On lower bounds for integration of multivariate permutation-invariant functions

In this note we study multivariate integration for permutation-invariant functions from a certain Banach space _Ed,α$E_{d,\alpha }$ of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d$d$ every cubature rule which reduces the initial error necessarily uses at least d+1$d+1$ 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 Wo?niakowski (1997) [3]. Moreover, for large smoothness parameters α$\alpha$ 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.