Efnisyfirlit

• Half-title
• Series-title
• Title
• Copyright
• Contents
• 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 Positive definite functions
• 6.1 Definition and basic properties
• 6.2 Bochner?s characterization
• 6.3 Radial functions
• 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 Bochner?s characterization
• 8.3 Examples of generalized Fourier transforms
• 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
• 11.3 Estimates for popular basis functions
• 11.4 Spectral convergence for Gaussians and (inverse) multiquadrics
• 11.5 Improved error estimates
• 11.6 Sobolev bounds for functions with scattered zeros
• 11.7 Notes and comments
• 12 Stability
• 12.1 Trade-off principle
• 12.2 Lower bounds for Lambdamin
• 12.3 Change of basis
• 12.4 Notes and comments
• 13 Optimal recovery
• 13.1 Minimal properties of radial basis functions
• 13.2 Abstract optimal recovery
• 13.3 Notes and comments
• 14 Data structures
• 14.1 The fixed-grid method
• 14.2 kd-Trees
• 14.3 bd-Trees
• 14.4 Range trees
• 14.5 Notes and comments
• 15 Numerical methods
• 15.1 Fast multipole methods
• 15.2 Approximation of Lagrange functions
• 15.3 Alternating projections
• 15.4 Partition of unity
• 15.5 Multilevel methods
• 15.6 A greedy algorithm
• 15.7 Concluding remarks
• 15.8 Notes and comments
• 16 Generalized interpolation
• 16.1 Optimal recovery in Hilbert spaces
• 16.2 Hermite?Birkhoff interpolation
• 16.3 Solving PDEs by collocation
• 16.4 Notes and comments
• 17 Interpolation on spheres and other manifolds
• 17.1 Spherical harmonics
• 17.2 Positive definite functions on the sphere
• 17.3 Error estimates
• 17.4 Interpolation on compact manifolds
• 17.5 Notes and comments
• References
• Index