We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev-polynomials. If the dimension d of the lattice is a power of two, i.e. d = 2 m , m ∈ N, the resulting lattice is an admissible lattice in the sense of Skriganov [12]. These are related to the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates [18] in a broad range of Besov-Lizorkin-Triebel spaces. We prove that the resulting lattices are orthogonal and possess a lattice representation matrix with entries not larger than 2 (in modulus). This allows for an efficient enumeration of the Frolov cubature nodes in the d-cube [−1/2, 1/2] d up to dimension d = 16. *

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In this paper, we deal with several aspects of the universal Frolov cubature method, that is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating polynomials that promise a significant reduction of the integration error compared to the classical polynomials. Moreover, we develop a new algorithm to enumerate the Frolov points from non-orthogonal lattices for numerical cubature in the d-dimensional unit cube [0, 1] d . Finally, we study Sobolev spaces with anisotropic mixed smoothness and compact support in [0, 1] d and derive explicit formulas for their reproducing kernels. This allows for the simulation of exact worst-case errors which numerically validate our theoretical results.