  We consider the problem of interpolating a function given on scattered points
using Hermite-Birkhoff formulas on the sphere and other manifolds. We express
each proposed interpolant as a linear combination of basis functions, the
combination coefficients being incomplete Taylor expansions of the interpolated
function at the interpolation points. The basis functions have the following
features: (i) depend on the geodesic distance; (ii) are orthonormal with
respect to the point-evaluation functionals; and (iii) have all derivatives
equal zero up to a certain order at the interpolation points. Moreover, the
construction of such interpolants, which belong to the class of partition of
unity methods, takes advantage of not requiring any solution of linear systems.
