Approximation rates for the hierarchical tensor format in periodic Sobolev spaces

In this note we estimate the asymptotic rates for the L2L2-error decay and the storage cost when approximating 2π2π-periodic, dd-variate functions from isotropic and mixed Sobolev classes by the recent hierarchical tensor format as introduced by Hackbusch and Kühn. To this end, we survey some results on bilinear approximation due to Temlyakov. The approach taken in this paper improves and generalizes recent results of Griebel and Harbrecht for the bi-variate case.