On some cubature formulas on the sphere

AbstractWe construct interpolatory cubature rules on the two-dimensional sphere, using the fundamental system of points obtained by Laín Fernández [Polynomial Bases on the Sphere, Logos-Verlag, Berlin, 2003; Localized polynomial bases on the sphere, Electron. Trans. Numer. Anal. 19 (2005) 84–93]. The weights of the cubature rules are calculated explicitly. We also discuss the cases when this cubature leads to positive weights. Finally, we study the possibility to construct spherical designs and the degree of exactness

Construction of spherical cubature formulas using lattices

We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on the sphere of dimension n-1 for n=4, 8, 12, 14, 16, 20, 23, and 24, and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming

Numerical cubature using error-correcting codes

We present a construction for improving numerical cubature formulas with equal weights and a convolution structure, in particular equal-weight product formulas, using linear error-correcting codes. The construction is most effective in low degree with extended BCH codes. Using it, we obtain several sequences of explicit, positive, interior cubature formulas with good asymptotics for each fixed degree $t$ as the dimension $n \to \infty$. Using a special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain an equal-weight $t$-cubature formula on the $n$-cube with O(n^{\floor{t/2}}) points, which is within a constant of the Stroud lower bound. We also obtain $t$-cubature formulas on the $n$-sphere, $n$-ball, and Gaussian $\R^n$ with $O(n^{t-2})$ points when $t$ is odd. When $\mu$ is spherically symmetric and $t=5$, we obtain $O(n^2)$ points. For each $t \ge 4$, we also obtain explicit, positive, interior formulas for the $n$-simplex with $O(n^{t-1})$ points; for $t=3$, we obtain O(n) points. These constructions asymptotically improve the non-constructive Tchakaloff bound. Some related results were recently found independently by Victoir, who also noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of their 40th anniversary. This version has a major improvement for the n-cub

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups

Numerical hyperinterpolation over nonstandard planar regions

We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes