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

Abstract:

This paper shows that, given a doubling weight $w$ on the unit sphere $\mathbb {S}^{d-1}$ of $\mathbb {R}^d$, there exists a positive constant $K_{w,d}$ such that, for each positive integer $n$ and each integer $N\ge \max _{x\in {\mathbb {S}^{d-1}}} \frac {K_{w,d}} {w(B(x, n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1$, …, $z_N$ on $\mathbb {S}^{d-1}$ for which \begin{equation} \tag {$\ast $} \frac {1}{w({\mathbb {S}^{d-1}})} \int _{{\mathbb {S}^{d-1}}} f(x) w(x)\, d\sigma _d(x)=\frac 1N \sum _{j=1}^N f(z_j),\qquad \forall f\in \Pi _n^d, \end{equation} where $d\sigma _d$, $B(x,r)$, and $\Pi _n^d$ denote the surface Lebesgue measure on ${\mathbb {S}^{d-1}}$, the spherical cap with center $x\in \mathbb {S}^{d-1}$ and radius $r>0$, and the space of all spherical polynomials of degree at most $n$ on ${\mathbb {S}^{d-1}}$, respectively, and $w(E)=\int _E w(x) \, d\sigma _d(x)$ for $E\subset {\mathbb {S}^{d-1}}$. If, in addition, $w\in L^\infty ({\mathbb {S}^{d-1}})$, then the above set of nodes can be chosen to be well separated: \[ \min _{1\leq i\neq j\leq N}\arccos (z_i\cdot z_j)\geq c_{w,d} N^{-\frac 1{d-1}}>0.\] It is further proved that the minimal number of nodes $\mathcal {N}_{n} (wd\sigma _d)$ required in ($\ast$) for a doubling weight $w$ on ${\mathbb {S}^{d-1}}$ satisfies \[ \mathcal {N}_n (wd\sigma _d) \sim \max _{x\in {\mathbb {S}^{d-1}}} \frac 1 {w(B(x, n^{-1}))},\qquad n=1,2,\ldots .\] 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 $N$. Similar results are also established on the unit ball and the standard simplex of $\mathbb {R}^d$.

Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs.