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Sampling inequalities for anisotropic tensor product grids

We derive sampling inequalities for discrete point sets that are of anisotropic tensor product form. Such sampling inequalities can be used to prove convergence for arbitrary stable reconstruction processes. As usual in the context of high-dimensional problems, our sampling inequalities are expressed in terms of the number of data sites, i.e., the number of points in the sparse grid. To this end, new bounds on specific monotone sets and on the number of points in an anisotropic sparse grid are derived.