ABSTRACT

In this talk, I will present the recent joint work with Feng Dai on the strict Chebyshevtype cubature formula (CF) (i.e., equal weighted CF) for doubling weights on the unit sphere equipped with the usual surface Lebesgue measure and geodesic distance. Our main interest is on the minimal number of nodes required in a strict Chebyshev-type CF. Precisely, given a normalized doubling weight on unit sphere, we will establish the sharp asymptotical estimates of the minimal number of distinct nodes which admits a strict Chebyshev-type CF. If, in addition, the weight function is essentially bounded, the nodes involved can be configured well-separately in some sense. The proofs of these results rely on constructing new convex partitions of the unit sphere that are regular with respect to the weight. The weighted results on the unit sphere also allow us to establish similar results on strict Chebyshev-type CFs on the unit ball and the standard simplex.