Abstract
We investigate dimension-dependent estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids parametrized by m∈N, of periodic d-variate functions from the space with Hölder–Zygmund mixed smoothness α>0. We consider two subsets of the unit ball in this space. The first one is the subset of functions with homogeneous condition, and the second one is the subset of functions depending on ν active variables (1≤ν≤d). For these classes, we prove some upper bounds and lower bounds (for α≤2) of the error of the optimal sampling recovery on Smolyak grids, explicit in d, ν, m when d and m may be large.
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