离散函数的极值分布
Extremal Distributions of Discrepancy functions
摘要：
单位区间内$N$点分布的不规则性通常用各种不同的差异概念来衡量。差异函数可以相对于$[0,t)\subset [0,1)$形式的区间或单位区间的任意子区间来定义。在前一种情况下，在差异理论中，具有最低$L_2$或$L_{\infty}$范数的$N$元点集是中心规则网格$$ Γ_N:=\left\{\frac{2n+1}{2N}: n=0,1,\dots,N-1\right\}. $$，这是一个众所周知的事实。我们对$[0,1]$中点集的差异函数的分布给出了一个更强的结果。其基本上表示$Γ_N$的差异函数的分布在某种意义上在所有$N$元素点集中是最小的。因此，我们可以将上述结果推广到相关不变范数，包括$L_p$，Orlicz和Lorentz范数.我们研究了同样的问题的差异概念与任意的子区间。在这种情况下，我们将观察到，我们必须处理函数卷积的积分。为此，我们证明了一个一般的上界这样的表达式，这可能是独立的利益。
英文摘要：
The irregularities of a distribution of $N$ points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form $[0,t)\subset [0,1)$ or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the $N$-element point set in the with the lowest $L_2$ or $L_{\infty}$ norm of the discrepancy function is the centered regular grid $$ Γ_N:=\left\{\frac{2n+1}{2N}: n=0,1,\dots,N-1\right\}. $$ We show a stronger result on the distribution of discrepancy functions of point sets in $[0,1]$, which basically says that the distribution of the discrepancy function of $Γ_N$ is in some sense minimal among all $N$-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including $L_p$, Orlicz and Lorentz norms. We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.
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