On lower bounds for integration of multivariate permutation-invariant functions

In this note we study multivariate integration for permutationinvariant functions from a certain Banach space Ed,a of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d every cubature rule which reduces the initial error necessarily uses at least d + 1 function values. Since this holds independently of the number of permutation-invariant coordinates, this shows that the integration problem can never be strongly polynomially tractable in this setting. Our assertions generalize results due to Sloan and Woiniakowski (1997) [3]. Moreover, for large smoothness parameters a our bound cannot be improved. Finally, we extend our results to the case of permutation-invariant functions from Korobov-type spaces equipped with product weights. (C) 2013 Published by Elsevier Inc.