Splines play an important role in applied mathematics since they possess high flexibility to approximate efficiently, even nonsmooth functions which are given explicitly or only implicitly, e.g. by differential equations. The aim of this book is to analyse in a unified approach basic theoretical and numerical aspects of interpolation and best approximation by splines in one variable. The first part on spaces ofpolynomials serves as a basis for investigating the more complex structure of spline spaces. Given in the appendix are brief introductions to the theory of splines with free knots (an algorithm is described in the main part), to splines intwo variables and to spline collocation for differential equations.A large number of new results presented here cannot be found in earlier books on splines. Researchers will find several references to recent developments. The book is an indispensable aid for graduate courses on splines or approximation theory. Students with a basic knowledge of analysis and linear algebra will be able to read the text. Engineers will find various pactical interpolation and approximation methods.
I. Polynomials and Chebyshev Spaces.- 1. Interpolation by Chebyshev Spaces.- 1.1. Lagrange Interpolation by Chebyshev Spaces.- 1.2. Hermite Interpolation by Extended Chebyshev Spaces.- 1.3. Characterization of Extended Complete Chebyshev Spaces.- 1.4. Further Properties of Chebyshev Spaces.- 1.5. Variation Diminishing Property of Order Complete Chebyshev Spaces.- 2. Interpolation by Polynomials and Divided Differences.- 2.1. Divided Differences.- 2.2. Newton Form of Interpolating Polynomials.- 2.3. Nearly Optimal Interpolation Points.- 3. Best Uniform Approximation by Chebyshev Spaces.- 3.1. Best Approximation in Normed Linear Spaces.- 3.2. Characterization of Best Uniform Approximations.- 3.3. Global Unicity and Strong Unicity of Best Uniform Approximations.- 3.4. Algorithm.- 3.5. Approximation Power of Polynomials.- 4. Best L1-Approximation by Chebyshev Spaces.- 4.1. Global Unicity of Best L1-Approximations.- 4.2. Interpolation at Canonical Points.- 5. Best One-Sided L1-Approximation by Chebyshev Spaces and Quadrature Formulas.- 5.1. Unicity of Best One-Sided L1-Approximations.- 5.2. Gauss Quadrature Formulas for Chebyshev Spaces.- 6. Best L2-Approximation.- II. Splines and Weak Chebyshev Spaces.- 1. Weak Chebyshev Spaces.- 1.1. Basic Properties.- 1.2. Best Uniform Approximation by Weak Chebyshev Spaces.- 1.3. Spline Spaces.- 2. B-Splines.- 2.1. Basic Properties.- 2.2. B-Spline Basis.- 2.3. Recurrence Relations.- 2.4. Variation Diminishing Property.- 3. Interpolation by Splines.- 3.1. Lagrange and Hermite Interpolation by Splines.- 3.2. Interpolation by Complete Splines, Periodic Splines and Natural Splines.- 3.3. Quasi-Interpolation.- 4. Best Uniform Approximation by Splines.- 4.1. Characterization, Unicity and Strong Unicity of Best Uniform Approximations.- 4.2. Algorithm (Fixed Knots).- 4.3. Algorithm (Free Knots).- 4.4. Approximation Power of Splines.- 5. Continuity of the Set Valued Metric Projection for Spline Spaces….- 5.1. Upper Semi continuity.- 5.2. Lower Semi continuity.- 5.3. Continuous Selections.- 6. Best L1-Approximation by Weak Chebyshev Spaces.- 6.1. Unicity of Best L1-Approximations.- 6.2. Interpolation at Canonical Points.- 7. Best One-Sided L1-Approximation by Weak Chebyshev Spaces and Quadrature Formulas.- 7.1. Unicity of Best One-Sided L1-Approximations.- 7.2. Gauss Quadrature Formulas for Weak Chebyshev Spaces.- 8. Approximation of Linear Functionals and Splines.- 9. Spaces of Splines with Multiple Knots.- 1. Splines with Free Knots.- 2. Splines in Two Variables.- 2.1. Tensor Product and Blending.- 2.2. Finite Element Functions.- 2.3. Spline Functions.- 3. Spline Collocation and Differential Equations.- References.