Radial basis functions (RBFs) are probably best known for their applications to scattered data problems. Until the 1990s, RBF theory only involved functions that were scalar-valued. Matrix-valued RBFs were subsequently introduced by Narcowich and Ward in 1994, when they constructed divergence-free vector-valued functions that interpolate data at scattered points. In 2002, Lowitzsch gave the first error estimates for divergence-free interpolants. However, these estimates are only valid when the target function resides in the native space of the RBF. In this paper we develop Sobolev-type error estimates for cases where the target function is less smooth than functions in the native space. In the process of doing this, we give an alternate characterization of the native space, derive improved stability estimates for the interpolation matrix, and give divergence-free interpolation and approximation results for band-limited functions. Furthermore, we introduce a new class of matrix-valued RBFs that can be used to produce curl-free interpolants.

Radial basis functions (RBFs) are probably best known for their applications to scattered data problems. Until the 1990s, RBF theory only involved functions that were scalar-valued. Matrix-valued RBFs were subsequently introduced by Narcowich and Ward in 1994, when they constructed divergence-free vector-valued functions that interpolate data at scattered points. In 2002, Lowitzsch gave the first error estimates for divergence-free interpolants. However, these estimates are only valid when the target function resides in the native space of the RBF. In this paper we develop Sobolev-type error estimates for cases where the target function is less smooth than functions in the native space. In the process of doing this, we give an alternate characterization of the native space, derive improved stability estimates for the interpolation matrix, and give divergence-free interpolation and approximation results for band-limited functions. Furthermore, we introduce a new class of matrix-valued RBFs that can be used to produce curl-free interpolants.