The curse of dimensionality for numerical integration of smooth functions

Abstract

We prove the curse of dimensionality for multivariate integration of C r C^r functions: The number of needed function values to achieve an error ϵ \epsilon is larger than c r ( 1 + γ ) d c_r (1+\gamma )^d for ϵ ≤ ϵ 0 \epsilon \le \epsilon _0 , where c r , γ > 0 c_r,\gamma >0 . The proofs are based on volume estimates for r = 1 r=1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for r > 1 r>1 .