Optimal global rates of convergence for interpolation problems with random design

Given a sample of a dd-dimensional design variable XX and observations of the corresponding values of a measurable function m:Rd→Rm:Rd→R without additional errors, we are interested in estimating mm on whole RdRd such that the L1L1 error (with integration with respect to the design measure) of the estimate is small. Under the assumption that the support of XX is bounded and that mm is (p,C)(p,C)-smooth (i.e., roughly speaking, mm is pp-times continuously differentiable) we derive the minimax lower and upper bounds on the L1L1 error.