Characterization and construction of radial basis functions Robert Schaback and Holger Wendland March 21, 2000 Abstract We review characterizations of (conditional) positive definiteness and show how they apply to the theory of radial basis functions. We then give complete proofs for the (conditional) positive definiteness of all practically relevant basis functions. Furthermore, we show how some of these characterizations may lead to construction tools for positive definite functions. Finally, we give new construction tech- niques based on discrete methods which lead to non-radial, even non- translation invariant, local basis functions. 1 Introduction Radial basis functions are an efficient tool for solving multivariate scattered data interpolation problems. To interpolate an unknown function f ∈ C (Ω) whose values on a set X = {x 1 ,...,x N }⊂ Ω ⊂ IR d are known, a function of the form s f,X (x)= N  j =1 α j Φ(x, x j )+ p(x) (1) is chosen, where p is a low degree polynomial and Φ : Ω × Ω → IR is a fixed function. The numerical treatment can be simplified in the special situations 1. Φ(x, y )= φ(x − y ) with φ : IR d → IR (translation invariance), 2. Φ(x, y )= φ(‖x − y ‖ 2 ) with φ : [0, ∞) → IR (radiality), and this is how the notion of radial basis functions arose. The most prominent examples of radial basis functions are: φ(r) = r β , β> 0, β ∈ 2IN, 1