Under certain conditions, an element of a tensor product space can be identified with a compact operator and the singular value decomposition (SVD) applies to the latter. These conditions are not fulfilled in Sobolev spaces. In the previous part of this work (part I) [Z. Anal. Anwend. 39 (2020), 349–369], we introduced some preliminary notions in the theory of tensor product spaces. We analyzed low-rank approximations in $H^1$ and the error of the SVD performed in the ambient $L^2$ space. In this work (part II), we continue by considering variants of the SVD in norms stronger than the $L^2norm. Overall and, perhaps surprisingly, this leads to a more difficult control of the $H^1$-error. We briefly consider an isometric embedding of $H^1$ that allows direct application of the SVD to $H^1$-functions. Finally, we provide a few numerical examples that support our theoretical findings.

中文翻译：

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Sobolev空间中的奇异值分解：第二部分

在某些条件下，张量积空间的元素可以使用紧凑算子识别，并且奇异值分解（SVD）应用于后者。Sobolev空间中不满足这些条件。在本文的上一部分（第一部分）中[Z. 肛门 安文德 39（2020），349–369]，我们在张量积空间理论中引入了一些初步的概念。我们分析了$ H ^ 1 $中的低秩近似，以及在环境$ L ^ 2 $空间中执行的SVD的误差。在这项工作（第二部分）中，我们继续考虑比$ L ^ 2norm更强的SVD变体。总体而言，也许令人惊讶的是，这导致对$ H ^ 1 $错误的控制更加困难。我们简要考虑$ H ^ 1 $的等距嵌入，它允许将SVD直接应用于$ H ^ 1 $-函数。最后，