Some lower bounds for optimal sampling recovery of functions with mixed smoothness

Recently there was a substantial progress in the problem of sampling recovery on function classes with mixed smoothness. It was mostly done by proving new and sometimes optimal upper bounds for both linear sampling recovery and nonlinear sampling recovery. In this paper we address the problem of lower bounds for the optimal rates of nonlinear sampling recovery. In the case of linear recovery one can use the well-developed theory of estimating the Kolmogorov and linear widths to establish some lower bounds for the optimal rates. In the case of nonlinear recovery we cannot use the above approach. It seems like the only technique which is available now is based on some simple observations. We demonstrate how these observations can be used.