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We consider the problem of interpolating a function given on scattered pointsusing Hermite-Birkhoff formulas on the sphere and other manifolds. We expresseach proposed interpolant as a linear combination of basis functions, thecombination coefficients being incomplete Taylor expansions of the interpolatedfunction at the interpolation points. The basis functions have the followingfeatures: (i) depend on the geodesic distance; (ii) are orthonormal withrespect to the point-evaluation functionals; and (iii) have all derivativesequal zero up to a certain order at the interpolation points. Moreover, theconstruction of such interpolants, which belong to the class of partition ofunity methods, takes advantage of not requiring any solution of linear systems.

Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data on the Sphere and Other Manifolds