The Curse of Dimensionality for Numerical Integration of Smooth Functions
Abstract
We prove the curse of dimensionality for multivariate integration of Cr functions: The number of needed function values to achieve an error ε is larger than cr (1+γ)d for ε ε0, where cr,γ>0 and d is the dimension. 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.
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