Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to support courses in computer aided geometric design, and meshless methods for partial differential equations.

1. Applications and motivations; 2. Hear spaces and multivariate polynomials; 3. Local polynomial reproduction; 4. Moving least squares; 5. Auxiliary tools from analysis and measure theory; 6. Positive definite functions; 7. Completely monotine functions; 8. Conditionally positive definite functions; 9. Compactly supported functions; 10. Native spaces; 11. Error estimates for radial basis function interpolation; 12. Stability; 13. Optimal recovery; 14. Data structures; 15. Numerical methods; 16. Generalised interpolation; 17. Interpolation on spheres and other manifolds.