Abstract

We define a space with multiweighted derivatives on the half-axis. A multiweighted derivative of a function is an operation under which each subsequent derivative is taken of the function multiplied by some weight function. All weight functions involved in the definition of a multiweighted derivative are assumed to be sufficiently smooth; therefore, the set of compactly supported infinitely smooth functions belongs to the space with multiweighted derivatives, and the closure of this set in the norm of the space is a subspace of the latter. We study the mutual relation between these spaces depending on the integral behavior of the weight functions in the neighborhood of zero and infinity.

中文翻译：

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具有多重加权导数的空间中紧支持函数的密度

摘要

我们定义一个在半轴上具有多个加权导数的空间。函数的多重导数是一种运算，在该运算中，函数的每个后续导数均乘以某个权重函数。假定涉及多重加权导数的所有权重函数都足够平滑。因此，紧密支持的无限光滑函数的集合属于具有多重加权导数的空间，并且该集合在空间范数中的闭合是后者的子空间。我们根据零和无穷大附近权重函数的积分行为研究这些空间之间的相互关系。