 6. The Birman-Solomyak Theorem.- x 7. Entropy Numbers of Operators.- x8. Notes.- 16. Convergence of Sequences of Operators.- x 1. Introduction.- x 2. Simple Necessary and Sufficient Conditions.- x 3. Geometric Properties of Dominating Sets.- x 4. Strict Dominating Systems; Minimal Systems; Examples.- x 5. Shadows of Sets of Continuous Functions.- x 6. Shadows in Banach Function Spaces.- x 7. Positive Contractions.- x 8. Contractions.- x9. Notes.- 17. Representation of Functions by Superpositions.- x 1. The Theorems of Kolmogorov.- x2. Proof of the Theorems.- x 3. Functions Not Representable by Superpositions.- x 4. Linear Superpositions.- x5. Notes.- Appendix 1. Theorems of Borsuk and of Brunn-Minkowski.- x1. Borsuk's Theorem.- x2. The Brunn-Minkowski Inequality.- Appendix 2. Estimates of Some Elliptic Integrals.- Appendix 3. Hardy Spaces and Blaschke Products.- x 1. Hardy Spaces.- x 2. Conjugate Functions and Cauchy Integrals.- x 3. Atomic Decompositions in Hardy Spaces.- x 4. Blaschke Products.- Appendix 4. Potential Theory and Logarithmic Capacity.- x1. Logarithmic Potentials.- x 2. Equilibrium Distribution and Logarithmic Capacity.- x 3. The Dirichlet Problem and Green's Function.- x 4. Balayage Methods.- Author Index.