Constructive Approximation

In the last 30 years, Approximation Theory has undergone wonderful develop­ ment, with many new theories appearing in this short interval. This book has its origin in the wish to adequately describe this development, in particular, to rewrite the short 1966 book of G. G. Lorentz, "Approximation of Functions." Soon after 1980, R. A. DeVore and Lorentz joined forces for this purpose. The outcome has been their "Constructive Approximation" (1993), volume 303 of this series. References to this book are given as, for example rCA, p.201]. Later, M. v. Golitschek and Y. Makovoz joined Lorentz to produce the present book, as a continuation of the first. Completeness has not been our goal. In some of the theories, our exposition offers a selection of important, representative theorems, some other cases are treated more systematically. As in the first book, we treat only approximation of functions of one real variable. Thus, functions of several variables, complex approximation or interpolation are not treated, although complex variable methods appear often.

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.