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

We investigate the rate of convergence of linear sampling numbers of the embedding Hα,β(Td)↪Hγ(Td)Hα,β(Td)↪Hγ(Td). Here αα governs the mixed smoothness and ββ the isotropic smoothness in the space Hα,β(Td)Hα,β(Td) of hybrid smoothness, whereas Hγ(Td)Hγ(Td) denotes the isotropic Sobolev space. If γ>βγ>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