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Chebyshev-type cubature formulas for doubling weighted spheres, balls, and simplexes. (English) Zbl 1428.41036

Summary: 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 \(z_1,\ldots , z_N\) on \(\mathbb{S}^{d-1}\) for which
\[\frac {1}{w(\mathbb{S}^{d-1})} \int _{\mathbb{S}^{d-1}} f(x) w(x)d\sigma_d(x)=\frac{1}{N} \sum _{j=1}^N f(z_j),\quad \forall f\in \Pi_n^d, \tag{\(\ast\)}\]
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 \[\mathcal{N}_n (wd\sigma _d) \sim \max _{x\in{\mathbb{S}^{d-1}}} \frac 1 {w(B(x, n^{-1}))},\quad n=1,2,\ldots .\]
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.

MSC:

41A55 Approximate quadratures
41A63 Multidimensional problems
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
65D32 Numerical quadrature and cubature formulas
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