  We investigate the rate of convergence of linear sampling numbers of the
embedding $H^{\alpha,\beta} (\mathbb{T}^d) \hookrightarrow H^\gamma
(\mathbb{T}^d)$. Here $\alpha$ governs the mixed smoothness and $\beta$ the
isotropic smoothness in the space $H^{\alpha,\beta}(\mathbb{T}^d)$ of hybrid
smoothness, whereas $H^{\gamma}(\mathbb{T}^d)$ denotes the isotropic Sobolev
space. If $\gamma>\beta$ 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 D\~ung, Ullrich, where general linear
approximations have been considered. In addition, we study the embedding
$H^\alpha_{mix} (\mathbb{T}^d) \hookrightarrow H^{\gamma}_{mix}(\mathbb{T}^d)$
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 $H^\alpha_{mix} (\mathbb{T}^d) \hookrightarrow
L_q(\mathbb{T}^d)$ for $2<q\leq \infty$ 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 $\beta = \gamma$ (including the
embedding into $L_2(\mathbb{T}^d)$). The best what we could prove there is a
(probably) non-sharp results with a logarithmic gap between lower and upper
bound.
