New cubature formulae and hyperinterpolation in three variables

TL;DR

This paper introduces a new algebraic cubature formula for the product Chebyshev measure in three dimensions, enabling efficient polynomial hyperinterpolation and a novel cubature method using fast 3D FFT algorithms.

Contribution

It develops a new degree $2n+1$ cubature formula with fewer nodes, and applies it to fast hyperinterpolation and cubature in three variables, enhancing computational efficiency.

Findings

New cubature formula with approximately $n^d/2^{d-1}$ nodes

Fast algorithm for computing hyperinterpolation coefficients using 3D FFT

A new Clenshaw-Curtis type cubature formula in 3-cube

Abstract

A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube.