Recent Developments in Approximation via Positive De nite Functions

Positive and conditionally positive deenite functions, especially radial basis functions and similar functions for spheres, tori, and even Riemannian manifolds, are of interest because of the their well-known ability to synthesize a good surface t from scattered data. More recently, positive deenite basis functions have been employed to analyze scattered data. The methods used to do this involve constructing multiresolution analyses or multilevel approximations. This paper will discuss recent developments in the synthesis and analysis problems, point out new directions in their investigation, and remark on applications. x1. Introduction Positive deenite and conditionally positive deenite functions and kernels are used in areas that require tting a surface to data taken at scattered points in Euclidean space or on some surface, a sphere or torus, say. When the underlying space is Euclidean, radial basis functions (RBFs)| e.g., Gaussians, multiquadrics, and thin-plate splines|are employed. These are positive deenite and conditionally positive deenite kernels that are functions of only the distance between points. Because they retain their positivity no matter what the dimension of the space is, they are especially well suited for being activation functions in multi-input, feed-forward neural networks 5, 62]. They also produce pleasing, smooth surfaces 24] when used in tting noisy data taken at two-dimensional sites. Some of the rst RBF applications , which arose in geophysics 32, 33], were of this type. There are several good review articles on RBFs and related topics. Frequently cited ones are those written by Buhmann 9], Dyn 14], and Powell 63]. Cheney's article 11], although not speciically about RBFs, also includes a review of them. Finally, Schaback's article 71] contains a very nice review of many important properties of RBFs. Positive deenite functions on spheres, which Schoenberg 68] introduced, behave like RBFs, and have been called spherical (radial) basis functions (SBFs) All rights of reproduction in any form reserved.