arXiv subjects

William W. L. Chen

Publications and source records attributed to William W. L. Chen.

2 recordsLinked to original sources

arXiv2017

Discrepancy and numerical integration on metric measure spaces

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz-Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small $L^p$ discrepancy with respect to certain classes of subsets, for example metric balls.

math.AP↗

arXiv2010

An L^1 estimate for half-space discrepancy

For every unit vector $σ\inΣ_{d-1}$ and every $r\ge0$, let % % \begin{displaymath} P_{σ,r}=[-1,1]^d\cap\{t\in\Rr^d:t\cdotσ\le r\} \end{displaymath} % % denote the intersection of the cube $[-1,1]^d$ with a half-space containing the origin $0\in\Rr^d$. We prove that if $N$ is the $d$-th power of an odd integer, then there exists a distribution $\PPP$ of $N$ points in $[-1,1]^d$ such that % % \begin{displaymath} \sup_{r\ge0} \int_{Σ_{d-1}}\vert\card(\PPP\cap P_{σ,r})-N2^{-d} \vert P_{σ,r}\vert\vert\,\ddσ\le c_d(\log N)^d, \end{displaymath} % % generalizing an earlier result of Beck and the first author.

math.NT↗