Eignets for function approximation on manifolds

Let X be a compact, smooth, connected, Riemannian manifold without boundary, G : X×X → R be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form PM j=1 ajG(◦, yj), where aj ∈ R, yj ∈ X, 1 ≤ j ≤ M . We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in L(μ;X) for a general class of measures μ and kernels G. Our algorithm yields linear operators. Using the minimal separation amongst the centers yj as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every individual function. We also give estimates on the coefficients aj in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.