A well known result from functional analysis states that any compact operator between Hilbert spaces admits a singular value decomposition (SVD). This decomposition is a powerful tool that is the workhorse of many methods both in mathematics and applied fields. A prominent application in recent years is the approximation of high-dimensional functions in a low-rank format. This is based on the fact that, under certain conditions, a tensor can be identified with a compact operator and SVD applies to the latter. One key assumption for this application is that the tensor product norm is not weaker than the injective norm. This assumption is not fulfilled in Sobolev spaces, which are widely used in the theory and numerics of partial differential equations. Our goal is the analysis of the SVD in Sobolev spaces.

This work consists of two parts. In this manuscript (part I), we address low-rank approximations and minimal subspaces in $H^1$. We analyze the $H^1$-error of the SVD performed in the ambient $L^2$-space. In part II, we will address variants of the SVD in norms stronger than the $L^2$-norm. We will 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：第一部分

功能分析的一个众所周知的结果表明，希尔伯特空间之间的任何紧凑算子都承认奇异值分解（SVD）。这种分解是一个强大的工具，是数学和应用领域中许多方法的主力军。近年来，一个突出的应用是将低维格式的高维函数近似化。这是基于以下事实：在某些条件下，可以使用紧凑算子识别张量，而SVD应用于后者。此应用程序的一个关键假设是张量积范数不弱于内射范数。在偏微分方程的理论和数值中广泛使用的Sobolev空间中，无法满足该假设。我们的目标是对Sobolev空间中的SVD进行分析。

这项工作包括两个部分。在本手稿（第一部分）中，我们讨论了$ H ^ 1 $中的低秩逼近和最小子空间。我们分析了在环境$ L ^ 2 $空间中执行的SVD的$ H ^ 1 $误差。在第二部分中，我们将以比$ L ^ 2 $规范更强的规范来处理SVD的变体。我们将提供一些数值示例来支持我们的理论发现。