集合的细分格式与对称差度量下集值函数的逼近
Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric
AI总结：
本文在对称差度量下构造了R^n子集的细分格式，通过定义新的加权平均（包括正负权重）将Lane-Riesenfeld算法和4点插值细分格式推广到集值情形，并证明了单调性保持等性质。
摘要：
本文构造了细化R^n一般子集的细分格式，并研究了它们在集值函数逼近中的应用。与以往集值逼近的工作不同，我们的方法是在赋予对称差度量的勒贝格可测集度量空间中发展和分析的。集值细分格式的构造基于两个集合的新加权平均，该平均针对正权重（对应插值）以及当一个权重为负时（对应外推）定义。利用正权重的新平均，我们将由Lane-Riesenfeld算法计算的样条细分格式适应到集合上，该算法仅需要数对的平均。然后将数的平均替换为集合对的新平均。在所得集值细分格式的其他性质中，我们证明了它们的单调性保持性质。利用同时具有正负权重的集合新加权平均，我们将4点插值细分格式适应到集合上。最后，我们讨论了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求平均运算的一般度量空间。
英文摘要：
In this work we construct subdivision schemes refining general subsets of R^n and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane-Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally we discuss the extension of the results obtained in the metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties.
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