Multivariate Clenshaw-Curtis integration using Chebyshev lattices

摘要

We present a multivariate extension to Clenshaw-Curtis quadrature based on the hyperinterpolation theory. At the heart of it, a cubature rule for an integral with Chebyshev weight function is needed. Several point sets have been discussed in this context but we introduce Chebyshev lattices as generalizing framework. This has several advantages: (1) it has a very natural extension to higher dimensions, (2) allows for a systematic search for good point sets and (3) because of the construction, there is a direct link with the Fourier transform that can be used to reduce the computational cost. It will be shown that almost all known two- and three-dimensional point sets for this Chebyshev weighted setting fit into the framework. We give them a uniform description using the Chebyshev lattices and reveal some previously unobserved similarities. Blending, the not so commonly known extension to higher dimensions, also fits into this framework.