论文标题
积分运算符的界变异函数的度量近似
Metric approximation of set-valued functions of bounded variation by integral operators
论文作者
论文摘要
我们将积分近似运算符的改编介绍给设置值函数（SVF，多函数），将紧凑的间隔$ [a，b] $映射到$ {\ Mathbb r}^d $的紧凑型非空子集的空间中。通过用紧凑型图替换加权度量积分的Riemann积分来改编所有运算符。对于这样的设置值函数$ f $，我们在连续性点上获得了积分运算符序列的重点错误估计，从而导致在此处的收敛到$ f $。我们得出的估计值是在不连续的$ f $的不连续点上，该估算值将收敛性产生为一组，这是我们先前在公制傅立叶运算符上的工作中首先描述的。我们的分析最近使用的是在连续点处的不连续点和几个局部Lipschitz属性概念的单方面局部准模型。 我们还为误差界提供了全局方法。多功能$ f $由其所有度量选择的集合表示，而其近似（操作员下的图像）由操作员下方的这些度量选择的图像集表示。 $ l^1 $中这两组单值函数之间的Hausdorff距离限制了我们的全球估计。 通过介绍两个混凝土操作员的示例来说明该理论：Bernstein-Durrmeyer操作员和Kantorovich操作员。
We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$. All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a set-valued function $F$, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to $F$. At points of discontinuity of $F$, we derive estimates, which yield the convergence to a set, first described in our previous work on the metric Fourier operator. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction $F$ is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $L^1$ provides our global estimates. The theory is illustrated by presenting the examples of two concrete operators: the Bernstein-Durrmeyer operator and the Kantorovich operator.
