MiS Preprint Repository

Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to ${\ell }^2$ or $L^2$ norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require ${\ell }^{\mathrm{\infty }}$ or $L^{\mathrm{\infty }}$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of ${\Vert \cdot \Vert }_{\mathrm{\infty }}$ by ${\Vert \cdot \Vert }_2$ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to ${\Vert \cdot \Vert }_{\mathrm{\infty }}$ can be derived from the Gagliardo-Nirenberg inequality because of the special nature of the SVD\ truncation.