English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 $N\geq \max_{x\in\mathbb{S}^{d-1}} \frac {K_w} {w(B(x, n^{-1}))}$, there exists a set of $N$distinct nodes $z_1,\cdots, z_N$ on $\mathbb{S}^{d-1}$ which admits a strictChebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x)d\sigma_d(x)$, $$ \frac 1{w(\mathbb{S}^{d-1})} \int_{\mathbb{S}^{d-1}} f(x)w(x)\, d\sigma_d(x)=\frac 1N \sum_{j=1}^N f(z_j),\ \ \forall f\in\Pi_n^d, $$and which, if in addition $w\in L^\infty(\mathbb{S}^{d-1})$, satisfies$$\min_{1\leq i\neq j\leq N}\mathtt{d}(z_i,z_j)\geq c_{w,d} N^{-\frac1{d-1}}$$for some positive constant $c_{w,d}$. Here, $d\sigma_d$ 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 $r>0$, $w(E)=\int_E w(x) \, d\sigma_d(x)$ for$E\subset\mathbb{S}^{d-1}$, and $\Pi_n^d$ denotes the space of all sphericalpolynomials of degree at most $n$ on $\mathbb{S}^{d-1}$. It is also shown thatthe minimal number of nodes $\mathcal{N}_{n} (wd\sigma_d)$ in a strictChebyshev-type CF of degree $n$ for a doubling weight $w$ on $\mathbb{S}^{d-1}$satisfies $$\mathcal{N}_n (wd\sigma_d) \sim \max_{x\in \mathbb{S}^{d-1}} \frac1 {w(B(x, n^{-1}))},\ \ n=1,2,\cdots.$$ Proofs of these results rely on newconvex partitions of $\mathbb{S}^{d-1}$ that are regular with respect to agiven weight $w$ and integer $N$. Our results extend the recent results ofBondarenko, 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).

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