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Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes. (arXiv:1705.04864v2 [math.CA] UPDATED)
来源于:arXiv
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 strict
Chebyshev-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}^{ 查看全文>>