A guide to learning about developments in approximation theory that have come about since the late 1970s. The emphasis is on multivariate approximation theory.

The purpose of this book is to guide the reader in learning about new developments in approximation theory that have come about in the past 20 years. The emphasis is on multivariate approximation theory (the approximation of functions of several variables, in contrast to the classical theory of functions of one variable). The authors cover positive definite functions, radial basis interpolation, thin-plate splines, neural networks, ridge functions, box splines, approximation on spheres, and wavelets.

1. Introductory Discussion of Interpolation 2. Linear Interpolation Operators 3. Optimization of the Lagrange Operator 4. Multivariate Polynomials 5. Moving the Nodes 6. Projections 7. Tensor Product Interpolation 8. The Boolean Algebra of Projections 9. The Newton Paradigm for Interpolation 10. The Lagrange Paradigm for Interpolation 11. Interpolation by Translates of a Single Function 12. Positive Definite Functions 13. Strictly Positive Definite Functions 14. Completely Monotone Functions 15. The Schoenberg Interpolation Theorem 16. The Micchelli Interpolation Theorem 17. Positive Definite Functions of Spheres 18. Approximation by Positive Definite Functions 19. Approximate Reconstruction of Functions and Tomography 20. Approximation by Convolution 21. The Good Kernels 22. Ridge Functions 23. Ridge Function Approximation via Convolutions 24. Density of Ridge Functions 25. Artificial Neural Networks 26. Chebyshev Centers 27. Optimal Reconstruction of Functions 28. Algorithmic Orthogonal Projections 29. Cardinal B-Splines and the Sinc Function 30. The Golomb-Weinberger Theory 31. Hilbert Function Spaces, Reproducing Kernels 32. Spherical Splines 33. Box Splines 34. Wavelets, Part I 35. Wavelets, Part II 36. Quasi-Interpolation.