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Chebyshev lattices, a unifying framework for cubature with Chebyshev weight function

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Abstract

We present a multivariate extension to Clenshaw-Curtis quadrature based on Sloan’s hyperinterpolation theory. At the centre of it, a cubature rule for integrals with Chebyshev weight function is needed. We introduce so called Chebyshev lattices as a generalising framework for the multitude of point sets that have been discussed in this context. This framework provides a uniform notation that extends easily to higher dimensions. In this paper we describe many known point sets as Chebyshev lattices.

In the introduction we briefly explain how convergence results from hyperinterpolation can be used in this context. After introducing Chebyshev lattices and the associated cubature rules, we show how most of the two- and three-dimensional point sets in this context can be described with this notation. The not so commonly known blending formulae from Godzina, which explicitly describe point sets in any number of dimensions, also fit in perfectly.

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Authors and Affiliations

  1. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium

    Ronald Cools & Koen Poppe

Authors
  1. Ronald Cools
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  2. Koen Poppe
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Correspondence to Koen Poppe.

Additional information

Communicated by Lothar Reichel.

This Project has benefited from the financial support of the Fund for Scientific Research–Flanders (Belgium) through project grants. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its author(s).

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Cools, R., Poppe, K. Chebyshev lattices, a unifying framework for cubature with Chebyshev weight function. Bit Numer Math 51, 275–288 (2011). https://doi.org/10.1007/s10543-010-0300-6

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  • DOI: https://doi.org/10.1007/s10543-010-0300-6

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