Improved stability estimates and a characterization of the native space for matrix-valued RBFs (Q960001)

The author obtains several results that treat matrix-valued radial basis functions. He introduces a class of functions of this kind that can be used to construct curl-free interpolants, characterizes the native spaces associated with divergence-free and curl-free kernels, and studies the stability of the interpolation matrix. In particular, he shows that, as in the scalar theory, certain kernels have native spaces the elements of which have a certain smoothness, and, with regard to the stability question, the author provides a lower bound for the lowest eigenvalue of the matrix. An erratum in the demonstration of the latter result has been observed by Professor Stefan Schrödl, and, subsequently, the author has provided the necessary correction, [Adv. Comput. Math. 29, No. 3, 311--313 (2008; Zbl 1165.41308)].