The curse of dimensionality for numerical integration of smooth functions.

We prove the curse of dimensionality for multivariate integration of C-r functions: The number of needed function values to achieve an error epsilon is larger than c(r)(1 + gamma)(d) for epsilon <= epsilon(0), where cr, gamma > 0. The proofs are based on volume estimates for r = 1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for r > 1.