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March 2019 Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes
Kosuke Suzuki, Takehito Yoshiki
Hiroshima Math. J. 49(1): 139-159 (March 2019). DOI: 10.32917/hmj/1554516041

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.

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Kosuke Suzuki. Takehito Yoshiki. "Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes." Hiroshima Math. J. 49 (1) 139 - 159, March 2019. https://doi.org/10.32917/hmj/1554516041

Information

Received: 20 March 2018; Revised: 8 January 2019; Published: March 2019
First available in Project Euclid: 6 April 2019

zbMATH: 07090067
MathSciNet: MR3936651
Digital Object Identifier: 10.32917/hmj/1554516041

Subjects:
Primary: 11P21, 11Y16, 65D30, 65D32

Rights: Copyright © 2019 Hiroshima University, Mathematics Program

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Vol.49 • No. 1 • March 2019
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