Nonlinear Approximation Theory

I. Preliminaries.- § 1. Some Notation, Definitions and Basic Facts.- A. Functional Analytic Notation and Terminology.- B. The Approximation Problem. Definitions and Basic Facts.- C. An Invariance Principle.- D. Divided Differences.- § 2. A Review of the Characterization of Nearest Points in Linear and Convex Sets.- A. Characterization via the Hahn-Banach Theorem and the Kolmogorov Criterion.- B. Special Function Spaces.- § 3. Linear and Convex Chebyshev Approximation.- A. Haar’s Uniqueness Theorem. Alternants.- B. Haar Cones.- C. Alternation Theorem for Haar Cones.- §4. L1-Approximation and Gaussian Quadrature Formulas.- A. The Hobby-Rice Theorem.- B. Existence of Generalized Gaussian Quadrature Formulas.- C. Extremal Properties.- II. Nonlinear Approximation: The Functional Analytic Approach.- §1. Approximative Properties of Arbitrary Sets.- A. Existence.- B. Uniqueness from the Generic Viewpoint.- §2. Solar Properties of Sets.- A. Suns. The Kolmogorov Criterion.- B. The Convexity of Suns.- C. Suns and Moons in C(X).- § 3. Properties of Chebyshev Sets.- A. Approximative Compactness.- B. Convexity and Solarity of Chebyshev Sets.- C. An Alternative Proof.- III. Methods of Local Analysis.- §1. Critical Points.- A. Tangent Cones and Critical Points.- B. Parametrizations and C1-Manifolds.- C. Local Strong Uniqueness.- §2. Nonlinear Approximation in Hilbert Spaces.- A. Nonlinear Approximation in Smooth Banach Spaces.- B. A Classification of Critical Points.- C. Continuity.- D. Functions with Many Local Best Approximations.- § 3. Varisolvency.- A. Varisolvent Families.- B. Characterization and Uniqueness of Best Approximations.- C. Regular and Singular Points.- D. The Density Property.- §4. Nonlinear Chebyshev Approximation: The Differentiable Case.- A. The Local Kolmogorov Criterion.- B. The Local Haar Condition.- C. Haar Manifolds.- D. The Local Uniqueness Theorem for C1-Manifolds.- §5. The Gauss-Newton Method.- A. General Convergence Theory.- B. Numerical Stabilization.- IV. Methods of Global Analysis.- §1. Preliminaries. Basic Ideas.- A. Concepts for the Classification of Critical Points.- B. An Example with Many Critical Points.- C. Local Homeomorphisms.- §2. The Uniqueness Theorem for Haar Manifolds.- A. The Deformation Theorem.- B. The Mountain Pass Theorem.- C. Perturbation Theory.- §3. An Example with One Nonlinear Parameter.- A. The Manifold $$E_n^c\backslash E_{n - 1}^c$$.- B. Reduction to One Parameter.- C. Improvement of the Bounds.- V. Rational Approximation.- §1. Existence of Best Rational Approximations.- A. The Existence Problem in C(X).- B. Rational Lp-Approximation. Degeneracy.- §2. Chebyshev Approximation by Rational Functions.- A. Uniqueness and Characterization of Best Approximations.- B. Normal Points.- C. The Lethargy Theorem and the Lip 1 Conjecture.- §3. Rational Interpolation.- A. The Cauchy Interpolation Problem.- B. Rational Functions with Real Poles.- C. Comparison Theorems.- §4. Padé Approximation and Moment Problems.- A. Padé Approximation.- B. The Stieltjes and the Hamburger Moment Problem.- §5. The Degree of Rational Approximation.- A. Approximation of ex on [?1, +1].- B. Approximation of e?x on [0, ?] by Inverses of Polynomials.- C. Rational Approximation of e?x on [0,?).- D. Rational Approximation of ?x.- E. Rational Approximation of ?x?.- §6. The Computation of Best Rational Approximations.- A. The Differential Correction Algorithm.- B. The Remes Algorithm.- VI. Approximation by Exponential Sums.- §1. Basic Facts.- A. Proper and Extended Exponential Sums.- B. The Descartes’ Rule of Signs.- §2. Existence of Best Approximations.- A. A Bound for the Derivatives of Exponential Sums.- B. Existence.- §3. Some Facts on Interpolation and Approximation.- A. Interpolation by Exponential Sums.- B. The Speed of Approximation of Completely Monotone Functions.- VII. Chebyshev Approximation by ?-Polynomials.- §1. Descartes Families.- A. ?-Polynomials.- B. Sign-Regular and Totally Positive Kernels.- C. The Generalized Descartes’ Rule.- D. Further Generalizations.- E. Examples.- §2. Approximation by Proper ?-Polynomials.- A. Varisolvency.- B. Sign Distribution.- C. Positive Sums.- §3. Approximation by Extended ?-Polynomials: Elementary Theory.- A. Non-Uniqueness, Characterization of Best Approximations.- B. ?Polynomials of Order 2.- §4. The Haar Manifold Gn\Gn?1.- A. Simple Parametrizations.- B. The Differentiable Structure.- C. Families with Bounded Spectrum.- §5. Local Best Approximations.- A. Characterization of Local Best Approximations.- B. The Generic Viewpoint.- §6. Maximal Components.- A. Introduction of Maximal Components.- B. The Boundary of Maximal Components.- §7. The Number of Local Best Approximations.- A. The Construction of Local Best Approximations.- B. Completeness of the Standard Construction.- VIII. Approximation by Spline Functions with Free Nodes.- §1. Spline Functions with Fixed Nodes.- A. Chebyshevian Spline Functions.- B. Zeros of Spline Functions.- C. Characterization of Best Uniform Approximations.- §2. Chebyshev Approximation by Spline Functions with Free Nodes.- A. Existence..- B. Continuity and Differentiability Properties.- C. Characterization of Best Approximations.- §3. Monosplines of Least L?-Norm.- A. The Family $$S_{n,k}^ +$$.- B. Monosplines.- C. The Fundamental Theorem of Algebra for Monosplines.- D. Monosplines with Multiple Nodes of Least L?-Norm.- E. Perfect Splines and Generalized Monosplines.- §4. Monosplines of Least L1-Norm.- A. Examples of Nonuniqueness.- B. Duality.- C. The Improvement Operator.- D. Proof of Lemma 4.3.- §5. Monosplines of Least Lp-Norm.- A. The Rodriguez Function for the Lp-Norms.- B. The Degree of the Mapping ?.- C. The Uniqueness Theorem.- Appendix. The Conjectures of Bernstein and Erdös.