Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Let Ω i ⊂ mathvariant="double-struck">R n \Omega _i\subset \mathbb {R}^{n_i} , alttext="i equals 1 comma ellipsis m"> = 1 , …<!-- … <mml:mi>m encoding="application/x-tex">i=1,\ldots ,m be given domains. In this article, we study the low-rank approximation with respect to alttext="upper L squared left-parenthesis normal times midline-horizontal-ellipsis m right-parenthesis"> L 2 ( ×<!-- × <mml:mo>⋯<!-- ⋯ stretchy="false">) encoding="application/x-tex">L^2(\Omega _1\times \dots \times \Omega _m) of functions from Sobolev spaces dominating mixed smoothness. To end, first estimate rank a bivariate approximation, i.e., continuous singular value decomposition. comparison case isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] 39 (2019), 1652–1671], obtain improved results due additional This convergence result is then used tensor train decomposition as method construct multivariate approximations We show that approach able beat curse dimension.