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Recently, error estimates have been made available for divergencefree radial basis function (RBF) interpolants. However, these results are only valid for functions within the associated reproducing kernel Hilbert space (RKHS) of the matrix-valued RBF. Functions within the associated RKHS, also known as the "native space" of the RBF, can be characterized as vector fields having a specific smoothness, making the native space quite small. In this paper we develop Sobolev-type error estimates whendoi:10.1090/s0025-5718-07-02096-0 fatcat:c4w6wi4ifrb5bg3uye4666ulqu