| Preface 1 Applications and motivations 1.1 Surface reconstruction! 1.2 Fluid-structure interaction in aeroelasticity 1.3 Grid-free semi-Lagrangian advection 1.4 Learning from splines 1.5 Approximation and approximation orders 1.6 Notation 1.7 Notes and comments 2 Haar spaces and multivariate polynomials 2.1 The Mairhuber-Curtis theorem 2.2 Multivariate polynomials 3 Local polynomial reproduction 3.1 Definition and basic properties 3.2 Norming sets 3.3 Existence for regions with cone condition 3.4 Notes and comments 4 Moving least squares 4.1 Definition and characterization 4.2 Local polynomial reproduction by moving least squares 4.3 Generalizations 4.4 Notes and comments 5 Auxiliary tools from analysis and measure theory 5.1 Bessel functions 5.2 Fourier transform and approximation by convolution 5.3 Measure theory 6 Positivie definite functions 6.1 Definition and basic properties 6.2 Boehner’s characterization 6.3 Radial functions 6.4 Functions, kernels, and other norms 6.5 Notes and comments 7 Completely monotone functions 7.1 Definition and first characterization 7.2 The Bernstein-Hausdorff-Widder characterization 7.3 Schoenberg's characterization 7.4 Notes and comments 8 Conditionally positive definite functions 8.1 Definition and basic properties 8.2 An analogue of Buchner’s characterization 8.3 Examples of generalized Fourier transform 8.4 Radial conditionally positive definite functions 8.5 Interpolation by conditionally positive definite functions 8.6 Notes and comments 9 Compactly supported functions 9.1 General remarks 9.2 Dimension walk 9.3 Piecewise polynomial functions with local support 9.4 Compactly supported functions of minimal degree 9.5 Generalizations 9.6 Notes and comments 10 Native spaces 10.1 Reproducing-kernel Hilbert spaces 10.2 Native spaces for positive definite kernels 10.3 Native spaces for conditionally positive definite kernels 10.4 Further characterizations of native spaces 10.5 Special cases of native spaces 10.6 An embedding theorem 10.7 Restriction and extension 10.8 Notes and comments 11 Error estimates for radial basis function interpolation 11.1 Power function and first estimates 11.2 Error estimates in terms of the fill distance …… 12 Stability 13 Optimal recovery 14 Data strutures 15 Numerical methods 16 Generalized interpolation 17 Interpolation on spheres and other manifolds References Index |
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