Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes

Abstract

This paper proves that given a doubling weight $w$ on the unit sphere ${\mathbb{S}}^{d-1}$ of ${\mathbb{R}}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\ge {\max }_{x\in {\mathbb{S}}^{d-1}}\frac{K_w}{w(B(x,n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1,\cdots ,z_N$ on ${\mathbb{S}}^{d-1}$ which admits a strict Chebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x)d\ddot{I}{ƒ}_d(x)$, $$\frac{1}{w({\mathbb{S}}^{d-1})}{\int }_{{\mathbb{S}}^{d-1}}f(x)w(x)d\ddot{I}{ƒ}_d(x)=\frac{1}{N}\sum _{j=1}^{N}f(z_j),\forall f\in {\hat{I}}_n^d,$$ and which, if in addition $w\in L^{\infty }({\mathbb{S}}^{d-1})$, satisfies $$\underset{1\le i\ne j\le N}{\min }\mathtt{d}(z_i,z_j)\ge c_{w,d}{N}^{-\frac{1}{d-1}}$$ for some positive constant $c_{w,d}$. Here, $d\ddot{I}{ƒ}_d$ and $\mathtt{d}(\cdot ,\cdot )$ denote the surface Lebesgue measure and the geodesic distance on ${\mathbb{S}}^{d-1}$ respectively, $B(x,r)$ denotes the spherical cap with center $x\in {\mathbb{S}}^{d-1}$ and radius $r>0$, $w(E)={\int }_{E}w(x)d\ddot{I}{ƒ}_d(x)$ for $E\subset {\mathbb{S}}^{d-1}$, and ${\hat{I}}_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on ${\mathbb{S}}^{d-1}$. It is also shown that the minimal number of nodes ${\mathcal{N}}_n(wd\ddot{I}{ƒ}_d)$ in a strict Chebyshev-type CF of degree $n$ for a doubling weight $w$ on ${\mathbb{S}}^{d-1}$ satisfies $${\mathcal{N}}_n(wd\ddot{I}{ƒ}_d)\sim \underset{x\in {\mathbb{S}}^{d-1}}{\max }\frac{1}{w(B(x,n^{-1}))},n=1,2,\cdots .$$ Proofs of these results rely on new convex partitions of ${\mathbb{S}}^{d-1}$ that are regular with respect to a given weight $w$ and integer $N$. Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs ({\it Ann. of Math. (2)} {\bf 178}(2013), no. 2, 443--452,{\it Constr. Approx.} {\bf 41}(2015), no. 1, 93--112).