Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles (Q692136)

The authors calculate exactly integrals \[ \int_{\Delta} T \, d{\mu}_2\,, \] where \(T\) is a bivariate trigonometric polynomial with coordinatewise degree at least \(n\), \(\Delta\) is a spherical triangle in the unit sphere \({\mathbb S}^2 \subset {\mathbb R}^3\), and \({\mu}_2\) is the area measure of \({\mathbb S}^2\). This method is developed into a specific algorithm which is tested on spherical triangles of various shapes. For an integer \(q\geq 2\), this integration method is generalized to quadrature formulas for \(q\)-variate trigonometric polynomials on a compact subset of the unit sphere \({\mathbb S}^q \subset {\mathbb R}^{q+1}\). Analogous exact quadrature formulas for multivariate algebraic polynomials are given by \textit{K. Hesse} and \textit{R. S. Womersley} [Adv. Comput. Math. 36, No. 3, 451--483 (2012; Zbl 1256.65024)].