Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in [formula omitted

Abstract

We investigate the rate of convergence of linear sampling numbers of the embedding Hα,β(Td)↪Hγ(Td). Here α governs the mixed smoothness and β the isotropic smoothness in the space Hα,β(Td) of hybrid smoothness, whereas Hγ(Td) denotes the isotropic Sobolev space. If γ>β we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on “energy-norm based sparse grids” for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding Hmixα(Td)↪Hmixγ(Td) and achieve optimality for Smolyak’s algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding Hmixα(Td)↪Lq(Td) for 2<q≤∞ where again Smolyak’s algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation β=γ (including the embedding into L2(Td)).