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 $(\Omega ,\mu )$ a measure space, in $L^p(\Omega ,\mu )$ with $1\le p\le \infty$ and for all $n\ge 1$, compact families of characteristic functions of sets (of finite measure for $p<\infty$) generate approximatively compact $n$-fold linear spans. Results are illustrated by examples of continuously parametrized sets.

中文翻译：

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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$（\Omega ，\mu ）$ 一个测量空间 ${\mathrm{大号}}^p（\Omega ，\mu ）$ 与 $\mathrm{1个}\le p\le \infty$ 并为所有人 $ñ\ge \mathrm{1个}$集的特征函数的紧族（对于 $p<\infty$）产生近似紧凑的 $ñ$倍线性范围。结果通过连续参数化集合的例子说明。