The Theory of Splines and Their Applications

PrefaceChapter I Introduction 1.1. What is a Spline? 1.2. Recent Developments in the Theory of SplinesChapter II The Cubic Spline 2.1. Introduction 2.2. Existence, Uniqueness, and Best Approximation 2.3. Convergence 2.4. Equal Intervals 2.5. Approximate Differentiation and Integration 2.6. Curve Fitting 2.7. Approximate Solution of Differential Equations 2.8. Approximate Solution of Integral Equations 2.9. Additional Existence and Convergence TheoremsChapter III Intrinsic Properties of Cubic Splines 3.1. The Minimum Norm Property 3.2. The Best Approximation Property 3.3. The Fundamental Identity 3.4. The First Integral Relation 3.5. Uniqueness 3.6. Existence 3.7. General Equations 3.8. Convergence of Lower-Order Derivatives 3.9. The Second Integral Relation 3.10. Raising the Order of Convergence 3.11. Convergence of Higher-Order Derivatives 3.12. Limits on the Order of Convergence 3.13. Hilbert Space Interpretation 3.14. Convergence in Norm 3.15. Canonical Mesh Bases and their Properties 3.16. Remainder Formulas 3.17. Transformations Defined by a Mesh 3.18. A Connection with Space TechnologyChapter IV The Polynomial Spline 4.1. Definition and Working Equations 4.2. Equal Intervals 4.3. Existence 4.4. Convergence 4.5. Quintic Splines of Deficiency 2, 3 4.6. Convergence of Periodic Splines on Uniform MeshesChapter V Intrinsic Properties of Polynomial Splines of Odd Degree 5.1. Introduction 5.2. The Fundamental Identity 5.3. The First Integral Relation 5.4. The Minimum Norm Property 5.5. The Best Approximation Property 5.6. Uniqueness 5.7. Defining Equations 5.8. Existence 5.9. Convergence of Lower-Order Derivatives 5.10. The Second Integral Relation 5.11. Raising the Order of Convergence 5.12. Convergence of Higher-Order Derivatives 5.13. Limits on the Order of Convergence 5.14. Hilbert Space Interpretation 5.15. Convergence in Norm 5.16. Canonical Mesh Bases and their Properties 5.17. Kernels and Integral Representations 5.18. Representation and Approximation of Linear FunctionalsChapter VI Generalized Splines 6.1. Introduction 6.2. The Fundamental Identity 6.3. The First Integral Relation 6.4. The Minimum Norm Property 6.5. Uniqueness 6.6. Defining Equations 6.7. Existence 6.8. Best Approximation 6.9. Convergence of Lower-Order Derivatives 6.10. The Second Integral Relation 6.11. Raising the Order of Convergence 6.12. Convergence of Higher-Order Derivatives 6.13. Limits on the Order of Convergence 6.14. Hilbert Space Interpretation 6.15. Convergence in Norm 6.16. Canonical Mesh Bases 6.17. Kernels and Integral Representations 6.18. Representation and Approximation of Linear FunctionalsChapter VII The Doubly Cubic Spline 7.1. Introduction 7.2. Partial Splines 7.3. Relation of Partial Splines to Doubly Cubic Splines 7.4. The Fundamental Identity 7.5. The First Integral Relation 7.6. The Minimum Norm Property 7.7. Uniqueness and Existence 7.8. Best Approximation 7.9. Cardinal Splines 7.10. Convergence Properties 7.11. The Second Integral Relation 7.12. The Direct Product of Hilbert Spaces 7.13. The Method of Cardinal Splines 7.14. Irregular Regions 7.15. Surface Representation 7.16. The Surfaces of CoonsChapter VIII Generalized Splines in Two Dimensions 8.1. Introduction 8.2. Basic Definition 8.3. The Fundamental Identity 8.4. Types of Splines 8.5. The First Integral Relation 8.6. Uniqueness 8.7. Existence 8.8. Convergence 8.9. Hilbert Space TheoryBibliographyIndex