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Perturbed kernel approximation on homogeneous manifolds. (English) Zbl 1103.41002

This paper addresses the loss of positive-definiteness and symmetry which may occur in the implementation of kernel interpolation methods on homogeneous manifolds. First, for kernels arising from compact perturbations of an integral operator it is proved that the corresponding interpolation problems have a unique solution converging at the same rate as that of the standard positive-definite kernel. Second, perturbations by Dunkl’s intertwining operators on zonal positive-definite kernels on spheres are shown to be invariant only under a finite reflexion group and positive-definite on spheres of lower dimensions.

MSC:

41A05 Interpolation in approximation theory
41A25 Rate of convergence, degree of approximation
41A29 Approximation with constraints
41A63 Multidimensional problems
65D05 Numerical interpolation
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