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Transactions of the American Mathematical Society

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


Authors: Feng Dai and Han Feng
Journal: Trans. Amer. Math. Soc. 372 (2019), 7425-7460
MSC (2010): Primary 41A55, 65D32; Secondary 41A63, 52C17, 52C99
DOI: https://doi.org/10.1090/tran/7924
Published electronically: August 28, 2019
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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,\ldots , z_N$ on $ {\mathbb{S}^{d-1}}$ for which

$\displaystyle \frac 1{w({\mathbb{S}^{d-1}})} \int _{{\mathbb{S}^{d-1}}} f(x) w(... ...)=\frac 1N \sum _{j=1}^N f(z_j),\qquad \forall f\in \Pi _n^d, \leqno {(\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:

$\displaystyle \min _{1\leq i\neq j\leq N}\arccos (z_i\cdot z_j)\geq c_{w,d} N^{-\frac 1{d-1}}>0.$

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

$\displaystyle \mathcal {N}_n (wd\sigma _d) \sim \max _{x\in {\mathbb{S}^{d-1}}} \frac 1 {w(B(x, n^{-1}))},\qquad n=1,2,\ldots .$

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
Email: fdai@ualberta.ca

Han Feng
Affiliation: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, People’s Republic of China
Email: hanfeng@cityu.edu.hk

DOI: https://doi.org/10.1090/tran/7924
Keywords: Chebyshev-type cubature formulas for doubling weights, spherical designs, spherical harmonics, convex partitions of the unit spheres
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.
Article copyright: © Copyright 2019 American Mathematical Society