Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:431

Article

Brauchart, J. S.^1,2  Dick, J.²  Saff, E. B.³  Sloan, I. H.²  Wang, Y. G.²  Womersley, R. S.² 

[1] Graz Univ Technol, Inst Anal & Computat Number Theory, A-8010 Graz, Austria

[2] Univ New S Wales, Sch Math & Stat, Sydney, NSW 2052, Australia

[3] Vanderbilt Univ, Dept Math, Ctr Construct Approximat, Nashville, TN 37240 USA

关键词: Covering radius;    Cubature on the sphere;    Quasi-Monte Carlo design sequences;    Spherical design;    Sobolev space;    Worst-case error;   

DOI  :  10.1016/j.jmaa.2015.05.079

来源: Elsevier

【 摘 要 】

We prove that the covering radius of an N-point subset X-N of the unit sphere S-d subset of Rd+1 is bounded above by a power of the worst-case error for equal weight cubature 1/N Sigma(x is an element of XN) f(x) approximate to integral(Sd) f d sigma(d) for functions in the Sobolev space W-p(s)(S-d), where sigma(d) denotes normalized area measure on S-d. These bounds are close to optimal when s is close to d/p. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for W-p(s)(S-d), which have previously been introduced only in the Hilbert space setting p = 2. We say that a sequence (X-N) of N-point configurations is a QMC-design sequence for W-p(s)(S-d) with s > d/p provided the worst-case equal weight cubature error for X-N has order N-s/d as N -> infinity, a property that holds, in particular, for a sequence of spherical t-designs in which each design has order t(d) points. For the case p = 1, we deduce that any QMC-design sequence (X-N) for W-1(s) (S-d) with s > d has the optimal covering property; i.e., the covering radius of X-N has order N-1/d as N -> infinity. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of X-N. As a consequence we prove that any QMC-design sequence for W-p(s)(S-d) is also a QMC-design sequence for W-p'(s), (S-d) for all 1 <= p < p' <= infinity and, furthermore, if (X-N) is a quasi-uniform QMC-design sequence for W-p(s), (S-d), then it is also a QMC-design sequence for W-p(s)', (S-d) for all s > s' > d/p. (C) 2015 Elsevier Inc. All rights reserved.

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