Before revision:  The approximation error is measured in the norm of the isotropic Sobolev space W^{1,p \mathbb I ^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,
After revision:  The approximation error is measured in the norm of the isotropic Sobolev space W^{1,p \mathbb I . For every function f from the H\"older-Zygmund space . Namely,