Journal of Approximation Theory ( IF 0.9 ) Pub Date : 2020-04-30 , DOI: 10.1016/j.jat.2020.105435 Paul C. Kainen , Věra Kůrková , Andrew Vogt
Best approximation by the set of all n-fold linear combinations of a family of characteristic functions of measurable subsets is investigated. Such combinations generalize Heaviside-type neural networks. Existence of best approximation is studied in terms of approximative compactness, which requires convergence of distance-minimizing sequences. We show that for a measure space, in with and for all , compact families of characteristic functions of sets (of finite measure for ) generate approximatively compact -fold linear spans. Results are illustrated by examples of continuously parametrized sets.
中文翻译:
特征函数的线性组合的近似紧致性
研究了一组可测量子集的特征函数的所有n折线性组合的最佳逼近。这样的组合推广了Heaviside型神经网络。最佳逼近的存在是根据逼近紧度来研究的,这需要最小化距离序列的收敛。我们证明了 一个测量空间 与 并为所有人 集的特征函数的紧族(对于 )产生近似紧凑的 倍线性范围。结果通过连续参数化集合的例子说明。