  We study the problem of reconstructing a function on a manifold satisfying
some mild conditions, given data on the values and some derivatives of the
function at arbitrary points on the manifold. While the problem of finding a
polynomial of two variables with total degree $\le n$ given the values of the
polynomial and some of its derivatives at exactly the same number of points as
the dimension of the polynomial space is sometimes impossible, we show that
such a problem always has a solution in a very general situation if the degree
of the polynomials is sufficiently large. We give estimates on how large the
degree should be, and give explicit constructions for such a polynomial even in
a far more general case. As the number of sampling points at which the data is
available increases, our polynomials converge to the target function on the set
where the sampling points are dense. Numerical examples in single and double
precision show that this method is stable and of high-order.
