Estimates for Local Approximations of Functions on Differential Manifold

We obtain an estimate for the convergence rate of approximation of functions on a differentiable manifold We consider two approaches to approximation of such functions. The first is based on approximate relations for the manifold and is independent of a given set of plane approximations. The second is based on approximations of functions in the plane. We consider the approximation of Courant type on the n-dimensional sphere, in the projective space, and also for the Zlamal approximation and interpolation splines.