On weak tractability of the Clenshaw–Curtis Smolyak algorithm

We consider the problem of integration of dd-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw–Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.