Abstract
The authors generalize Wahba’s theory [SIAM J. Sci. Statist. Comput., 2 (1981), pp. 5–16] on spline interpolation and smoothing on the surface of the two-dimensional unit sphere to arbitrary dimensional hyperspheres. As a consequence, practical solutions to minimum norm interpolation and smoothing problems on hyperspheres are provided in terms of certain hyperspherical splines. In addition, Wahba's results for powers of the Laplace–Beltrami operator are extended to more general operators, and Wahba’s Hilbert space of constant functions is expanded to allow more than just constant functions. Extensive curve fitting calculations are made for some two- and higher-dimensional test problems using hyperspherical harmonics and hyperspherical splines. It is found that hyperspherical splines yield better fits than hyperspherical harmonics for test functions that possess no symmetry and are not infinitely differentiable. Tests have been run using several continuous functions; satisfactory absolute errors can be obt...
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