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
