Monatshefte für Mathematik

, Volume 184, Issue 3, pp 425–441 | Cite as

On the orthogonality of the Chebyshev–Frolov lattice and applications

  • Christopher Kacwin
  • Jens Oettershagen
  • Tino Ullrich
Article

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 \(d=16\).

Keywords

Numerical integration Frolov cubature formula Admissible lattices Orthogonal lattices Lattice points in convex bodies 

Mathematics Subject Classification

11K06 11J71 65D30 65D32 06D50 52C07 

Notes

Acknowledgements

The authors acknowledge the fruitful discussions with D. Bazarkhanov, A. Hinrichs, W. Sickel, V.N. Temlyakov and M. Ullrich on the topic of this paper. Tino Ullrich gratefully acknowledges support by the German Research Foundation (DFG) and the Emmy-Noether programme, Ul-403/1-1. Jens Oettershagen was supported by the DFG via project GR-1144/21-1 and the CRC 1060.

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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Institute for Numerical SimulationHausdorff-Center for MathematicsBonnGermany

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