## Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions.(English)Zbl 1120.41022

Authors’ summary: Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not “large” enough to be applicable to a number of practical situations. In this paper the authors obtain Sobolev-type error estimates on compact regions of $$\mathbf {R}^{n}$$ when the RBFs have Fourier transforms that decay algebraically. In addition, they derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter.

### MSC:

 41A25 Rate of convergence, degree of approximation 41A05 Interpolation in approximation theory 41A63 Multidimensional problems
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