Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
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Let Ω_i⊂ℝ^n_i, i=1,…,m, be given domains. In this article, we study the low-rank approximation with respect to L^2(Ω_1×…×Ω_m) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare <cit.>, we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.
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