Singular Value Decomposition in Sobolev Spaces: Part II

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-2-norm. 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.