On the orthogonality of the Chebyshev–Frolov lattice and applications

Abstract

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. $$d=2^m, m \in \mathbb {N}$$ , the resulting lattice is an admissible lattice in the sense of Skriganov. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube $$[-1/2,1/2]^d$$ up to dimension $$d=16$$ .