On weak tractability of the Smolyak algorithm for approximation problems

We consider the problems of LpLp-approximation of dd-variate analytic functions defined on the cube with directional derivatives of all orders bounded by 1. For 1≤p<∞1≤p<∞, it is shown that the Smolyak algorithm based on polynomial interpolation at the extrema of the Chebyshev polynomials leads to weak tractability of these problems. This gives an affirmative answer to one of the open problems raised recently by Hinrichs et al. (2014). Our proof uses the polynomial exactness of the algorithm and an explicit bound on the operator norm of the algorithm.