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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Chebyshev-type cubature formulas for doubling weighted spheres, balls, and simplexes
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by Feng Dai and Han Feng PDF
Trans. Amer. Math. Soc. 372 (2019), 7425-7460 Request permission


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.

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Additional Information
  • Feng Dai
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
  • MR Author ID: 660750
  • Email:
  • Han Feng
  • Affiliation: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, People’s Republic of China
  • MR Author ID: 997105
  • Email:
  • Received by editor(s): June 13, 2017
  • Received by editor(s) in revised form: July 12, 2017, and May 23, 2019
  • Published electronically: August 28, 2019
  • Additional Notes: This work was supported by NSERC Canada under grant RGPIN 04702.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7425-7460
  • MSC (2010): Primary 41A55, 65D32; Secondary 41A63, 52C17, 52C99
  • DOI:
  • MathSciNet review: 4024557