Optimal global rates of convergence for interpolation problems with random design

Given a sample of a d-dimensional design variable X and observations of the corresponding values of a measurable function m:Rd→R without additional errors, we are interested in estimating m on whole Rd such that the L1 error (with integration with respect to the design measure) of the estimate is small. Under the assumption that the support of X is bounded and that m is (p,C)-smooth (i.e., roughly speaking, m is p-times continuously differentiable) we derive the minimax lower and upper bounds on the L1 error.