Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

Abstract

For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $\mathbb Z$-lattice in $\mathbb {R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when $d=2^n$ for a non-negative integer $n$, since the points are used as the nodes of Frolov’s cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. Kacwin, Oettershagen and Ullrich suggested an enumeration algorithm for such points and later Kacwin improved it, which are claimed to be e‰cient up to dimension $d = 16$. In this paper we suggest a new algorithm which enumerates such points in realistic time for $d = 2^n$, up to $d = 32$. Our algorithm is faster than theirs by a constant factor.