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Scattered data interpolation on spheres: error estimates and locally supported basis functions. (English) Zbl 1055.41007

The paper works with the \(n\)-sphere as the underlying manifold for interpolation problem. There are obtained Sobolev-type error estimates for interpolating function \(f\in C^{(2k)}(S^n)\)from ”shifts” by using of a smoother positive definite function \(\phi\)on \(S^n\). It is shown that the estimates are close to the optimal ones in order. It is also studied the class of locally supported positive definite functions on \(S^n\), further functions based on Wendland’s compactly supported radia basis functions.

MSC:

41A25 Rate of convergence, degree of approximation
41A05 Interpolation in approximation theory
41A63 Multidimensional problems
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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