Operators associated with conditionally positive definite kernels

Conditionally positive definite kernels are frequently used in multi-dimensional data fitting. Starting with a single conditionally positive definite kernel on a domain Omega and a set of interpolation points in Omega, the interpolant consists of a linear combination of translates by the interpolation points of the kernel plus a low degree polynomial. A major problem is to identify which conditionally positive definite kernels can be used to carry out the interpolation. In this paper, we study linear and nonlinear operators that leave the class of conditionally positive definite kernels invariant. The operators are useful in the description of the class of strictly conditionally positive definite kernels with respect to a polynomial space P on Omega, when a description of the class of strictly conditionally positive definite kernels with respect to a subspace of P is available.