Before revision:  Furthermore, the approximant is shown to be smooth and of approximation order of O(h^{m+1}) for non-noisy data , where h is the mesh size with respect to the manifold domain, and m is the degree of a local polynomial approximation utilized in our algorithm.
After revision:  Furthermore, the approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is O(h^{m+1}) for non-noisy data , where h is the mesh size with respect to the manifold domain, and m is the degree of a local polynomial approximation utilized in our algorithm.