对称差度量下集值函数的Bernstein型逼近
Bernstein-type approximation of set-valued functions in the symmetric difference metric
AI总结：
通过引入新的加权平均集，将基于正样本的实值函数逼近算子（如Bernstein算子）推广到集值函数，在对称差度量下证明了Weierstrass逼近定理的集值类比，并研究了Hölder连续集值函数的逼近速率。
摘要：
我们研究了通过将基于正样本的实值函数逼近算子适应到集值函数（SVFs）来逼近单变量和多变量集值函数。为此，我们引入了一种新的多个集合的加权平均并研究了其性质。逼近结果是在具有对称差度量的Lebesgue可测集空间中获得的。特别地，我们将新的集合平均应用于将经典Bernstein逼近算子适应到SVFs，并得到了Weierstrass逼近定理的集值类比。研究了Bernstein算子对Hölder连续SVFs的逼近速率，并证明其渐近等于实值函数的逼近速率。最后，将集合度量空间中获得的结果推广到具有满足某些性质的平均的度量空间。
英文摘要：
We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive samples-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric. In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and obtain a set-valued analog of the Weierstrass approximation theorem. The rate of approximation of Hölder continuous SVFs by Bernstein operators is studied and shown to be asymptotically equal to that for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.
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