We investigate approximations of functions from the H\"older-Zygmund space of mixed smoothness H^\alpha_\infty( \mathbb I%DIFDELCMD < }%%% ^d) defined on the d-dimensional unit cube \mathbb I ^d:=%DIFDELCMD < [%%% 0,1 ^d, by linear algorithms of sparse-grid sampling recovery and by deep ReLU (Rectified Linear Unit) neural networks when the dimension d may be very large. The approximation error is measured in the norm of the isotropic Sobolev space W^{1,p \mathbb I%DIFDELCMD < }%%% ^d). The optimality of this sampling recovery is studied in terms of sampling n-widths. Optimal linear sampling algorithms are constructed on sparse grids using the piece-wise linear B-spline interpolation representation. We prove some tight dimension-dependent bounds of the sampling n-widths explicit in d and n. Based on the results on sampling recovery, we investigate the expressive power of deep ReLU neural networks to approximate functions in H\"older-Zygmund space . Namely, for any function f\in H^\alpha_\infty( \mathbb I%DIFDELCMD < }%%% ^d), we explicitly construct a deep ReLU neural network having an output that approximates f in the W_0^{1,p \mathbb I%DIFDELCMD < }%%% ^d)-norm with a prescribed accuracy \varepsilon, and prove tight dimension-dependent  bounds of the computation complexity of this approximation, characterized as the number of weights and the depth of this deep ReLU neural network, explicitly in d and \varepsilon. Moreover, we show that under a certain restriction the curse of dimensionality can be avoided in the approximations by sparse-grid sampling recovery and deep ReLU neural networks .
We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the H\"older-Zygmund space of mixed smoothness %DIFDELCMD < }%%% defined on the d-dimensional unit cube %DIFDELCMD < [%%% when the dimension d may be very large. The approximation error is measured in the norm of isotropic Sobolev space %DIFDELCMD < }%%% . For every function f from the H\"older-Zygmund space %DIFDELCMD < }%%% of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates f %DIFDELCMD < }%%% with a prescribed accuracy \varepsilon, and prove tight dimension-dependent upper and lower bounds of the computation complexity of this approximation, characterized as the size and the depth of this deep ReLU neural network, explicitly in d and \varepsilon. The proof of these results are in particular, relied on the approximation by sparse-grid sampling recovery based on the Faber series .
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