Nonparametric estimation of a function from noiseless observations at random points

In this paper we study the problem of estimating a function from n noiseless observations of function values at randomly chosen points. These points are independent copies of a random variable whose density is bounded away from zero on the unit cube and vanishes outside. The function to be estimated is assumed to be (p,C)-smooth, i.e., (roughly speaking) it is p times continuously differentiable. Our main results are that the supremum norm error of a suitably defined spline estimate is bounded in probability by {ln(n)∕n}p∕d for arbitrary p and d and that this rate of convergence is optimal in minimax sense.