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

摘要

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.