It is shown in [10] that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.

中文翻译：

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有效区间和正则 Dirichlet 子空间

在[10]中表明，区间上的规则和局部狄利克雷形式可以用具有尺度函数的所谓有效区间表示。本文重点讨论如何在有效区间上进行操作以获得规则的狄利克雷子空间。第一个结果是 Dirichlet 形式的完整表征为这种 Dirichlet 形式在有效间隔方面的正则子空间。然后我们给出了一个明确的路线图，如何通过对上述表示中的有效区间的一系列直观操作，从一个区间上的局部和正则 Dirichlet 形式获得所有正则 Dirichlet 子空间。最后应用之前的结果，我们将证明每个正则和局部 Dirichlet 形式都有一个特殊的标准核，由连续且严格递增的函数生成。