Constructive approximation : advanced problems
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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. Pade Approximation.- 1. The Pade 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 ?.- 7. Problems.- 8. Notes.- 11. Muntz Polynomials.- 1. Definitions and Simple Properties.- 2. Muntz-Jackson Theorems.- 3. An Inverse Muntz-Jackson Theorem.- 4. The Index of Approximation.- 5. Markov-Type Inequality for Muntz 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.