Interpolation by radial basis functions on Sobolev space

Interpolation by translates of suitable radial basis functions is an important approach towards solving the scattered data problem. However, for a large class of smooth basis functions (including multiquadrics φ(x) = (|x|²+λ²)^m-d/2, m>d/2, 2m-d ∉ 2ℤ), the existing theories guarantee the interpolant to approximate well only for a very small class of very smooth approximands. The approximands f need to be extremely smooth. Hence, the purpose of this paper is to study the behavior of interpolation by smooth radial basis functions on larger spaces, especially on the homogeneous Sobolev spaces.