Conditionally positive definite kernels on Euclidean domains ✩

This paper characterizes several classes of conditionally positive definite kernels on a domain Ω of either Rl or Cl . Among the classes is that composed of strictly conditionally positive definite kernels. These kernels are known to be useful in the solution of variational interpolation problems on Ω. Our study covers the case in which Ω is the sphere Sl−1 of Rl or a similar manifold. Among other things, our results imply that the characterization of (strict) conditional positive definiteness on Ω can be obtained from a characterization of (strict) positive definiteness on Ω. The bi-zonal strictly conditionally positive definite kernels on Sl−1, l 3, are described.  2004 Elsevier Inc. All rights reserved.