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MiS Preprint
17/2012
$L^{\infty}$ estimation of tensor truncations
Wolfgang Hackbusch
Abstract
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 $\ell^{\infty}$ or $L^{\infty}$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $\left\Vert \cdot\right\Vert _{\infty}$ by $\left\Vert \cdot\right\Vert _{2}$ 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 $\left\Vert \cdot\right\Vert _{\infty}$ can be derived from the Gagliardo-Nirenberg inequality because of the special nature of the SVD\ truncation.
