Approximation by Spline Functions

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 of polynomials 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 in two 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.

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I. Polynomials and Chebyshev Spaces.- 1. Interpolation by Chebyshev Spaces.- 2. Interpolation by Polynomials and Divided Differences.- 3. Best Uniform Approximation by Chebyshev Spaces.- 4. Best L1-Approximation by Chebyshev Spaces.- 5. Best One-Sided L1-Approximation by Chebyshev Spaces and Quadrature Formulas.- 6. Best L2-Approximation.- II. Splines and Weak Chebyshev Spaces.- 1. Weak Chebyshev Spaces.- 2. B-Splines.- 3. Interpolation by Splines.- 4. Best Uniform Approximation by Splines.- 5. Continuity of the Set Valued Metric Projection for Spline Spaces....- 6. Best L1-Approximation by Weak Chebyshev Spaces.- 7. Best One-Sided L1-Approximation by Weak Chebyshev Spaces and Quadrature Formulas.- 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.