Uniform recovery of high-dimensional [formula omitted]-functions

Abstract

We consider functions on the d-dimensional unit cube whose partial derivatives up to order r are bounded by one. It is known that the minimal number of function values that is needed to approximate the integral of such functions up to the error ε is of order (d∕ε)d∕r. Among other things, we show that the minimal number of function values that is needed to approximate such functions in the uniform norm is of order (dr∕2∕ε)d∕r whenever r is even.