کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4615108 | 1339308 | 2015 | 30 صفحه PDF | دانلود رایگان |
We prove that the covering radius of an N -point subset XNXN of the unit sphere Sd⊂Rd+1Sd⊂Rd+1 is bounded above by a power of the worst-case error for equal weight cubature 1N∑x∈XNf(x)≈∫Sdfdσd for functions in the Sobolev space Wps(Sd), where σdσd denotes normalized area measure on SdSd. These bounds are close to optimal when s is close to d/pd/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 Wps(Sd), which have previously been introduced only in the Hilbert space setting p=2p=2. We say that a sequence (XN)(XN) of N -point configurations is a QMC-design sequence for Wps(Sd) with s>d/ps>d/p provided the worst-case equal weight cubature error for XNXN has order N−s/dN−s/d as N→∞N→∞, a property that holds, in particular, for a sequence of spherical t -designs in which each design has order tdtd points. For the case p=1p=1, we deduce that any QMC-design sequence (XN)(XN) for W1s(Sd) with s>ds>d has the optimal covering property; i.e., the covering radius of XNXN has order N−1/dN−1/d as N→∞N→∞. 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 XNXN. As a consequence we prove that any QMC-design sequence for Wps(Sd) is also a QMC-design sequence for Wp′s(Sd) for all 1≤p
s′>d/ps>s′>d/p.
Journal: Journal of Mathematical Analysis and Applications - Volume 431, Issue 2, 15 November 2015, Pages 782–811