  In this paper we develop a discrete Hierarchical Basis (HB) to efficiently
solve the Radial Basis Function (RBF) interpolation problem with variable
polynomial order. The HB forms an orthogonal set and is adapted to the kernel
seed function and the placement of the interpolation nodes. Moreover, this
basis is orthogonal to a set of polynomials up to a given order defined on the
interpolating nodes. We are thus able to decouple the RBF interpolation problem
for any order of the polynomial interpolation and solve it in two steps: (1)
The polynomial orthogonal RBF interpolation problem is efficiently solved in
the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR
preconditioner. (2) The residual is then projected onto an orthonormal
polynomial basis. We apply our approach on several test cases to study its
effectiveness, including an application to the Best Linear Unbiased Estimator
regression problem.
